Pythagorean Theorem & Distance Formula Cheat Sheet

A printable reference covering the Pythagorean Theorem, distance formula, coordinate distance, Pythagorean triples, and converse for grades 8-10.

Download PNG

Related Tools

Related Labs

Related Worksheets

Related Infographics

The Pythagorean Theorem connects the side lengths of right triangles and is one of the most useful tools in geometry. This cheat sheet helps students identify right triangles, find missing side lengths, and connect triangle geometry to the coordinate plane. It is especially useful when solving multi-step problems involving diagrams, grids, and real-world distances.

Key Facts

In a right triangle, the Pythagorean Theorem is $a^2 + b^2 = c^2$ , where $c$ is the hypotenuse.
The hypotenuse is always the side across from the right angle and is always the longest side of a right triangle.
To find a missing leg, use $a = \sqrt{c^2 - b^2}$ or $b = \sqrt{c^2 - a^2}$ .
To find the hypotenuse, use $c = \sqrt{a^2 + b^2}$ .
The distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ is $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ .
The distance formula comes from making a right triangle on the coordinate plane with legs $|x_2 - x_1|$ and $|y_2 - y_1|$ .
Common Pythagorean triples include $3,4,5$ , $5,12,13$ , $8,15,17$ , and multiples such as $6,8,10$ .
The converse of the Pythagorean Theorem says that if $a^2 + b^2 = c^2$ for the longest side $c$ , then the triangle is a right triangle.

Vocabulary

Right Triangle: A triangle with one angle that measures $90^\circ$ .
Hypotenuse: The side across from the right angle in a right triangle, and the longest side of the triangle.
Leg: One of the two sides that form the right angle in a right triangle.
Pythagorean Triple: A set of three positive integers that satisfies $a^2 + b^2 = c^2$ .
Distance Formula: A formula used to find the straight-line distance between two coordinate points: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ .
Converse: A reversed statement used here to test whether side lengths form a right triangle.

Common Mistakes to Avoid

Using the wrong side as $c$ is wrong because $c$ must be the hypotenuse, which is across from the right angle and is the longest side.
Adding the side lengths instead of squaring them is wrong because the theorem uses areas of squares, so the correct relationship is $a^2 + b^2 = c^2$ .
Forgetting the square root in the final step is wrong because solving $c^2 = 25$ gives $c = \sqrt{25} = 5$ , not $25$ .
Subtracting coordinates in an inconsistent order is wrong if it leads to expression errors, so use $(x_2 - x_1)^2$ and $(y_2 - y_1)^2$ carefully.
Assuming any three side lengths make a right triangle is wrong because they must satisfy $a^2 + b^2 = c^2$ with $c$ as the longest side.

Practice Questions

1 A right triangle has legs of length $9$ and $12$ . Find the hypotenuse $c$ .
2 Find the missing leg $a$ if a right triangle has hypotenuse $13$ and one leg $5$ .
3 Find the distance between the points $(-2, 4)$ and $(6, -2)$ .
4 Explain why the distance formula is really an application of the Pythagorean Theorem on the coordinate plane.