Key takeaway: The Pythagorean theorem says that in any right triangle, the area of the square on the longest side equals the combined areas of the squares on the two shorter sides.
Grade 88.G.B.7Geometry and Measurement
Definition of Pythagorean Theorem
The Pythagorean theorem connects the three side lengths of every right triangle. Build a square on each side: the two smaller squares, built on the legs, have a combined area exactly equal to the area of the big square built on the longest side.
That longest side is the hypotenuse, the one facing the right angle. With the theorem you can find one missing side when you know the other two, and you can test whether a triangle with given sides has a right angle. It only works for right triangles; other triangles need different tools.
Formula and notation
Name the legs \(a\) and \(b\), and the hypotenuse \(c\). The Pythagorean theorem is the equation
\[a^{2} + b^{2} = c^{2}\]
Solving for the hypotenuse gives \(c = \sqrt{a^{2} + b^{2}}\); solving for a leg gives \(a = \sqrt{c^{2} – b^{2}}\). Keep every length in one unit; the squares are then in that unit squared.
Key facts
- The hypotenuse is always the longest side of a right triangle.
- If two shorter sides squared and added match the longest side squared, the triangle is right.
- Whole-number side sets such as 5-12-13 are called Pythagorean triples.
- The Pythagorean theorem also measures straight-line distances between points on a grid.
Pythagorean Theorem example
A zip-line cable runs from the top of a 16-foot pole down to a stake on the ground 63 feet from the base of the pole. How long is the cable?
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The pole and the ground are the legs of a right triangle: \(a = 16\) and \(b = 63\).
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Square and add: \(16^{2} + 63^{2} = 256 + 3969 = 4225\).
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Take the square root: \(c = \sqrt{4225} = 65\).
The cable is 65 feet long.
Common mistake
Watch out
Keisha stops after adding the squares and says the cable is \(4225\) feet long.
Why it is wrong
The sum \(a^{2} + b^{2}\) equals \(c^{2}\), not \(c\). A cable almost a mile long attached to a 16-foot pole makes no sense. Always finish with the square root: \(\sqrt{4225} = 65\) feet.
Related terms and lessons
Browse the glossary: A–Z list · Grade 8 vocabulary · Geometry and Measurement