problem solving with angles calculator

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Triangle Calculator

Please provide values for any three of the six fields below. At least one of those values must be a side length.

Hello there!

About the Triangle Calculator

This triangle calculator lets you solve a triangle. It calculates the missing measurements of a triangle if you know any one side and any two from the remaing five mesurements.

The calculator will give you not just the answers, but also a step-by-step solution.

Usage Guide

I. valid inputs.

The triangle calculator requires exactly three of the six inputs — one side-length and any two of the remaining inputs.

The inputs themselves must be non-negative real numbers and can be in any format — integers, decimals, fractions, or even mixed numbers. Here are a few examples.

Whole numbers or decimals → 2 \hspace{0.2em} 2 \hspace{0.2em} 2 , − 4.25 \hspace{0.2em} -4.25 \hspace{0.2em} − 4.25 , 0 \hspace{0.2em} 0 \hspace{0.2em} 0 , 0.33 \hspace{0.2em} 0.33 \hspace{0.2em} 0.33
Fractions → 2 / 3 \hspace{0.2em} 2/3 \hspace{0.2em} 2/3 , − 1 / 5 \hspace{0.2em} -1/5 \hspace{0.2em} − 1/5
Mixed numbers → 5 1 / 4 \hspace{0.2em} 5 \hspace{0.5em} 1/4 \hspace{0.2em} 5 1/4

Finally, of course, the inputs shouldn't violate any of the properties of triangles. For example, sum of angles must not exceed 180 ° \hspace{0.2em} 180 \degree \hspace{0.2em} 180° .

ii. Example

If you would like to see an example of the calculator's working, just click the "example" button.

iii. Solutions

As mentioned earlier, the calculator won't just tell you the answer but also the steps you can follow to do the calculation yourself. The "show/hide solution" button would be available to you after the calculator has processed your input.

We would love to see you share our calculators with your family, friends, or anyone else who might find it useful.

By checking the "include calculation" checkbox, you can share your calculation as well.

Here's a quick overview of what it means to solve a triangle and a few related concepts to help you make sense of the solutions provided by the triangle calculator.

For those interested, we have a more comprehensive tutorial on solving triangles .

Solving a Triangle

There are six values describing the six parts of a triangle — three sides and three angles. Now, if we know one side and any two of the other five values, we can use that information to find the remaining three.

Finding the unknown measurements of a triangles from what is known is referred to as solving triangles .

Important Concepts

Let's look at a few of the important concepts that help us solve triangles.

Angle Sum Property

The sum of the three internal angles of a triangle is 180 ° \hspace{0.2em} 180 \degree 180° .

The sine rule states that the ratio of side length to the sine of opposite angle is the same for all sides in a triangle.

Cosine Rule

The cosine rule gives the relationship between the side lengths of a triangle and the cosine of any of its angles. It says —

Re-framing the formula for other sides, we have

For cases where we need to find angles using the cosine rule, the three formulas can be rearranged as —

When it comes to solving triangles, there are five different types of problems depending on which three of the triangle's measurements we know.

S S S \hspace{0.2em} SSS \hspace{0.2em} SSS — all three sides are known
S A S \hspace{0.2em} SAS \hspace{0.2em} S A S — two sides and the included angle
S S A \hspace{0.2em} SSA \hspace{0.2em} SS A — two sides and a non-included angle
A S A \hspace{0.2em} ASA \hspace{0.2em} A S A — two angles and the included side
A A S \hspace{0.2em} AAS \hspace{0.2em} AA S — two angles and the non-included side

While every problem can be solved using the fundamentals discussed earlier and a basic knowledge of triangles, each type has a sequence of steps that you can use to solve problems of that type.

Let me show you what I mean using an example.

The lengths of the three sides of a triangle are 6 \hspace{0.2em} 6 \hspace{0.2em} 6 , 7 \hspace{0.2em} 7 \hspace{0.2em} 7 , and 8 \hspace{0.2em} 8 \hspace{0.2em} 8 . Solve the triangle.

The question gives us the three sides of the triangle. So the problem is of type S S S \hspace{0.2em} SSS \hspace{0.2em} SSS . Solving the triangle would mean calculating its three angles.

Step 0. We start by drawing a rough sketch of the triangle and labeling the information given in the question. It’s not necessary but often makes things easier and helps avoid silly mistakes.

Step 1. Use the Cosine Rule to find the largest angle

When we know all the side lengths, we can use the Cosine Rule to find any of the angles.

It's best to find the largest angle first — the angle opposite to the longest side.

That's because if there is an obtuse angle ( > 90 ° ) \hspace{0.2em} (>90 \degree) \hspace{0.2em} ( > 90° ) in the triangle, it has to be this angle. So in the next step, we don't need to worry about the obtuse solutions when taking sine inverse.

Here the largest angle would be C \hspace{0.2em} C \hspace{0.2em} C . So using the formula for cos ⁡ C \hspace{0.2em} \cos C \hspace{0.2em} cos C , we have

Taking cos inverse on both sides.

Step 2. Use the Sine Rule for one of the remaining angles

Now that we know the three sides and one angle, we can use the Sine Rule to find any of the remaining two angles. Let's calculate A \hspace{0.2em} A \hspace{0.2em} A .

According to the sine rule

Substituting the known values and solving for B, we have

Step 3. Use the Angle Sum Property to find the third angle

And we have solved the triangle.

