Geometry

Proving the Pythagorean Theorem

Understanding why a² + b² = c² for a right triangle.

Advanced20 min lesson3 min readUpdated August 12, 2026Author not yet attributed

Why a² + b² = c²

For any right triangle, squaring each of the two shorter sides and adding the results always equals the square of the longest side. Before reading on, test this on a right triangle with legs 3 and 4: what is 3² + 4², and how does it compare to 5²?

Definition — The Pythagorean Theorem

For a right triangle with legs a and b and hypotenuse c (the side opposite the right angle, always the longest side): a² + b² = c².

One way to see why this must be true: take four identical copies of a right triangle with legs a and b, and arrange them inside a large square of side length (a + b), each triangle's hypotenuse forming one side of a smaller, tilted square in the middle. The large square's total area can be found two ways — directly, as (a + b)², or as the four triangles' combined area plus the tilted inner square's area, c². Since both descriptions measure the exact same total area, (a + b)² must equal 4 triangle-areas plus c² — and working through that algebra reduces to exactly a² + b² = c².

Worked Example — Verifying the Theorem Numerically

Check the theorem for a right triangle with legs 3 and 4 and hypotenuse 5. a² + b² = 3² + 4² = 9 + 16 = 25. c² = 5² = 25. Both sides match, confirming the relationship.

Worked Example — Finding a Hypotenuse

A right triangle has legs 6 and 8. Find the hypotenuse. a² + b² = c²: 6² + 8² = 36 + 64 = 100. Taking the square root: c = √100 = 10.

a^{2} + b^{2} = c^{2}

Geometry Canvas

Point mode (click to add a point)LabelsMeasurementsSnap to grid

Construct

Tool

Objects

1.
Color
P1
P1: a free point, draggable on the plane
2.
Color
P2
P2: a free point, draggable on the plane
3.
Color
P3
P3: a free point, draggable on the plane
4.
Color
poly1
poly1: the polygon through P1, P2, P3

Measurements

poly1area = 15perimeter = 17.66

Tip

Always identify the hypotenuse first — it's the longest side, always located directly opposite the right angle — before deciding which two values to square and add, and which one to isolate.

Common Mistakes

Adding the squares of all three sides, or squaring the hypotenuse and adding it to a leg instead of isolating it correctly.
The theorem specifically combines the two legs (a² + b²) to equal the hypotenuse squared (c²) — the hypotenuse is never added into the sum on the left side.
Forgetting to take the square root at the end when solving for a side length, leaving the squared value as the final answer.
Since the theorem gives an equation in terms of squares, the very last step to find an actual side length is always taking a square root.

Key Takeaways

The Pythagorean Theorem relates a right triangle's two legs and hypotenuse: a² + b² = c².
One proof rearranges four copies of the triangle inside a square, comparing the square's area calculated two different ways.
The hypotenuse is always the longest side, located opposite the right angle.

Summary

Understanding why the Pythagorean Theorem holds, not just memorizing the formula, makes it a genuinely usable tool. The next lesson applies it to real distance problems, including distances on the coordinate plane.

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