What is the Pythagorean Theorem?

The Pythagorean Theorem states that the sum of the squared sides of a right triangle equals the length of the hypotenuse squared.

You might recognize this theorem in the form of the Pythagorean equation:

a^{2} + b^{2} = c^{2}

If you know the length of any 2 sides of a right triangle you can use the Pythagorean equation formula to find the length of the third side.

Equation for sides a or b, or the hypotenuse c. The hypotenuse is the side of the triangle opposite the right angle.

For right triangles only, enter any two values to find the third. See the solution with steps using the Pythagorean Theorem formula.

This calculator also finds the area A of the right triangle with sides a and b. The formula for area of a right triangle is:

A = \frac{a b}{2}

Example

Hypotenuse Calculator

The hypotenuse is the longest side of a right-angled triangle, opposite the right angle. It can be calculated using the Pythagorean theorem.

The general approach to finding the hypotenuse includes:

Recognizing the triangle as a right triangle.
Using the Pythagorean theorem: \( c^2 = a^2 + b^2 \), where \( c \) is the hypotenuse, and \( a \) and \( b \) are the legs of the triangle.
Taking the square root of the sum of the squares of the legs to find the hypotenuse.

Calculating the Hypotenuse

The hypotenuse can be calculated when the lengths of the two legs of the triangle are known. The general form of the Pythagorean theorem is:

\[ c^2 = a^2 + b^2 \]

Example:

If the lengths of the legs are 3 and 4, the solution is:

Step 1: Square the legs: \( 3^2 = 9 \) and \( 4^2 = 16 \).
Step 2: Add the squared values: \( 9 + 16 = 25 \).
Step 3: Take the square root of the sum: \( \sqrt{25} = 5 \).

Finding a Leg Length (when the Hypotenuse is known)

If the hypotenuse and one leg are known, the missing leg can be calculated by rearranging the Pythagorean theorem:

\[ a^2 = c^2 - b^2 \]

Example:

If the hypotenuse is 10 and one leg is 6, the solution is:

Step 1: Square the hypotenuse and the known leg: \( 10^2 = 100 \) and \( 6^2 = 36 \).
Step 2: Subtract the square of the known leg from the square of the hypotenuse: \( 100 - 36 = 64 \).
Step 3: Take the square root of the result: \( \sqrt{64} = 8 \).

Real-life Applications of the Hypotenuse Calculation

Calculating the hypotenuse has many practical applications, such as:

Determining the diagonal distance across a rectangular object (e.g., a TV screen or floor space).
Measuring the length of a ladder needed to reach a certain height.
Finding the distance across a slope, such as in construction or landscaping projects.

Common Operations with the Hypotenuse Formula

Pythagorean Theorem: \( c^2 = a^2 + b^2 \)

Leg Length Calculation: \( a^2 = c^2 - b^2 \)

Modifying Equations: Additional operations like squaring, taking square roots, or rearranging the equation may be needed to solve for missing values.

Hypotenuse Calculator Examples Table
Problem Type	Description	Steps to Solve	Example
Basic Hypotenuse Calculation	Finding the hypotenuse of a right triangle when the lengths of the two legs are known.	Use the Pythagorean theorem: \( c^2 = a^2 + b^2 \), where \( c \) is the hypotenuse, and \( a \) and \( b \) are the lengths of the two legs. Take the square root of the sum of the squares of the two legs to find the hypotenuse.	For a triangle with legs of length 3 and 4, apply the Pythagorean theorem: \( c^2 = 3^2 + 4^2 = 9 + 16 = 25 \). Take the square root of 25 to get \( c = 5 \).
Finding the Leg Length	Finding the length of one leg of a right triangle when the hypotenuse and the other leg are known.	Rearrange the Pythagorean theorem to solve for the missing leg: \( a^2 = c^2 - b^2 \). Take the square root of the result to find the leg length.	For a triangle with hypotenuse 10 and one leg of length 6, apply the formula: \( a^2 = 10^2 - 6^2 = 100 - 36 = 64 \). Take the square root of 64 to get \( a = 8 \).
Real-life Applications	Applying the hypotenuse formula to real-world situations such as determining the distance across a sloped surface.	Use the Pythagorean theorem to calculate the diagonal distance across a rectangular object (like a screen or floor).	If the length of a screen is 15 meters and the height is 20 meters, use the Pythagorean theorem to find the diagonal: \( c^2 = 15^2 + 20^2 = 225 + 400 = 625 \), so \( c = 25 \) meters.