Pythagorean Theorem Formula

Pythagorean theorem is in a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides.

The Formula

a2+b2=c2a^2+b^2=c^2

When to use: If you draw squares on each side of a right triangle, the two smaller squares fill the big one exactly.

Quick Example

Sides 3 and 4: 32+42=9+16=25=523^2 + 4^2 = 9 + 16 = 25 = 5^2, so the hypotenuse is 5.

Notation

cc is the hypotenuse, the side opposite the right angle.

What This Formula Means

In a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides.

If you draw squares on each side of a right triangle, the two smaller squares fill the big one exactly.

Formal View

In △ABC\triangle ABC with ∠C=90°\angle C = 90°: ∣AB∣2=∣AC∣2+∣BC∣2|AB|^2 = |AC|^2 + |BC|^2, equivalently c2=a2+b2c^2 = a^2 + b^2 where c=∣AB∣c = |AB|

Worked Examples

Example 1

easy
A right triangle has legs of length 33 and 44. Find the hypotenuse.

Answer

c=5c = 5

First step

1
Apply the Pythagorean theorem: a2+b2=c2a^2 + b^2 = c^2.

Full solution

  1. 2
    Substitute: 32+42=c2⇒9+16=c2⇒c2=253^2 + 4^2 = c^2 \Rightarrow 9 + 16 = c^2 \Rightarrow c^2 = 25.
  2. 3
    Take the positive square root: c=25=5c = \sqrt{25} = 5.
The Pythagorean theorem relates the three sides of a right triangle. The hypotenuse (opposite the right angle) is always the longest side. The 33-44-55 triple is the most common Pythagorean triple.

Example 2

medium
A right triangle has a hypotenuse of 1313 and one leg of length 55. Find the other leg.

Example 3

easy
Find the distance between (1,2)(1,2) and (4,6)(4,6).

Common Mistakes

  • Using the theorem without a right angle — confirm the triangle is right first.
  • Putting a leg in the hypotenuse spot — the hypotenuse is opposite the right angle and is longest.
  • Forgetting to take the square root — solving for a side length requires undoing the square.

Why This Formula Matters

This theorem links geometry, algebra, distance, and square roots. Students must recognize right-triangle structure before using the formula; otherwise the equation gives meaningless results. Recognizing it by "Do I know which side is the hypotenuse?" — rather than by familiar numbers — is what lets a student tell it apart from distance formula and triangle inequality in a mixed problem set.

Frequently Asked Questions

What is the Pythagorean Theorem formula?

In a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides.

How do you use the Pythagorean Theorem formula?

If you draw squares on each side of a right triangle, the two smaller squares fill the big one exactly.

What do the symbols mean in the Pythagorean Theorem formula?

cc is the hypotenuse, the side opposite the right angle.

Why is the Pythagorean Theorem formula important in Math?

This theorem links geometry, algebra, distance, and square roots. Students must recognize right-triangle structure before using the formula; otherwise the equation gives meaningless results. Recognizing it by "Do I know which side is the hypotenuse?" — rather than by familiar numbers — is what lets a student tell it apart from distance formula and triangle inequality in a mixed problem set.

What do students get wrong about Pythagorean Theorem?

The procedure for pythagorean theorem is the easy part; the trap is using the theorem without a right angle. Asking "Do I know which side is the hypotenuse?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

What should I learn before the Pythagorean Theorem formula?

Before studying the Pythagorean Theorem formula, you should understand: triangles, exponents, square roots.

← Back to Pythagorean Theorem See All Examples Practice Problems