The Pythagorean Theorem: How to Use It

In a right triangle, the side opposite the right angle is called the hypotenuse (the longest side). The other two sides are the legs. The Pythagorean theorem relates all three:

where a and b are the legs and c is the hypotenuse.

Finding the Hypotenuse

When you know both legs, add their squares and take the square root.

Example: Legs are 3 and 4. Find the hypotenuse.

Hypotenuse = 5

The 3-4-5 triangle is a classic. Other common ones: 5-12-13, 8-15-17.

Finding a Leg

When you know the hypotenuse and one leg, subtract the known leg's square from the hypotenuse's square, then take the square root.

Example: Hypotenuse = 10, one leg = 6. Find the other leg.

Other leg = 8

When to Use It

• Right triangles only — the theorem does not apply to other triangles
• Finding distance between two points (treat the difference in x and y as the legs)
• Checking if a triangle is right (if a² + b² = c², it's right)

Distance Formula (Application)

The distance between points (x₁, y₁) and (x₂, y₂) comes from the Pythagorean theorem:

The horizontal and vertical differences are the legs; the distance is the hypotenuse.

Common mistakes

Using a² + b² = c² on non-right triangles: This theorem ONLY works for right triangles. Check for the 90° angle first.

Confusing legs with hypotenuse: The hypotenuse (c) is always the longest side, opposite the right angle. The two shorter sides are the legs (a and b).

Forgetting the square root: After computing c² = 25, you need c = √25 = 5, not c = 25.

More examples

Example: Legs are 5 and 12. Find the hypotenuse.

This is the famous 5-12-13 Pythagorean triple.

Example: Hypotenuse = 15, one leg = 9. Find the other.

Try Arithmia