How the Pythagorean Theorem Works

a² + b² = c² may be the most recognized formula in mathematics, and it earns the fame: it converts the two short sides of any right triangle into the long one, underlies the distance formula, sets the size of every TV screen, and once scandalized the Greeks by producing a number that refused to be written as a fraction. Here is what it actually says, why it is true, and where it earns its keep.

What the formula says (and when it applies)

In a right triangle — one with a 90° corner — the two sides that meet at the right angle are the legs (call them a and b), and the long side across from the right angle is the hypotenuse (c). The theorem states:

a² + b² = c²
c = √(a² + b²)

The condition matters: the triangle must contain a right angle. For any other triangle the equation does not hold (an acute triangle's hypotenuse-style longest side is shorter than √(a² + b²), an obtuse one's is longer — which is precisely how the converse test below works). The formula answers three practical questions: given two legs, find the diagonal distance; given the hypotenuse and one leg, find the missing leg as √(c² − a²); given all three sides, check whether the corner is square.

The 3-4-5 triangle and why it matters

The smallest whole-number example is the classic:

3² + 4² = 9 + 16 = 25 = 5²    →   legs 3 and 4, hypotenuse 5

This is the carpenter's trick for checking a corner: mark 3 units along one edge and 4 along the other; if the diagonal measures exactly 5, the corner is a true 90°. Construction crews scale it up as 6-8-10 or 9-12-15, which work for the same reason — multiplying every side by the same factor preserves the relationship (6² + 8² = 4 × 25). Whole-number solutions like this are called Pythagorean triples; other favorites include 5-12-13, 8-15-17, and 7-24-25 (check: 49 + 576 = 625 ✓).

Worked examples: ladder and TV

The ladder problem. A 13-foot ladder leans against a wall with its base 5 feet out. How high does it reach? The ladder is the hypotenuse:

height = √(13² − 5²) = √(169 − 25) = √144 = 12 feet

Notice the minus sign: given hypotenuse and one leg, you subtract — a common slip is adding instead.

The TV size problem. Screens are advertised by diagonal: a "55-inch" TV has a corner-to-corner hypotenuse of 55 inches. At the standard 16:9 aspect ratio, the width and height are in proportion 16:9, so write them as 16k and 9k:

(16k)² + (9k)² = 55²
256k² + 81k² = 337k² = 3025
k = √(3025/337) ≈ 2.996
width ≈ 47.9 in,  height ≈ 27.0 in

So the screen area you actually watch is about 47.9 × 27.0 ≈ 1,293 square inches — the diagonal alone never tells you that; the Pythagorean theorem in reverse does.

The distance formula is Pythagoras wearing coordinates

The single most reused consequence of the theorem is the distance between two points on a grid. The horizontal gap and vertical gap between the points form the legs of a right triangle; the straight-line distance is the hypotenuse:

distance = √((x₂ − x₁)² + (y₂ − y₁)²)

(1, 2) to (4, 6):  √((4−1)² + (6−2)²) = √(9 + 16) = √25 = 5

Every "as the crow flies" distance ever computed — in mapping apps, game engines, machine learning — is this formula running. It also connects to slope: a line's slope is rise over run, the ratio of the two legs. Slope and distance answer different questions about the same slanted segment — slope says how steep, the theorem says how far — which is why a slope calculator and a distance calculation so often appear in the same homework problem, each in charge of one leg of the reasoning.

The converse: testing for a right angle

The theorem runs both ways. If three side lengths satisfy a² + b² = c² (with c the longest), the angle between a and b is exactly 90°. This is the ancient surveyor's rope trick in mathematical form: knot a rope in 12 equal parts, pull it taut as a 3-4-5 loop, and the corner is square. Numerically:

Is a triangle with sides 8, 15, 17 right-angled?
8² + 15² = 64 + 225 = 289 = 17²   →  yes, exactly 90°

The same test doubles as a classifier: if a² + b² < c² the angle is obtuse (wider than 90°); if a² + b² > c² it is acute. A triangle calculator runs this logic for any three sides, reporting the angles so you can see which case you have.

From sides to area — and to a scandal

Once a triangle's sides are known, its area follows. For a right triangle it is simply half the product of the legs — the legs serve as base and height because the right angle guarantees they are perpendicular:

area = ½ × a × b = ½ × 3 × 4 = 6 square units

For non-right triangles, where no side plays the perpendicular-height role directly, Heron's formula gets the area from the three sides alone: s = (a + b + c)/2, then area = √(s(s−a)(s−b)(s−c)). For 3-4-5: s = 6, so area = √(6 × 3 × 2 × 1) = √36 = 6 — same triangle, same answer, as it should be. The Triangle Area Calculator handles both routes, plus the base-and-height method.

The theorem also triggered an ancient crisis. Take the simplest square: side 1, diagonal c, and the theorem demands 1² + 1² = c², so c = √2. The Pythagoreans believed every length was a ratio of whole numbers — but √2 is irrational, provably not any fraction, a discovery (and legend of its discoverer being drowned for it) that forced the Greeks to separate the ideas of number and length and reshaped the foundations of mathematics.

Try it yourself

Enter any three triangle measurements into the Triangle Calculator — it solves the remaining sides and angles and tells you whether the triangle is right, acute, or obtuse. The Triangle Area Calculator computes area from sides (Heron's formula), base-and-height, or two legs of a right triangle. And to connect slope with distance, the Slope Calculator converts two points into the rise, run, and angle of the line between them — the legs and hypotenuse angle of the very triangle the theorem describes.