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Triangle Angle Calculator

The Triangle Angle Calculator is a tool that calculates the angles of a triangle given the length of its sides.

To use the calculator, the user simply inputs the length of the three sides of the triangle and the calculator will use the law of cosines to determine the angles. The results are displayed in degrees and can be used to solve problems in geometry, engineering, and other fields.

$c^2=a^2+b^2-2ab cos \gamma$

Triangle angles for given triangle sides

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Triangle angles calculator

Triangle angles

A triangle is a three-sided polygon. The sum of the angles of a triangle is always 180 degrees. Each angle in a triangle is called a vertex angle, and there are three vertex angles in every triangle. Knowing the measures of the vertex angles of a triangle can help you solve many mathematical problems involving triangles.

Triangle Angles Formula

The formula for finding the measure of each vertex angle in a triangle is:

Angle(γ) = 180⁰ - (Angle(α) + Angle(β))

For example, if two of the vertex angles in a triangle are 40 degrees and 60 degrees, the formula would be:

Angle = 180 - 60 - 40 = 80 degrees

Also triangle angles can be found by knowing all three sides of the triangle through the cosine theorem . The formula is:

cos(\alpha) = \dfrac{a^2 + c^2 - b^2}{2ac} cos(\beta) = \dfrac{a^2 + b^2 - c^2}{2ab} cos(\gamma) = \dfrac{c^2 + b^2 - a^2}{2cb}

How to use triangle angles calculator?

Our triangle angles calculator makes it easy for you to find the measures of the vertex angles in your triangle. Simply enter the lengths of the three sides, and the calculator will automatically calculate the measure of the three angles. This tool is perfect for students, teachers, and anyone else who needs to calculate triangle angles quickly and accurately.

Suppose you have a triangle with sides length of a = 30 cm b = 50 cm, and c = 60 cm. What is the measure of the angles?

cos(\alpha) = \dfrac{30^2 + 60^2 - 50^2}{2*30*60} = 0.55 \implies \alpha = 56.25^0 cos(\beta) = \dfrac{30^2 + 50^2 - 60^2}{2*30*50} = −0.06 \implies \beta = 93.82^0 cos(\gamma) = \dfrac{60^2 + 50^2 - 30^2}{2*60*50} = 0.86 \implies \gamma = 29.92^0

Therefore, the measure of the angle are α = 56.25, β = 93.82, and γ = 29.92 degrees.

Suppose you have a triangle with sides a = 10, b = 15, c = 17. What is the measure of the triangle angles?

cos(\alpha) = \dfrac{10^2 + 17^2 - 15^2}{2*10*17} = 0.482 \implies \alpha = 61.16^0 cos(\beta) = \dfrac{10^2 + 15^2 - 17^2}{2*10*15} = 0.12 \implies \beta = 83.1^0 cos(\gamma) = \dfrac{17^2 + 15^2 - 10^2}{2*17*15} = 0.81 \implies \gamma = 35.73^0

Therefore, the measure of the angles are α = 61.16, β = 83.1, and γ = 35.73 degrees.

In conclusion, the measures of vertex angles in a triangle are important for solving mathematical problems involving triangles. Our triangle angles calculator makes it easy to find these measures quickly and accurately. Whether you are a student, teacher, or anyone else who needs to calculate triangle angles, our calculator will help you get the job done. So, use our calculator today and make your calculations easier!

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Calculators
Calculators: Geometry
Geometry Calculator

Triangle Calculator

Solve triangles step by step.

The calculator will try to find all sides and angles of the triangle (right triangle, obtuse, acute, isosceles, equilateral), as well as its perimeter and area, with steps shown.

If the calculator did not compute something or you have identified an error, or you have a suggestion/feedback, please write it in the comments below.

Our Triangle Calculator easily finds the sides and angles of a triangle with just a few inputs. If you want to deepen your understanding of triangles, our online tool is here.

How to Use the Triangle Calculator?

Enter the known measurements of the triangle. This could include side lengths or angles.

Calculation

Once you've entered all the available information, click the "Calculate" button.

The calculator will instantly provide results based on your input. This will include the missing side lengths, angles, perimeter, and area.

What Is a Triangle?

Equilateral Triangle: All three sides are of equal length, and all three angles are 60 degrees.
Isosceles Triangle: It has two sides of equal length, and the angles opposite those sides are also equal.
Scalene Triangle: All sides have different lengths and all angles have different measures.

equilateral, isosceles, scalene triangles

Acute Triangle: All three angles are less than 90 degrees.
Obtuse Triangle: One of the angles is greater than 90 degrees.
Right-Angled Triangle: One of the angles is exactly 90 degrees, making it a right angle.

Triangles are fundamental shapes in geometry and have various properties and theorems associated with them.

What Are Some Key Facts, Theorems, and Laws Related to Triangles?

Facts about Triangles:

Angle Sum: The sum of the interior angles of any triangle is always 180 degrees.
Exterior Angle: An exterior angle of a triangle is equal to the sum of the two opposite interior angles.

