Pythagorean Theorem

Geometry
rule

Also known as: a² + b² = c², right triangle theorem

Grade 6-8

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In a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides. Essential for distance, navigation, and countless applications.

Definition

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

💡 Intuition

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

🎯 Core Idea

A fundamental relationship between sides of right triangles.

Example

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

Formula

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

Notation

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

🌟 Why It Matters

Essential for distance, navigation, and countless applications.

💭 Hint When Stuck

Ask yourself: which side is across from the 90-degree angle? Label that one c, then plug the other two into a-squared plus b-squared.

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|

🚧 Common Stuck Point

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

⚠️ Common Mistakes

  • Using the wrong side as the hypotenuse — c must be the longest side, opposite the 90° angle
  • Forgetting to take the square root after computing a^2 + b^2 to find c
  • Applying the theorem to non-right triangles — it only works when one angle is exactly 90°

Frequently Asked Questions

What is Pythagorean Theorem in Math?

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

What is the Pythagorean Theorem formula?

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

When do you use Pythagorean Theorem?

Ask yourself: which side is across from the 90-degree angle? Label that one c, then plug the other two into a-squared plus b-squared.

How Pythagorean Theorem Connects to Other Ideas

To understand pythagorean theorem, you should first be comfortable with triangles, exponents and square roots. Once you have a solid grasp of pythagorean theorem, you can move on to distance formula and trigonometric functions.

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Interactive Playground

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