Right Triangles

Rules, Formula and more

Pythagorean Theorem

The sum of the squares of the lengths of the legs equals the square of the length of the hypotenuse.

Usually, this theorem is expressed as $$ A^2 + B^2 = C^2 $$.

Diagram 1 a squared plus b squared equals c squared
Diagram 2 Pythagorean Theorem

A Right Triangle's Hypotenuse

The hypotenuse is the largest side in a right triangle and is always opposite the right angle.

(Only right triangles have a hypotenuse). The other two sides of the triangle, AC and CB are referred to as the 'legs'. Hypotenuse

In the triangle above, the hypotenuse is the side AB which is opposite the right angle, $$ \angle C $$.

Hypotenuse Calculator

Online tool calculates the hypotenuse (or a leg) using the Pythagorean theorem.

(Also draws a free downloadable picture of your right Triangle!).

Practice Problems

Below are several practice problems involving the Pythagorean theorem, you can also get more detailed lesson on how to use the Pythagorean theorem here.

Problem 1

Find the length of side t in the triangle on the left.

5, 12, 13 right triangle

Substitute the two known sides into the Pythagorean theorem's formula:
A² + B² = C²

right triangle equation
Problem 2

What is the value of x in the picture on the left?

pythagorean theorem

Set up the Pythagorean Theorem: 142 + 482 = x2
2,500 = X2

$$ x = \sqrt{2500} = 50 $$

Problem 3

What is the value of x in the picture on the left?

Diagram, Pythagorean Theorem

$$ x^2 = 21^2 + 72^2 \\ x^2= 5625 \\ x = \sqrt{5625} \\ x =75 $$

Problem 4

Find the length of side X in the triangle on on the left?

3, 4, 5 right triangle

Substitue the two known sides into the pythagorean theorem's formula:
$$ A^2 + B^2 = C^2 \\ 8^2 + 6^2 = x^2 \\ x = \sqrt{100}=10 $$

Problem 5

What is x in the triangle on the left?

pythagorean image

x2 + 42 = 52
x2 + 16 = 25
x2 = 25 - 16 = 9
x = 3

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