Pythagorean Theorem: a² + b² = c²

The Pythagorean theorem is the most famous result in elementary geometry: in any right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides.

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

The visualization shows three squares built on the three sides of a right triangle. The two smaller squares (built on the legs) have a combined area exactly equal to the larger square (built on the hypotenuse). It's a relationship purely about areas — and that's the key to its many beautiful proofs.

Step-by-Step Solution

Given: A right triangle with legs $a = 3$ and $b = 4$.

Find: The hypotenuse $c$ using the Pythagorean theorem, and verify by computing the three areas.

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Step 1 — Apply the theorem.

$c^{2} = a^{2} + b^{2} = 3^{2} + 4^{2} = 9 + 16 = 25$

Step 2 — Take the square root.

$c = \sqrt{25} = 5$

So the hypotenuse is exactly 5.

Step 3 — Verify with the areas.

• Area of square on leg $a$: $A_a = a^{2} = 9$
• Area of square on leg $b$: $A_b = b^{2} = 16$
• Sum: $A_a + A_b = 25$
• Area of square on hypotenuse $c$: $A_c = c^{2} = 25$

The two small squares add up to exactly the large square. ✓

Step 4 — Famous Pythagorean triples.

When all three sides are integers, we have a "Pythagorean triple":

• $(3, 4, 5)$ — the smallest
• $(5, 12, 13)$ — $25 + 144 = 169$
• $(8, 15, 17)$ — $64 + 225 = 289$
• $(7, 24, 25)$ — $49 + 576 = 625$
• $(20, 21, 29)$ — $400 + 441 = 841$

Each one is a right triangle with whole-number sides. There are infinitely many.

Step 5 — A famous algebraic proof (one of dozens).

Take a square of side $(a + b)$, and tile it with four copies of the right triangle plus a tilted square of side $c$ in the middle. The total area is:

$(a + b)^{2} = 4 \times \tfrac{1}{2}ab + c^{2}$

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

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

Done. There are over 400 known proofs of the Pythagorean theorem — by mathematicians, by US President Garfield, by Leonardo da Vinci, and many others.

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Answer: For a right triangle with legs 3 and 4:

$\boxed{\;a^{2} + b^{2} = c^{2}\;\Longrightarrow\;9 + 16 = 25\;\Longrightarrow\;c = 5\;}$

The two squares on the legs together have the same total area as the square on the hypotenuse — this is the geometric content of the theorem, and it's true for every right triangle.

Try It

• Adjust the leg lengths $a$ and $b$ — the hypotenuse and all three squares update.
• The HUD shows the area of each square and their relationship.
• Try $(3, 4)$, $(5, 12)$, $(8, 15)$ — all are integer-sided right triangles.