Area Calculation: The area $$$ A $$$ of a triangle with the side $$$ b $$$ and the height $$$ h $$$ dropped to this side can be found using the following formula:

Triangle Theorems:

Pythagorean Theorem (For Right Triangles): In a right triangle, the square of the length of the hypotenuse $$$ c $$$ is equal to the sum of the squares of the lengths of the other two sides (called legs) $$$ a $$$ and $$$ b $$$ . This can be represented as

Isosceles Triangle Theorem: In an isosceles triangle, the angles opposite the equal sides are also equal.
Base Angles Theorem: The base angles are congruent in an isosceles triangle.
Converse of the Base Angles Theorem: If two angles of a triangle are congruent, then the sides opposite those angles are also congruent, making it an isosceles triangle.

Triangle Laws

Law of Sines: For any triangle, the ratio of the length of a side to the sine of its opposite angle is constant. This can be written as

Law of Cosines: This rule relates the lengths of the sides of a triangle to the cosine of one of its angles. It can be used for any triangle, not just right triangles. Mathematically, it can be represented as

Law of Tangents: For any triangle,

Understanding and applying these facts, theorems, and laws is critical to anyone who studies geometry. They provide insight into the properties of triangles and provide fundamental knowledge for more complex geometric concepts and real-world applications.

Why Choose Our Triangle Calculator?

Our calculator uses advanced algorithms to guarantee accuracy every time, giving you confidence in the results.

User-Friendly Interface

Built with user preferences in mind, our platform makes it easy to solve triangles, even for people new to geometry.

Versatility

Regardless of the type of triangle (scalene, isosceles, acute, or obtuse), our calculator will help you.

Fast Results

Our calculator provides answers almost instantly, making it much faster to solve multiple problems.

How accurate are the results from the Triangle Calculator?

Our calculator uses advanced algorithms to ensure accurate and correct results.

Which types of triangles can this calculator handle?

Our Triangle Calculator is versatile and can handle various types of triangles, including scalene, isosceles, right, acute, and obtuse triangles.

What formulas does the Triangle Calculator use?

The calculator uses a variety of geometric formulas depending on the input, such as the Pythagorean theorem for right triangles, and the law of sines and cosines for others.

What is the Pythagorean Theorem?

The Pythagorean Theorem is a fundamental theorem in Euclidean geometry that describes the relationship between the three sides of a right triangle. It states that the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.

Angle Calculator

[fstyle] Angle Calculator Line1 Slope * Line2 Slope * Angle If you are human, leave this field blank. Calculate [/fstyle]

Welcome, geometry enthusiasts, DIY aficionados, and curious minds! Ever found yourself in a situation where you need to figure out an angle but just can’t seem to wrap your head around it? Fear not! This guide will walk you through the magical world of angle calculators. Whether you’re a seasoned pro or a complete novice, this guide promises to make angle calculations as enjoyable as a cup of hot cocoa on a chilly day. Let’s dive in and unravel the mysteries of angles, with a sprinkle of wit to keep things lively!

Table of Contents

What is an Angle Calculator?

An angle calculator is a nifty tool that helps you determine angles with ease and precision. Whether you’re working on a geometry problem, designing a piece of furniture, or adjusting the slope of your roof, an angle calculator can save you time and effort. It does the heavy lifting for you by computing the angles based on the values you provide, ensuring your calculations are spot-on.

Key Concepts to Understand

Types of angles.

Before we get into the nitty-gritty of angle calculators, let’s brush up on the types of angles you might encounter:

Acute Angle: Less than 90 degrees. Think of it as the angle that’s still waking up and hasn’t had its coffee yet.
Right Angle: Exactly 90 degrees. The perfect balance – not too sleepy, not too hyper.
Obtuse Angle: More than 90 degrees but less than 180 degrees. The angle that’s just lounging around.
Straight Angle: Exactly 180 degrees. A straight shooter, no nonsense here.
Reflex Angle: More than 180 degrees. The drama queen of angles, always going over the top.

Angle Relationships

Understanding how angles relate to each other is crucial for accurate calculations:

Complementary Angles: Two angles that add up to 90 degrees. They’re like best friends always hanging out together.
Supplementary Angles: Two angles that add up to 180 degrees. Think of them as a dynamic duo, completing each other.
Adjacent Angles: Angles that share a common side and vertex. Neighbors in the angle world.
Vertical Angles: Angles opposite each other when two lines intersect. Always equal, like twins separated by a line.

Measuring Angles

Angles can be measured in degrees (°) or radians (rad). Degrees are more common in everyday use, while radians are often used in higher mathematics and engineering.

Why Use an Angle Calculator?

Why should you bother with an angle calculator? Here are a few compelling reasons:

Accuracy: Ensures precise calculations, reducing the risk of errors.
Efficiency: Saves time and effort, especially for complex problems.
Versatility: Useful for a wide range of applications, from geometry to construction.
Simplicity: User-friendly interfaces make angle calculations accessible to everyone.

Common Mistakes vs Tips (Table Format)

Common Mistakes	Tips for Success
	Always convert between degrees and radians as needed.
	Use complementary and supplementary angles to simplify problems.
	Double-check your values before entering them.
	Break down complex shapes into simpler components.
	Clearly label angles in your diagrams to avoid confusion.
	Leverage an angle calculator to avoid manual errors.
	Brush up on basic geometry concepts regularly.
	Use reference materials to confirm your calculations.

Step-by-Step Guide to Using an Angle Calculator

☑️ Identify the Type of Angle Calculation Needed: Determine whether you need to calculate an angle, find complementary or supplementary angles, or solve for an angle in a geometric shape.

