Math

The Pythagorean Theorem

2 min readEasy5-question drill

Right triangles show up everywhere on the test — in geometry, coordinate plane distance problems, and word problems about ladders and ramps. The Pythagorean Theorem is the single tool that unlocks almost all of them.

Where this lesson fits

You are here

→Right Triangles and Trigonometry →Coordinate Geometry →Lines, Angles, and Triangles

A right triangle is a triangle with one 90° angle (a square corner). The two sides that form that square corner are called legs, and the side directly across from the right angle — always the longest side — is the hypotenuse.

The Pythagorean Theorem says that for any right triangle:

a² + b² = c²

where a and b are the two legs and c is the hypotenuse. The key rule: c is ALWAYS the hypotenuse — the side opposite the right angle. Never plug a leg in for c.

Loading…

A 3-4-5 right triangle: legs a and b, hypotenuse c opposite the right angle.

How to use it: If you know any two sides, you can find the third.

Finding the hypotenuse: add the squares of the legs, then take the square root. Legs 3 and 4 → 3² + 4² = 9 + 16 = 25, so c = √25 = 5.
Finding a leg: subtract. If the hypotenuse is 13 and one leg is 5, then 5² + b² = 13² → 25 + b² = 169 → b² = 144 → b = 12.

Pythagorean triples are whole-number side sets that work perfectly. Memorize these — they appear constantly:

3, 4, 5 (and multiples: 6, 8, 10; 9, 12, 15)
5, 12, 13
8, 15, 17

Special right triangles have fixed side ratios so you don't even need the full theorem:

A 45-45-90 triangle has two equal legs and a hypotenuse √2 times longer. Ratio: leg : leg : hypotenuse = 1 : 1 : √2. So if a leg is 7, the hypotenuse is 7√2. If the hypotenuse is 10√2, each leg is 10.
A 30-60-90 triangle has sides in the ratio 1 : √3 : 2 (short leg : long leg : hypotenuse).

Loading…

A 45-45-90 triangle: equal legs and a hypotenuse √2 times longer.

When you see a right triangle and two known sides, reach for a² + b² = c². When you see 45-45-90 or 30-60-90, use the ratio shortcut instead — it's faster.

Quick check

A right triangle has legs of length 6 and 8. What is the length of the hypotenuse?

Worked examples

Example 1

A right triangle has legs of length 9 and 12. What is the length of the hypotenuse?

Example 2

In a right triangle, the hypotenuse is 17 and one leg is 8. What is the length of the other leg?

Example 3

In a 45-45-90 triangle, the hypotenuse has length 6√2. What is the length of each leg?

Common pitfalls

Plugging a leg in for c

c must be the hypotenuse — the side OPPOSITE the right angle and the longest side. If you put a leg where c goes, every number comes out wrong. Always locate the right angle first.

Forgetting to take the square root

After computing a² + b² you get c², not c. Students who stop at 225 instead of √225 = 15 pick a trap answer. Finish the square root.

Adding instead of subtracting to find a leg

When the hypotenuse is given and you need a leg, you must SUBTRACT: b² = c² - a². Adding gives a number bigger than the hypotenuse, which is impossible for a leg.

Mixing up the 45-45-90 and 30-60-90 ratios

45-45-90 is 1 : 1 : √2 (two equal legs). 30-60-90 is 1 : √3 : 2. Match the ratio to the angles you actually see, and remember the hypotenuse is always the biggest number in the ratio.

Key takeaways

The Pythagorean Theorem is a² + b² = c², where c is always the hypotenuse (opposite the right angle).
To find the hypotenuse, add the squares and take the root; to find a leg, subtract then take the root.
Memorize the triples 3-4-5, 5-12-13, 8-15-17 and their multiples to save time.
45-45-90 triangles use the ratio 1 : 1 : √2; 30-60-90 triangles use 1 : √3 : 2.
Always finish by taking the square root — don't stop at c².

Tracks your progress across lessons.

Try it yourself

5 practice questions on The Pythagorean Theorem, drawn from the question bank. The tutor is one click away if you get stuck.

Where this lesson fits

The Pythagorean Theorem

You are here

→Right Triangles and Trigonometry →Coordinate Geometry →Lines, Angles, and Triangles