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📐 The Pythagorean Theorem

In every right triangle, a² + b² = c². Discover this 2,500-year-old rule, use it to find missing sides, and spot the famous 3-4-5 triangle everywhere.

lessons

~15 min

to learn

🔢 Math

subject

Teens

level

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What you’ll learn

The Theorem and the Right TriangleIdentify the parts of a right triangle and state the Pythagorean theorem.A right triangle has one 90° angle, two legs that form it, and a hypotenuse — the longest side opposite the right angle. The Pythagorean theorem states a² + b² = c²: the squares of the legs add up to the square of the hypotenuse. The 3-4-5 triangle is the classic example (9 + 16 = 25 = 5²), and whole-number sets like it are called Pythagorean triples.
Using PythagorasUse the Pythagorean theorem to find missing sides and test for right angles.To find a hypotenuse, square both legs, add, and take the square root: legs 6 and 8 give √100 = 10. To find a leg, subtract instead: a² = c² − b², so hypotenuse 13 and leg 5 give √144 = 12. The theorem also works in reverse — if a² + b² equals c² for a triangle's sides, the triangle must be right-angled. These skills power construction, navigation, and design.

Questions this course answers

In a right triangle, which side is the hypotenuse?

The hypotenuse is the longest side and always sits opposite the 90° right angle.

What does the Pythagorean theorem state?

The theorem states a² + b² = c²: the squares of the two legs add up to the square of the hypotenuse.

For the 3-4-5 triangle, what is 3² + 4²?

3² + 4² = 9 + 16 = 25, which equals 5², confirming the right angle.

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

6² + 8² = 36 + 64 = 100, and √100 = 10, so the hypotenuse is 10.

A right triangle has hypotenuse 13 and one leg 5. What is the other leg?

To find a leg, subtract: 13² − 5² = 169 − 25 = 144, and √144 = 12.

Which set of side lengths does NOT form a right triangle?

For 2, 3, 4: 2² + 3² = 4 + 9 = 13, but 4² = 16, so it fails a² + b² = c².

Grounded in trusted sources

Khan Academy — Pythagorean theorem
National Council of Teachers of Mathematics
BBC Bitesize — Pythagoras
Britannica — Pythagorean theorem

Every Wunder lesson is built from real, reputable sources — never invented.

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