☑️ Gather Necessary Measurements: Collect all relevant measurements, such as side lengths, other angles, and any given values.

☑️ Choose the Right Angle Calculator: Select an angle calculator that fits your needs, whether it’s for basic angle calculation, trigonometry, or geometry.

☑️ Input Values: Enter the known values into the calculator. Ensure you input them correctly to avoid errors.

☑️ Select the Desired Function: Choose the function you need, such as calculating an unknown angle, finding complementary angles, or determining the angle in a triangle.

☑️ Review the Results: Check the calculated angle to ensure it makes sense within the context of your problem.

☑️ Convert Units if Necessary: If your problem requires, convert the angle measurement between degrees and radians.

☑️ Apply the Results: Use the calculated angle in your project, whether it’s solving a math problem, designing something, or adjusting an angle.

☑️ Double-Check: Always double-check your calculations and results to ensure accuracy.

Frequently Asked Questions (FAQs)

Q: can i use an angle calculator for any type of angle.

A: Yes, angle calculators can handle all types of angles, from acute to reflex, and can solve for various angle relationships like complementary and supplementary angles.

Q: What’s the difference between degrees and radians?

A: Degrees measure angles based on dividing a circle into 360 parts, while radians measure angles based on the radius of the circle. One full circle is 360 degrees or (2\pi) radians.

Q: How do I convert between degrees and radians?

A: To convert degrees to radians, multiply by (\pi/180). To convert radians to degrees, multiply by (180/\pi).

Q: Why do my angle calculations sometimes seem off?

A: Common reasons include incorrect input values, not converting units, or misunderstanding angle relationships. Double-check your inputs and use an angle calculator for accuracy.

Q: Can angle calculators help with trigonometry?

A: Absolutely! Many angle calculators can solve trigonometric problems, such as finding angles in right triangles using sine, cosine, and tangent functions.

Q: Are there online angle calculators available?

A: Yes, numerous online angle calculators are available for free. They range from basic tools to advanced calculators for trigonometry and geometry.

Q: How can I improve my angle calculation skills?

A: Practice regularly, brush up on basic geometry concepts, and use angle calculators to check your work and learn from any mistakes.

Q: Do professionals use angle calculators?

A: Yes, professionals in fields such as engineering, construction, and design frequently use angle calculators to ensure precise measurements and efficient workflow.

Tips for Accurate Angle Calculations

Use Clear Diagrams: Draw clear and labeled diagrams to visualize the problem.
Double-Check Inputs: Always verify the values you input into the calculator.
Understand Relationships: Familiarize yourself with angle relationships like complementary, supplementary, and adjacent angles.
Practice Regularly: Regular practice helps reinforce concepts and improve accuracy.
Leverage Technology: Use angle calculators and other tools to enhance your calculations.
Stay Updated: Keep up with the latest tools and techniques in geometry and trigonometry.
Consult References: Use reference materials to confirm your calculations and learn new methods.
Seek Help: Don’t hesitate to ask for help or consult experts if you’re stuck on a problem.

Real-World Application: A Case Study

Let’s walk through a practical example to illustrate the process of using an angle calculator. Emily is working on a DIY project to build a custom bookshelf with angled supports.

Step-by-Step Process:

Identify the Type of Angle Calculation Needed: Emily needs to calculate the angle between the base of the bookshelf and the angled supports.
Gather Necessary Measurements: Emily measures the height of the bookshelf as 72 inches and the length of the base as 36 inches.
Choose the Right Angle Calculator: Emily selects an online angle calculator that can handle trigonometric functions.
Input Values: Emily enters the height (72 inches) and the base length (36 inches) into the calculator.
Select the Desired Function: Emily chooses the function to calculate the angle using the tangent function: (\tan(\theta) = \text{opposite} / \text{adjacent}).
Review the Results: The calculator provides the angle as approximately 63.4 degrees.
Convert Units if Necessary: Since Emily’s project uses degrees, no conversion is needed.
Apply the Results: Emily uses the calculated angle to cut the supports at the correct angle, ensuring a perfect fit.
Double-Check: Emily double-checks the angle and measurements before cutting the supports.

Emily successfully builds her custom bookshelf with perfectly angled supports, thanks to the angle calculator. The bookshelf is sturdy, and Emily’s calculations were spot-on, making her DIY project a success.

For further reading and reliable information, check out these resources:

National Institute of Standards and Technology (NIST): www.nist.gov
U.S. Department of Education: www.ed.gov
Khan Academy: www.khanacademy.org

And there you have it – a comprehensive, fun, and engaging guide to mastering the angle calculator! Remember, with the right tools and a bit of practice, you can conquer any angle calculation challenge. So go ahead, grab that calculator, and start solving angles like a pro. Happy calculating!

Online Triangle Calculator

Enter any valid input (3 side lengths, 2 sides and an angle or 2 angle and a 1 side) and our calculator will do the rest.

Triangle App
Triangle Animated Gifs

This is the acute triangle in Quadrant I, for more information on this topic, check out the law of sines ambiguous case .

Sides	Angles

This is the obtuse triangle in Quadrant II, for more information on this topic, check out the law of sines ambiguous case .

Sides	Angles

Status: Calculator waiting for input

Why is the calculator saying there's an error when there shouldn't be?

