The Pythagorean Theorem

One of mathematics' greatest hits

The Pythagorean theorem is over 2,500 years old, named after the Greek mathematician Pythagoras, yet builders, engineers, game developers, and navigators use it every single day. It describes a beautiful and exact relationship between the three sides of a right-angled triangle, and once you know it, you can find a missing distance you cannot measure directly.

That is the magic: with two sides, you can always calculate the third — no ruler needed.

Right triangles and the hypotenuse

The theorem works on right-angled triangles only — triangles that contain one 90° angle (the little square symbol in the corner). It does not apply to other triangles, so the first thing to check is always: is there a right angle?

A right triangle has three sides with special names:

• The two shorter sides that form the right angle are the legs (sometimes called a and b).
• The longest side, opposite the right angle, is the hypotenuse (called c).

The hypotenuse is always the longest side and always sits across from the 90° corner. Identifying it correctly is the most common place students slip, so take a moment with every problem.

The theorem

The relationship is wonderfully simple:

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

In words: the square of the hypotenuse equals the sum of the squares of the two legs. The squaring here is exactly the exponent idea from Exponents and Powers — a² just means a × a.

Why is it true?

You do not have to take it on faith. Picture a square literally built on each side of the triangle. The theorem says the area of the big square on the hypotenuse equals the combined areas of the two squares on the legs.

Here is one clean way to see it. Take four identical right triangles with legs a and b. Arrange them inside a large square of side (a + b) so they leave a tilted square hole of side c in the middle.

• Area of the big square = (a + b)² = a² + 2ab + b².
• Area as parts = the four triangles + the tilted square = 4 × (½ab) + c² = 2ab + c².

Set the two expressions equal:

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

Cancel the 2ab from both sides and you are left with a² + b² = c². The relationship is not a coincidence — it falls straight out of comparing areas. (Working with areas of squares connects directly to Area and Perimeter.)

Finding the hypotenuse

When you know both legs and want the hypotenuse, plug in and take the square root at the end.

Worked example 1. A right triangle has legs of 3 and 4. Find the hypotenuse c.

$$ c^2 = 3^2 + 4^2 = 9 + 16 = 25 $$ $$ c = \sqrt{25} = 5 $$

So the hypotenuse is 5. This famous 3–4–5 triangle is the simplest example.

Worked example 2. Legs of 5 and 12.

$$ c^2 = 5^2 + 12^2 = 25 + 144 = 169 $$ $$ c = \sqrt{169} = 13 $$

The hypotenuse is 13.

Finding a missing leg

Sometimes you know the hypotenuse and one leg, and need the other leg. Now you subtract, because you are rearranging the formula. The balancing skill is the same one from Solving Linear Equations.

Rearrange a² + b² = c² to get:

$$ b^2 = c^2 - a^2 $$

Worked example 3. The hypotenuse is 13 and one leg is 5. Find the other leg.

$$ b^2 = 13^2 - 5^2 = 169 - 25 = 144 $$ $$ b = \sqrt{144} = 12 $$

The missing leg is 12. Notice we subtract here because the unknown is a leg, not the hypotenuse — getting this direction right is essential.

Pythagorean triples

Most triangles give a messy, irrational hypotenuse like √20 ≈ 4.47. But a few special sets of whole numbers fit perfectly. These are called Pythagorean triples.

Triple (a, b, c)	Check
3, 4, 5	9 + 16 = 25
5, 12, 13	25 + 144 = 169
8, 15, 17	64 + 225 = 289
6, 8, 10	36 + 64 = 100
7, 24, 25	49 + 576 = 625

Notice 6, 8, 10 is just 3, 4, 5 doubled. Multiply any triple by a whole number and you get another one. Recognising these saves time, because you can spot the answer without a calculator.

A real-world example

A ladder leans against a wall. Its foot is 6 m from the wall, and it reaches 8 m up the wall. How long is the ladder?

The wall and the ground meet at a right angle, so the ladder is the hypotenuse.

$$ c^2 = 6^2 + 8^2 = 36 + 64 = 100 \;\Rightarrow\; c = 10 \text{ m} $$

The ladder is 10 metres long — a distance you could not easily measure with a tape, found purely with arithmetic.

Using the converse

The theorem also runs backwards. If a triangle's sides satisfy a² + b² = c², then the triangle must have a right angle. Builders use this "3–4–5 trick" to check that corners are square: measure 3 units along one edge, 4 along the other, and if the diagonal is exactly 5, the corner is a true 90°.

Practice activity

Work these out. Leave answers as exact square roots where they are not whole numbers.

1. Legs 9 and 12 — find the hypotenuse.
2. Hypotenuse 25, one leg 7 — find the other leg.
3. Legs 1 and 1 — find the hypotenuse (this gives an irrational number).
4. Is a triangle with sides 8, 15, 17 right-angled? Show why.
5. A TV screen is 16 inches wide and 12 inches tall. How long is its diagonal?

Answers: 1) c = √225 = 15 2) b = √(625 − 49) = √576 = 24 3) c = √2 ≈ 1.41 4) 8² + 15² = 64 + 225 = 289 = 17², so yes 5) √(256 + 144) = √400 = 20 inches.

Why this matters

The Pythagorean theorem is the bridge between distance and right angles. It lets architects square buildings, lets surveyors measure across rivers, powers the distance formula in coordinate geometry, and underlies how 3D games calculate how far apart objects are. From one short equation — a² + b² = c² — flows an enormous amount of practical mathematics. Remember to check for the right angle, square carefully, and add to find the hypotenuse but subtract to find a leg, and you can unlock any right-triangle problem.