what is the pythagorean theorem

The Pythagorean theorem says that in any right-angled triangle, the square of the longest side (the hypotenuse) equals the sum of the squares of the other two sides: a2+b2=c2a^{2}+b^{2}=c^{2}a2+b2=c2.

Quick Scoop: What Is the Pythagorean Theorem?

Think of a right triangle (one angle is exactly 90 degrees). Label the short sides aaa and bbb, and the longest side opposite the right angle ccc (the hypotenuse).

The theorem says:

a2+b2=c2a^{2}+b^{2}=c^{2}a2+b2=c2

In words: the area of the square built on the hypotenuse equals the sum of the areas of the squares built on the other two sides.

Mini breakdown

• It only works for right-angled triangles, not for other triangle types.

• ccc is always the hypotenuse (the side opposite the 90° angle and the longest side).

• If you know any two sides of a right triangle, you can use the formula to find the third.

Tiny example

Suppose a right triangle has sides a=3a=3a=3 and b=4b=4b=4. Then:

• a2+b2=32+42=9+16=25a^{2}+b^{2}=3^{2}+4^{2}=9+16=25a2+b2=32+42=9+16=25
• So c2=25c^{2}=25c2=25, which means c=5c=5c=5.

That 3–4–5 triangle is a classic Pythagorean triple used in school problems and real-world construction.

Why people still talk about it

• It’s one of the most famous results in geometry, known for thousands of years.

• It underpins distance formulas in coordinate geometry, complex numbers, and trigonometry (like sin⁡2θ+cos⁡2θ=1\sin^{2}\theta +\cos^{2}\theta =1sin2θ+cos2θ=1).

• It shows up in everyday tasks: measuring the shortest ladder length to reach a wall, checking if corners are square in building, or finding distances on maps and screens.

TL;DR

• Right triangle only.
• Two short sides aaa and bbb, hypotenuse ccc.
• Relationship: a2+b2=c2a^{2}+b^{2}=c^{2}a2+b2=c2.

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