The most frequent reason for this is because you are rounding the sides and angles which can, at times, lead to results that seem inaccurate. In these cases, in actuality , the calculator is really producing correct results. However, it is then rounding them for you- which leads to seemingly inaccurate results and possible error warnings. To see if that is your problem, set the rounding to maximum accuracy .

Ultimate Math Solver (Free) Free Algebra Solver ... type anything in there!

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Triangle calculator

How does this calculator solve a triangle.

The expert phase is different for different tasks. The calculator tries to calculate the sizes of three sides of the triangle from the entered data. He gradually applies the knowledge base to the entered data, which is represented in particular by the relationships between individual triangle parameters. These are successively applied and combined, and the triangle parameters are calculated. Calculator iterates until the triangle has calculated all three sides. For example, the appropriate height is calculated from the given area of the triangle and the corresponding side. From the known height and angle, the adjacent side, etc., can be calculated. The calculator uses use knowledge, e.g., formulas and relations like the Pythagorean theorem, Sine theorem, Cosine theorem, and Heron's formula.
The second stage is the calculation of the properties of the triangle from the available lengths of its three sides.

Examples of how to enter a triangle:

Triangles in word problems:.

Look also at our friend's collection of math problems and questions:

right triangle
Heron's formula
The Law of Sines
The Law of Cosines
Pythagorean theorem
triangle inequality
similarity of triangles
The right triangle altitude theorem

Calculate Δ by 3 sides SSS
Δ SAS by 2 sides and 1 angle
Δ ASA by 1 side and 2 angles
Scalene triangle
Right-angled Δ
Equilateral Δ
Isosceles Δ
Δ by coordinates
List of triangles

	=			=			=
	=			=			=

Capital-letter variable names correspond to angle measures, opposite from each side length named by the lowercase-letter counterpart. All angle measure inputs are in degrees (pre-multiplied by 180:pi). The side lengths a , b and c are in arbitrary units and do not at all affect the triangle size or drawing in any way, other than to help compute any unknown angle measures.

Quadrant II (90 to 180 degrees)	Quadrant I (0 to 90 degrees)
element.

I also maintain a separate research project to derive many exact trigonometric ratios so that many of these answers can be expressed in exact form.

All JavaScript on this website is freedom-respecting software. Learn more.

Triangle Calculator

Lesson on Solving Triangles

Lesson contents, how to calculate the sides of a triangle.

If solving for a side length of a right triangle where know two side lengths, we may use the Pythagorean theorem. The Pythagorean theorem is given as:

Where a and b are the legs and c is the hypotenuse.

For non-right triangles, we must know three parameters of the triangle. The three known parameters may either be two side lengths and an angle or two angles and a side length.

There are several formulas we may use for solving side lengths. The most common and versatile are the law of cosines and the law of sines .

The law of cosines is split into three formulas. They are given as: a 2 = b 2 + c 2 – 2bc×cos(A) b 2 = a 2 + c 2 – 2ac×cos(B) c 2 = a 2 + b 2 – 2ab×cos(C) Where a , b , and c are the side lengths and A , B , and C are the internal angles.

The law of sines is given as: sin(A) ⁄ a = sin(B) ⁄ b = sin(C) ⁄ c Where a , b , and c are the side lengths and A , B , and C are the internal angles.

There are other formulas available for solving triangle sides, but the law of cosines and law of sines may be leveraged in combination to solve any triangle and therefore will most commonly be used.

How to Calculate the Angles of a Triangle

When solving for a triangle’s angles, a common and versatile formula for use is called the sum of angles . It is given as: A + B + C = 180 Where A , B , and C are the internal angles of a triangle.

If two angles are known and the third is desired, simply apply the sum of angles formula given above. If three side lengths are known, use the law of cosines. If an angle and two sides are known, use either the law of cosines or the law of sines, depending on the combination of angle and sides.

How the Calculator Works

The calculator on this page is written in the programming language JavaScript (JS) which allows it to run in your device’s internet browser JS engine. The JS engine running the code allows for instant answers at the click of a button.

When the calculate button is pressed, your selected “solving for” parameter is inputted, and the calculator determines which set of formulas will be used to solve. Then, your inputted side/angle parameters are converted to standard units and fed into the applicable formulas. This calculator is powered by the law of cosines and other basic triangle identities.

Once the desired output parameter has been calculated, the solution’s units are converted if the selected unit requires so. The final answer is rounded and then printed to the applicable output area on the calculator.

Triangle Sides & Angles Calculator

Calculate sides, angles of a triangle step-by-step.

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Trigonometry Calculator

Table of contents

This trigonometry calculator will help you in two popular cases when trigonometry is needed. If you want to find the values of sine, cosine, tangent, and their reciprocal functions, use the first part of the calculator. Are you searching for the missing side or angle in a right triangle using trigonometry? Our tool is also a safe bet! Type 2-3 given values in the second part of the calculator, and you'll find the answer in a blink of an eye. Scroll down if you want to learn about trigonometry and where you can apply it.

There are many other useful tools when dealing with trigonometry problems. Check out two popular trigonometric laws with the law of sines calculator and our law of cosines calculator , which will help you to solve any kind of triangle.

How to use this trigonometry calculator

This trigonometry calculator has two sections that perform two different functions using trigonometry. The following instructions should help you work your way around this calculator with ease:

The first section takes an angle input and gives you a list of trigonometric function values for this angle. To input in a different unit, click on the unit to change it, and then enter the angle.

