Right Triangles

The Good Old 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
$$ A^2 + B^2 = C^2 $$

Right Triangle Formulas

Right triangle picture

A right triangle has one $$ 90^{\circ} $$ angle ($$ \angle $$ B in the picture on the left) and a variety of often-studied formulas such as:

SOHCAHTOA only applies to right triangles (more here )


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'.

In the triangle on the left, 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!)

The Pythagorean Theorem

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.

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

Problem 2

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

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?

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

Problem 4

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

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?

X2 +42=52
X2 +16=25
X2 = 25 - 16 = 9
X= 3

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