Pythagorean Theorem Formula

The Formula

a^2 + b^2 = c^2 (c is the hypotenuse)

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: 3^2 + 4^2 = 9 + 16 = 25 = 5^2, so the hypotenuse is 5.

Notation

a, b are the legs; c is the hypotenuse (longest 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 \triangle ABC with \angle C = 90°: |AB|^2 = |AC|^2 + |BC|^2, equivalently c^2 = a^2 + b^2 where c = |AB|

Worked Examples

Example 1

easy
A right triangle has legs of length 3 and 4. Find the hypotenuse.

Solution

  1. 1
    Apply the Pythagorean theorem: a^2 + b^2 = c^2.
  2. 2
    Substitute: 3^2 + 4^2 = c^2 \Rightarrow 9 + 16 = c^2 \Rightarrow c^2 = 25.
  3. 3
    Take the positive square root: c = \sqrt{25} = 5.

Answer

c = 5
The Pythagorean theorem relates the three sides of a right triangle. The hypotenuse (opposite the right angle) is always the longest side. The 3-4-5 triple is the most common Pythagorean triple.

Example 2

medium
A right triangle has a hypotenuse of 13 and one leg of length 5. Find the other leg.

Common Mistakes

  • Using wrong side as hypotenuse
  • Forgetting to take square root for final answer

Why This Formula Matters

Essential for distance, navigation, and countless applications.

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?

a, b are the legs; c is the hypotenuse (longest side, opposite the right angle)

Why is the Pythagorean Theorem formula important in Math?

Essential for distance, navigation, and countless applications.

What do students get wrong about Pythagorean Theorem?

c must be the longest side (hypotenuse), opposite the right angle.

What should I learn before the Pythagorean Theorem formula?

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

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