For example, to find the value of sin(45°), we merely enter 45 degrees as the angle. The calculator instantly tells you that sin(45°) = 0.70710678. It also gives the values of other trig functions, such as cos(45°) and tan(45°).

The second section uses trigonometry to determine the missing parameters of a right-angled triangle:

First, select what parameters are known about the triangle. You can choose between " two sides ", " an angle and one side ", and " area and one side ".

Refer to the diagram at the bottom of the calculator to understand the parameter labels. Enter the parameters based on your choice. If you need to input a parameter in a different unit, change the unit before entering a value.

The tool will calculate the missing parameters using trigonometry.

For example, if we know two sides, a = 7 cm and b = 12 cm , we enter them in this calculator. Right away, we see that c = 13.892 cm , α = 30.256° , and β = 59.74° .

We can use this section in reverse, too! Say we know the angle β = 30° , b = 10 in , and c = 20 in . First, enter β = 30° , and instantly we learn that α = 60° . Now, enter b = 10 in (ensuring the unit is inches first). Similarly, input c = 20 in , and we see that a = 17.32 in .

If you're wondering how trigonometry can help you learn so much about triangles, continue reading this article. It answers many questions that people new to trigonometry should know.

What is trigonometry?

Trigonometry is a branch of mathematics. The word itself comes from the Greek trigōnon (which means "triangle") and metron ("measure"). As the name suggests, trigonometry deals primarily with angles and triangles ; in particular, it defines and uses the relationships and ratios between angles and sides in triangles. The primary application is thus solving triangles, precisely right triangles, and any other type of triangle you like.

🔎 Trigonometric functions (sin, cos, tan) are all ratios. Therefore, you can find the missing terms using nothing else but our ratio calculator !

Trigonometry has plenty of applications: from everyday life problems such as calculating the height or distance between objects to the satellite navigation system, astronomy, and geography. Also, sine and cosine functions are fundamental for describing periodic phenomena - thanks to them, we can describe oscillatory movements (as in our simple pendulum calculator ) and waves like sound, vibration, or light.

trigonometric functions in a right triangle

Many fields of science and engineering use trigonometry and trigonometric functions, namely: music, acoustics, electronics, medicine and medical imaging, biology, chemistry, meteorology, electrical, mechanical, civil engineering, and even economics... The trigonometric functions are really all around us!

Trigonometry calculator as a tool for solving right triangle

To find the missing sides or angles of the right triangle, all you need to do is enter the known variables into the trigonometry calculator. You need only two given values in the case of:

one side and one angle
area and one side

Remember that if you know two angles, it's not enough to find the sides of the triangle. Two triangles having the same shape (which means they have equal angles) may be of different sizes (not the same side length) - that kind of relationship is called triangle similarity . If the sides have the same length, then the triangles are congruent .

Meet the creator of our trigonometry calculator

Hi, I'm Hanna, the brainchild of this trigonometry calculator, and I'm a doctor of mechanical engineering and a maestro at creating scientific tools. I knew a good trigonometry calculator would help me get the right depth of field for my photography, so I sat down to create it. Thanks to this, I have captured some crisp photos of colorful birds in the wild!

We put extra care into the quality of our content so that they are as accurate and reliable as possible. Each tool is peer-reviewed by a trained expert and then proofread by a native speaker. To learn more about our standards, please check the Editorial Policies page .

Trigonometry is the study of the relationships within a triangle . For right-angled triangles, the ratio between any two sides is always the same and is given as the trigonometry ratios, cos, sin, and tan. Trigonometry can also help find some missing triangular information , e.g., the sine rule.

How to do trigonometry?

Find which two of these you have: the hypotenuse, adjacent or opposite side, or angle.
Work out which of the remaining options you are trying to calculate.
Choose which relationship you need ( remember, SOHCAHTOA ).
Fill in the data you have into the equation.
Rearrange and solve for the unknown.
Check your answers with our trig calculator.

Is trigonometry hard?

Trigonometry can be hard at first, but after some practice, you will master it! Here are some trigonometry tips:

Label the hypotenuse, adjacent, and opposite sides on your triangle to help you figure out what identity to use.
Remember the mnemonic SOHCAHTOA for the trigonometric relationships!

What is trigonometry used for?

Trigonometry is used to find information about all triangles , and right-angled triangles in particular. Since triangles are everywhere in nature , trigonometry is used outside of math in fields such as construction, physics, chemical engineering, and astronomy.

Who invented trigonometry?

Since trigonometry is the relationship between angles and sides of a triangle, no one invented it , it would still be there even if no one knew about it! The first people to discover part of trigonometry were the Ancient Egyptians and Babylonians , but Euclid and Archemides first proved the identities, although they did it using shapes, not algebra.

What grade is trigonometry taught at?

Trigonometry is usually taught to teenagers aged 13-15 , which is grades 8 & 9 in the USA and years 9 & 10 in the UK. The exact age at which trigonometry is taught depends on the country, school, and pupils' ability.

How to convert decimal to degrees in trigonometry?

Find which trigonometric relationship you are using with SOHCAHTOA.
Take the inverse identity of your decimal, e.g., sin⁻¹(0.5).
The resulting number is the degree of your angle .
Check your results with our trigonometry calculators.

How to find the height of a triangle using trigonometry?

