Pythagorean Theorem
Master a² + b² = c² to find missing sides of right triangles, identify Pythagorean triples, and apply the theorem to real-world distance and diagonal problems.
The Pythagorean Theorem is one of the most used formulas in existence. A contractor checks that a room corner is square by measuring 3-4-5. A GPS calculates your distance from a cell tower using this exact formula. A game developer computes whether a bullet hits a target. Construction, navigation, physics, and data science all depend on \(a^2 + b^2 = c^2\). This single equation unlocks an enormous range of real problems.
- Apply \(a^2 + b^2 = c^2\) to find any missing side of a right triangle
- Recognize and use Pythagorean triples (3-4-5, 5-12-13, 8-15-17)
- Verify whether a triangle is a right triangle from its three sides
- Solve real-world distance and diagonal problems
In any right triangle, the sum of the squares of the two legs equals the square of the hypotenuse. To find a missing side:
- Find hypotenuse: \(c = \sqrt{a^2 + b^2}\)
- Find a leg: \(a = \sqrt{c^2 - b^2}\)
The 3-4-5 Right Triangle
Common Pythagorean triples: 3-4-5, 5-12-13, 8-15-17, 7-24-25
A right triangle has legs 6 cm and 8 cm. Find the hypotenuse.
\[ c = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10 \text{ cm} \]This is a 6-8-10 triangle — a multiple of the 3-4-5 triple (×2). Always check for multiples of known triples first.
A right triangle has hypotenuse 13 and one leg of 5. Find the other leg.
\[ a = \sqrt{13^2 - 5^2} = \sqrt{169 - 25} = \sqrt{144} = 12 \]The missing leg is 12. This is the 5-12-13 Pythagorean triple.
A TV screen is advertised as "55 inches" — that's the diagonal. The screen is 48 inches wide. How tall is it?
\[ h = \sqrt{55^2 - 48^2} = \sqrt{3025 - 2304} = \sqrt{721} \approx 26.8 \text{ inches} \]The screen is about 26.8 inches tall. TV "size" is always the diagonal — the Pythagorean Theorem decodes what that means in width and height.
- Find the hypotenuse of a right triangle with legs 9 and 12.
- Is a triangle with sides 7, 10, and 13 a right triangle?
- A 15-foot ladder leans against a wall, with its base 6 feet from the wall. How high does it reach?
▶ Show Answers
- \(\sqrt{81+144} = \sqrt{225} =\) 15 (a 9-12-15 triple, multiple of 3-4-5).
- \(7^2+10^2=49+100=149 \neq 169=13^2\) → Not a right triangle.
- \(h = \sqrt{15^2 - 6^2} = \sqrt{225-36} = \sqrt{189} \approx\) 13.7 feet.
- Squaring instead of square-rooting: \(c^2 = 100\) means \(c = 10\), not \(c = 100\). The final answer requires taking the square root.
- Adding instead of subtracting to find a leg: To find a leg: \(a^2 = c^2 - b^2\) (subtract). Only finding the hypotenuse uses addition.
- Using on non-right triangles: \(a^2+b^2=c^2\) only works for right triangles. For other triangles, use the Law of Cosines.
- \(\mathbf{a^2 + b^2 = c^2}\) — the two legs squared and added equal the hypotenuse squared.
- Find hypotenuse: \(c = \sqrt{a^2+b^2}\). Find a leg: \(a = \sqrt{c^2-b^2}\).
- Pythagorean triples (3-4-5, 5-12-13, 8-15-17) give whole-number answers — recognize multiples.
- Converse: If \(a^2+b^2=c^2\), the triangle is a right triangle.
Contractors use the 3-4-5 rule to ensure corners are perfectly square — measure 3 feet along one wall, 4 feet along the adjacent wall, and if the diagonal is 5 feet, the corner is exactly 90°. GPS systems compute straight-line distance using the Pythagorean Theorem extended to three dimensions. Any time you need the "straight-line distance" between two points — in construction, gaming, robotics, or mapping — this theorem is the tool.
Calculator Connection
The Pythagorean Theorem Calculator finds any missing side of a right triangle — enter two known values and it solves for the third with step-by-step work shown.
Interactive Diagram
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Pythagorean Theorem — Quiz
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