The Pythagorean Theorem

The single most-cited theorem in all of mathematics. Twenty-five centuries after the Pythagorean school of Croton wrote it down, it still rules every triangle calculation, every distance measurement, every notion of length in spaces from the chalkboard to general relativity. Two thousand years before there was algebra, geometers already had a name for the surprising fact that the squares on the legs of a right triangle, taken together, exactly equal the square on the hypotenuse. This section explains why it must be true.

The statement

In a right triangle with legs $a$ and $b$ and hypotenuse $c$ (the side opposite the right angle, always the longest):

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

The squares here are literal, if you draw a square on each side of the triangle, the area of the square on $c$ equals the combined area of the squares on $a$ and $b$.

Why is this true?, the area-rearrangement proof

Take four identical copies of the right triangle. Arrange them inside a large square of side $a + b$ so that their hypotenuses face inward; the four triangles tile the border, and a smaller tilted square of side $c$ sits at the centre.

The area of the large square is $(a + b)^2 = a^2 + 2ab + b^2$. The four triangles contribute $4 \cdot \tfrac{1}{2}ab = 2ab$. The central square has area $c^2$. Therefore:

$a^2 + 2ab + b^2 \;=\; 2ab + c^2 \quad\Longrightarrow\quad a^2 + b^2 = c^2.$

The $2ab$ on each side cancels, and the theorem falls out. Notice how unexpectedly clean this is, no calculus, no trigonometry, just counting areas two ways.

• Construction: The "3-4-5 triangle" is the carpenter's right-angle check: stake distances of 3 and 4 units on two ropes; if the diagonal is exactly 5, the corner is perfectly square. This is a literal Pythagorean triple.
• Computer Graphics: Every collision-detection routine in a game engine ultimately tests $(x_1 - x_2)^2 + (y_1 - y_2)^2 \leq r^2$, the squared form of Pythagoras' theorem, kept squared to avoid the costly square root.
• Special Relativity: Spacetime intervals replace $a^2 + b^2 = c^2$ with $s^2 = t^2 - x^2$; the minus sign is the only twist, but the geometric reasoning is direct Pythagoras adapted to Minkowski space.

Drag the leg sliders $a$ and $b$. Squares grow on each side of the right triangle, and the area readouts confirm that $a^2 + b^2 = c^2$. Switch to "Rearrangement proof" mode to see the four-triangles-inside-a-square diagram, the visual heart of the area-counting proof.

The converse, flipping the implication

The converse is also true: if a triangle has sides $a, b, c$ satisfying $a^2 + b^2 = c^2$, then it is a right triangle, with the right angle opposite side $c$. This is what lets you check by arithmetic whether a triangle is right: compute $a^2 + b^2$; compute $c^2$; if they match, the angle is 90^\circ.

Pythagorean triples

A Pythagorean triple is a triple of positive integers $(a, b, c)$ that satisfies $a^2 + b^2 = c^2$. The most famous is $(3, 4, 5)$; others include $(5, 12, 13)$, $(8, 15, 17)$, $(7, 24, 25)$, and $(9, 40, 41)$. Any scalar multiple is also a triple: $(6, 8, 10) = 2(3, 4, 5)$. There are infinitely many primitive triples (ones whose three components share no common factor, Euclid gave a complete parametrisation in Elements Book X).

Connection to the distance formula

The distance formula $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ that you saw in §5.1 is literally Pythagoras' theorem rearranged for the hypotenuse: the legs are $\Delta x$ and $\Delta y$, and the hypotenuse $d$ pops out by taking the square root of both sides. Every time you compute a Euclidean distance, you are using Pythagoras.

Try it

• Set legs $a =$ and $b =$. Predict first: with legs $a = 3$ and $b = 4$, what is the hypotenuse $c$? Set the sliders and verify it shows $c = 5$.
• Set $a =$ and $b =$. Before sliding: with $a = 5$ and $b = 12$, what is $c$? Set the sliders and confirm it reads $c = 13$, another classic Pythagorean triple.
• Try $a =$ and $b =$. Before setting: with both legs $= 1$, what is the hypotenuse? Set the sliders and verify it reads $\sqrt{2} \approx 1.41$, the famous irrational that motivated the Greeks to invent the real numbers.

A trap to watch for

The Pythagorean theorem holds only for right triangles. If a triangle has sides $5, 6, 7$, then $5^2 + 6^2 = 61$ but $7^2 = 49$, so this is not a right triangle. The Greek geometers eventually generalised the fact into the law of cosines: $c^2 = a^2 + b^2 - 2ab\cos\theta$, where $\theta$ is the angle opposite $c$. When \theta = 90^\circ, $\cos\theta = 0$ and the formula reduces to Pythagoras. Do not apply Pythagoras to a non-right triangle.

What you now know

You can compute the missing side of a right triangle in either direction, recognise common Pythagorean triples, and connect the theorem back to the distance formula. The next chapter applies these tools to the most-studied curve in all of geometry: the circle.

Quick check

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References

• Lang, S. (1971). Basic Mathematics. Springer. Chapter 5, §3, the proof reproduced above, in his minimalist style.
• Euclid (c. 300 BCE). Elements, Book I, Proposition 47 (Heath translation, Dover, 1956). The original Greek proof, by similar triangles.
• Loomis, E. S. (1968). The Pythagorean Proposition, 2nd ed. NCTM. Contains hundreds of distinct proofs of the theorem.
• Hartshorne, R. (2000). Geometry: Euclid and Beyond. Springer. §1.5 places Pythagoras in its modern axiomatic context.