Draw your triangle and mark the height. You will have to split the triangle into two smaller triangles.
Solve either of these remaining triangles using regular trigonometry to find the height. The opposite or adjacent will now be the hypotenuse of the smaller triangle.
Check your answers with Omni Calculator.

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ol{padding-top:0;}.css-63uqft ul:not(:first-child),.css-63uqft ol:not(:first-child){padding-top:4px;} Trigonometric functions: sin, cos, tan...

To enter an angle, for example π/6, select π radians (× π rad) units from the dropdown list and input 1/6 .

...or trigonometry in right triangle

Side length a

Side length b

Side length c

Diagram illustrating the trigonometry in a right triangle.

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Calculators » Math » Right Triangle

Right Angle Triangle Calculator

Our online tools will provide quick answers to your calculation and conversion needs. On this page, you can solve math problems involving right triangles. You can calculate angle, side (adjacent, opposite, hypotenuse) and area of any right-angled triangle and use it in real world to find height and distances.

Enter any two values and press calculate to get the other values

Angle A (degree):

Angle B (degree):

Download: Use this right-triangle solver offline with our all-in-one calculator app for Android and iOS .

Right triangle calculation

Formulas used for calculations on this page:

Pythagoras' Theorem a 2 + b 2 = c 2

Trigonometric functions: sin(A) = a/c, cos(A) = b/c, tan(A) = a/b sin(B) = b/c, cos(B) = a/c, tan(B) = b/a

Area = a*b/2, where a is height and b is base of the right triangle.

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Triangle Calculator

Triangle calculator

This solver uses the Law of Sines, and the Law of Cosines to solve acute and obtuse triangles , i.e., to find missing angles or sides if you know any three of them. Provide any three triangle properties of an oblique triangle to find the missing side, angle or area. The calculator shows all the steps and gives a full explanation for each step.

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$$ A = \frac{a\,b\,\sin\gamma}{2} $$

area

$$ A = \frac{b\,c\,\sin\alpha}{2} $$

area

$$ A = \frac{a\,c\,\sin\beta}{2} $$

area

$$ c^2 = a^2 + b^2 - 2ab \cos \gamma $$

law of cosines

$$ b^2 = a^2 + c^2 - 2ac \cos \beta $$

law of cosines

$$ a^2 = b^2 + c^2 - 2 b c \cos \alpha $$

law of cosines

$$ \frac{a}{\sin\alpha} = \frac{b}{\sin\beta} = \frac{c}{\sin\gamma} $$

law of sines

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Pythagorean Theorem Calculator

Please provide any 2 values below to solve the Pythagorean equation: a 2 + b 2 = c 2 .

a =

√

√

Related Triangle Calculator | Right Triangle Calculator

Pythagorean Theorem

The Pythagorean Theorem, also known as Pythagoras' theorem, is a fundamental relation between the three sides of a right triangle. Given a right triangle, which is a triangle in which one of the angles is 90°, the Pythagorean theorem states that the area of the square formed by the longest side of the right triangle (the hypotenuse) is equal to the sum of the area of the squares formed by the other two sides of the right triangle:

In other words, given that the longest side c = the hypotenuse, and a and b = the other sides of the triangle:

a 2 + b 2 = c 2

This is known as the Pythagorean equation, named after the ancient Greek thinker Pythagoras. This relationship is useful because if two sides of a right triangle are known, the Pythagorean theorem can be used to determine the length of the third side. Referencing the above diagram, if

a = 3 and b = 4

the length of c can be determined as:

c = √ a 2 + b 2 = √ 3 2 +4 2 = √ 25 = 5

It follows that the length of a and b can also be determined if the lengths of the other two sides are known using the following relationships:

a = √ c 2 - b 2

b = √ c 2 - a 2

The law of cosines is a generalization of the Pythagorean theorem that can be used to determine the length of any side of a triangle if the lengths and angles of the other two sides of the triangle are known. If the angle between the other sides is a right angle, the law of cosines reduces to the Pythagorean equation.

There are a multitude of proofs for the Pythagorean theorem, possibly even the greatest number of any mathematical theorem.

Algebraic proof:

In the figure above, there are two orientations of copies of right triangles used to form a smaller and larger square, labeled i and ii, that depict two algebraic proofs of the Pythagorean theorem.

In the first one, i, the four copies of the same triangle are arranged around a square with sides c. This results in the formation of a larger square with sides of length b + a, and area of (b + a) 2 . The sum of the area of these four triangles and the smaller square must equal the area of the larger square such that:

(b + a) = c + 4

which yields:

c =	(b + a) - 2ab
=	b + 2ab + a - 2ab
=	a + b

which is the Pythagorean equation.

(b - a) + 4

Since the larger square has sides c and area c 2 , the above can be rewritten as:

c 2 = a 2 + b 2

which is again, the Pythagorean equation.

There are numerous other proofs ranging from algebraic and geometric proofs to proofs using differentials, but the above are two of the simplest versions.

