How do you find the hypotenuse?

Use the Pythagorean theorem c² = a² + b², where c is the hypotenuse and a and b are the legs.

Can this calculator solve missing angles?

Yes. The calculator uses trigonometric functions such as sine, cosine, and tangent to calculate unknown angles.

What formulas are used in a right triangle?

The calculator uses the Pythagorean theorem and trigonometric ratios including sine, cosine, and tangent.

Right Triangle Calculator & Solver With Steps

Q: What is a right triangle?

A right triangle is a triangle that contains one angle measuring exactly 90 degrees.

A right triangle calculator helps you find missing sides, angles, area, perimeter, and other measurements of a right triangle. Simply enter any two known values, and the calculator automatically solves the rest of the triangle while showing the formulas and steps used.

Whether you are studying geometry, preparing for exams, solving construction problems, or checking homework answers, this right triangle solver provides a quick and accurate solution.

Use the Right Triangle Calculator

Enter any two known values into the calculator above. Supported input combinations include:

Two sides
One side and one angle
One side and the area
Area and perimeter

The calculator automatically determines

Side (a)
Side (b)
Hypotenuse (c)
Both acute angles
Area
Perimeter
Semi-perimeter
Altitude to the hypotenuse

What Is a Right Triangle?

A right triangle is a triangle that contains one angle equal to 90°.

The side opposite the right angle is called the hypotenuse, and it is always the longest side of the triangle. The other two sides are known as the legs of the triangle.

In this calculator:

(a) = height (vertical leg)
(b) = base (horizontal leg)
(c) = hypotenuse

Right triangles are among the most important shapes in mathematics because they form the basis of trigonometry and many geometric calculations.

Pythagorean Theorem

The most important formula for a right triangle is the Pythagorean theorem: a² + b² = c²

Where:

(a) = first leg
(b) = second leg
(c) = hypotenuse

This theorem states that the sum of the squares of the two legs equals the square of the hypotenuse.

Example

Suppose:
Given: a = 3, b = 4
c = √(3² + 4²) c = √(9 + 16) c = √25 c = 5

This famous 3-4-5 triangle is a common example of a right triangle.

How to Find the Hypotenuse

If both legs are known, use:

\[
c=\sqrt{a^2+b^2}
\]

Example

\[
a=8,\quad b=15
\]

\[
c=\sqrt{8^2+15^2}
\]

\[
c=\sqrt{64+225}
\]

\[
c=\sqrt{289}
\]

\[
c=17
\]

How to Find a Missing Leg

If the hypotenuse and one leg are known, rearrange the Pythagorean theorem.

Find (a)

\[
a=\sqrt{c^2-b^2}
\]

Find (b)

\[
b=\sqrt{c^2-a^2}
\]

Example

\[
c=13,\quad b=5
\]

\[
a=\sqrt{13^2-5^2}
\]

\[
a=\sqrt{169-25}
\]

\[
a=\sqrt{144}
\]

\[
a=12
\]

How to Find Angles in a Right Triangle

The angles of a right triangle can be found using trigonometric ratios.

Sine

\[
\sin(\theta)=\frac{\text{opposite}}{\text{hypotenuse}}
\]

Cosine

\[
\cos(\theta)=\frac{\text{adjacent}}{\text{hypotenuse}}
\]

Tangent

\[
\tan(\theta)=\frac{\text{opposite}}{\text{adjacent}}
\]

To find an angle, use the inverse trigonometric functions:

\[
\theta=\sin^{-1}\left(\frac{\text{opposite}}{\text{hypotenuse}}\right)
\]

\[
\theta=\cos^{-1}\left(\frac{\text{adjacent}}{\text{hypotenuse}}\right)
\]

\[
\theta=\tan^{-1}\left(\frac{\text{opposite}}{\text{adjacent}}\right)
\]

Since the triangle already contains a 90° angle, the second acute angle is:

\[
90^\circ-\theta
\]

Area of a Right Triangle

The area of a right triangle is calculated using:

\[
A=\frac{1}{2}bh
\]

Since the two legs are perpendicular:

\[
A=\frac{1}{2}ab
\]

Example

\[
a=8,\quad b=5
\]

\[
A=\frac{1}{2}(8)(5)
\]

\[
A=20
\]

square units

Perimeter of a Right Triangle

The perimeter is the sum of all three sides:

\[
P=a+b+c
\]

Example

\[
a=3,\quad b=4,\quad c=5
\]

\[
P=3+4+5
\]

\[
P=12
\]

units

Altitude to the Hypotenuse

The altitude drawn from the right angle to the hypotenuse can be found using:

\[
h=\frac{ab}{c}
\]

Where:

(a) and (b) are the legs
(c) is the hypotenuse

This measurement is often used in geometry and engineering calculations.

Special Right Triangles

45°–45°–90° Triangle

In this triangle:

Both legs are equal.
The angles are 45°, 45°, and 90°.

Relationship:

\[
c=a\sqrt{2}
\]

If one leg is 6:

\[
c=6\sqrt{2}
\]

30°–60°–90° Triangle

This special triangle follows the ratio:

\[
1:\sqrt{3}:2
\]

If the shortest side is (x):

Longer leg = \(x\sqrt{3}\)
Hypotenuse = \(2x \)

The calculator automatically detects this triangle when appropriate.

Example: Solving a Complete Right Triangle

Suppose:

\[
a=6
\]

\[
b=8
\]

Step 1: Find the Hypotenuse

\[
c=\sqrt{6^2+8^2}
\]

\[
c=\sqrt{36+64}
\]

\[
c=\sqrt{100}
\]

\[
c=10
\]

Step 2: Find the Area

\[
A=\frac{1}{2}(6)(8)
\]

\[
A=24
\]

Step 3: Find the Perimeter

\[
P=6+8+10
\]

\[
P=24
\]

Step 4: Find the Angles

\[
\theta=\tan^{-1}\left(\frac{6}{8}\right)
\]

\[
\theta\approx36.87^\circ
\]

Second angle:

\[
90^\circ-36.87^\circ
\]

\[
53.13^\circ
\]

Common Mistakes

Students often make the following mistakes when solving right triangles:

Confusing the hypotenuse with a leg
Forgetting to take the square root after applying the Pythagorean theorem
Using the wrong trigonometric ratio
Mixing degrees and radians
Entering only one value into the calculator
Using a hypotenuse shorter than a leg

Always verify that the hypotenuse is the longest side of the triangle.

Real-Life Applications of Right Triangles

Right triangles appear in many practical situations, including:

Construction and roofing
Surveying and land measurement
Engineering design
Architecture
Navigation
Physics calculations
Distance and height problems
Computer graphics and game development

Because of these applications, right triangles are one of the most frequently used shapes in mathematics and science.

Why Use This Right Triangle Calculator?

This calculator offers several advantages:

Supports multiple input combinations
Solves the entire triangle instantly
Provides step-by-step explanations
Calculates area and perimeter automatically
Detects special right triangles
Includes a live triangle diagram
Works on desktop and mobile devices

Whether you need a quick answer or a detailed solution, the calculator simplifies the entire process.

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Final Summary

A right triangle calculator is a powerful geometry tool that combines the Pythagorean theorem and trigonometry to solve right triangles quickly and accurately. By entering any two known values, you can determine all missing sides, angles, area, perimeter, and other properties of the triangle.

Use the calculator above to solve homework problems, verify calculations, explore special triangles, and better understand the relationships between the sides and angles of a right triangle.