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The formula for finding the measure of each vertex angle in a triangle is: Angle (γ) = 180⁰ - (Angle (α) + Angle (β)) For example, if two of the vertex angles in a triangle are 40 degrees and 60 degrees, the formula would be: Angle = 180 - 60 - 40 = 80 degrees. Also triangle angles can be found by knowing all three sides of the triangle ...
Triangle Calculator
Law of Cosines: This rule relates the lengths of the sides of a triangle to the cosine of one of its angles. It can be used for any triangle, not just right triangles. Mathematically, it can be represented as. c 2 = a 2 + b 2 − 2 a b cos ⁡ (C) c^2=a^2+b^2-2ab\cos\left (C\right) c2 = a2 +b2 − 2abcos(C)
Angle Calculator
Q: Can angle calculators help with trigonometry? A: Absolutely! Many angle calculators can solve trigonometric problems, such as finding angles in right triangles using sine, cosine, and tangent functions. Q: Are there online angle calculators available? A: Yes, numerous online angle calculators are available for free.
Online Triangle Calculator
Free Algebra Solver ... type anything in there! Math Warehouse's popular online triangle calculator: Enter any valid combination of sides/angles (3 sides, 2 sides and an angle or 2 angle and a 1 side) , and our calculator will do the rest! It will even tell you if more than 1 triangle can be created.
Triangle calculator
The calculator solves the triangle specified by three of its properties. Each triangle has six main characteristics: three sides a, b, c, and three angles (α, β, γ). The classic trigonometry problem is to specify three of these six characteristics and find the other three.
Trig Triangle Calculator
The sum of the internal angles of a triangle will always be 360 ° 360\degree 360°. In a right triangle, there's a right angle (90 ° 90\degree 90°) and, therefore, the sum of the other two angles (α \alpha α and β \beta β) will be 90 ° 90\degree 90°. Therefore: Given β \beta β: α = 90 ° − β \alpha = 90\degree - \beta α = 90 ...
Right triangle calculator
The right triangle calculator finds the missing area, angle, leg, hypotenuse and height of a triangle. The calculator also provides steps on how to solve the most important right triangles: the 30-60-90 triangle and the 45-45-90 triangle. Special right triangle General right triangle.
Interactive Triangle Solver
All angle measure inputs are in degrees (pre-multiplied by 180:pi). The side lengths a, b and c are in arbitrary units and do not at all affect the triangle size or drawing in any way, other than to help compute any unknown angle measures. Quadrant II (90 to 180 degrees) Quadrant I (0 to 90 degrees)
Triangle Calculator
When solving for a triangle's angles, a common and versatile formula for use is called the sum of angles. It is given as: A + B + C = 180. Where A, B, and C are the internal angles of a triangle. If two angles are known and the third is desired, simply apply the sum of angles formula given above. If three side lengths are known, use the law ...
Triangle Sides & Angles Calculator
Free Triangle Sides & Angles Calculator - Calculate sides, angles of a triangle step-by-step. Solutions Graphing ... Each new topic we learn has symbols and problems we have never seen. ... Read More. Study Tools AI Math Solver Popular Problems Worksheets Study Guides Practice Cheat Sheets Calculators Graphing Calculator Geometry Calculator ...
Trigonometry Calculator
Trigonometry is a branch of mathematics. The word itself comes from the Greek trigōnon (which means "triangle") and metron ("measure"). As the name suggests, trigonometry deals primarily with angles and triangles; in particular, it defines and uses the relationships and ratios between angles and sides in triangles.The primary application is thus solving triangles, precisely right triangles ...
Right Angle Triangle Calculator
Right Angle Triangle Calculator. Our online tools will provide quick answers to your calculation and conversion needs. On this page, you can solve math problems involving right triangles. You can calculate angle, side (adjacent, opposite, hypotenuse) and area of any right-angled triangle and use it in real world to find height and distances.
Right Triangle Calculator
A right triangle is a type of triangle that has one angle that measures 90°. Right triangles, and the relationships between their sides and angles, are the basis of trigonometry. In a right triangle, the side that is opposite of the 90° angle is the longest side of the triangle, and is called the hypotenuse.
Triangle Calculator
Triangle calculator. This solver uses the Law of Sines, and the Law of Cosines to solve acute and obtuse triangles , i.e., to find missing angles or sides if you know any three of them. Provide any three triangle properties of an oblique triangle to find the missing side, angle or area. The calculator shows all the steps and gives a full ...
Pythagorean Theorem Calculator
a 2 + b 2 = c 2. This is known as the Pythagorean equation, named after the ancient Greek thinker Pythagoras. This relationship is useful because if two sides of a right triangle are known, the Pythagorean theorem can be used to determine the length of the third side. Referencing the above diagram, if. a = 3 and b = 4.
Triangle Solver
Lighting Calculators - 2. Math Calculators; Time Calculators; Wire Calculators; Travel Calculators - 3. Health Calculators; Engineering Calculators; Security Calculators; Date & Time Calculators ... Angle A (degrees): Angle B (degrees): Angle C (degrees): Solve Triangle. Related Math, Chemistry, Physics, Algebra & Geometry Calculators. Gas ...
Mathway
They do not store directly personal information, but are based on uniquely identifying your browser and internet device. If you do not allow these cookies, you will experience less targeted advertising. Free math problem solver answers your algebra homework questions with step-by-step explanations.

Triangle Calculator

About the Triangle Calculator

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Solving a Triangle

Important Concepts

Angle Sum Property

Cosine Rule

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Triangle Angle Calculator

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Triangle angles calculator

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Triangle Calculator

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Angle Calculator

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Q: How do I convert between degrees and radians?

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Q: Can angle calculators help with trigonometry?

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