Pythagorean Theorem

In any right triangle, the two legs squared add to the hypotenuse squared. It turns a right angle into arithmetic.

Draw a right triangle with legs 3 and 4. Build a square on each of the three sides.

Count the little tiles in each square: 9 on one leg, 16 on the other.

build a square on each side of a right triangle 9 16 25 the two small squares hold exactly as many tiles as the big one

Now count the square on the long slanted side. It holds exactly 25.

And 9 + 16 = 25. Not approximately — exactly, every time, for every right triangle.

Try it with legs 5 and 12: the squares hold 25 and 144, and the big one holds 169.

That is what a2 + b2 = c2 is saying. The letters look like algebra, but the statement is about areas of real squares — which is why the theorem is true rather than merely useful.

Squares built on the three sides of a 3-4-5 right triangle 9 16 25 3 4 5 a² + b² = c² 9 + 16 = 25 the two smaller squares, together, have exactly the area of the big one c is opposite the right angle and always the longest side

The statement, and what the letters mean

a2 + b2 = c2, where a and b are the legs and c is the hypotenuse.

c is always the side opposite the right angle, and always the longest. Getting this wrong is the main source of errors.

The theorem only applies to right triangles. No right angle, no formula.

Finding the hypotenuse

Legs 6 and 8: 62 + 82 = 36 + 64 = 100, so c = √100 = 10.

Square both legs, add, then take the square root. Three steps, in that order.

Sanity check

The hypotenuse must be longer than either leg, but shorter than their sum. For legs 6 and 8, expect between 8 and 14. A "hypotenuse" of 5 is impossible.

Finding a leg — subtract, do not add

If you know the hypotenuse, rearrange: a2 = c2 - b2.

Hypotenuse 13, one leg 5: a2 = 169 - 25 = 144, so a = 12.

Adding here instead of subtracting is the most common slip in this unit. Identify the hypotenuse first, and the direction follows.

A right triangle with hypotenuse 13 and leg 5: the unknown leg comes from 169 minus 25 spot the hypotenuse first: it is across from the right angle 5 a = ? c = 13 c² = 169 25a² = 144 b² a² = c² − b² = 169 − 25 = 144 a = √144 = 12 adding gives √194 ≈ 13.9 — longer than the hypotenuse: impossible

Triples worth memorising

3-4-5, 5-12-13, 8-15-17, 7-24-25.

Multiples count too: 6-8-10 and 9-12-15 are both just 3-4-5 scaled.

Spotting one saves all the arithmetic — and catching that 10-24-26 is 2 × (5-12-13) is faster than squaring anything.

The converse: is it a right triangle?

Test the three sides. If a2 + b2 = c2 with c the longest, the triangle is right-angled.

9, 12, 15: 81 + 144 = 225 = 152 ✓ — right-angled.

4, 5, 8: 16 + 25 = 41 but 64 — not right-angled. Since 41 < 64, the angle opposite 8 is obtuse.

Three triangles tested with the converse: a² + b² greater than, equal to, or less than c² means acute, right or obtuse acutesides 6, 7, 876c = 836 + 49 = 8585 > 64 → acuterightsides 9, 12, 15129c = 1581 + 144 = 225225 = 15² → rightobtusesides 4, 5, 854c = 816 + 25 = 4141 < 64 → obtuse

Where it actually gets used

The diagonal of a rectangle, the height of a ladder against a wall, the straight-line distance between two points on a grid.

On coordinates, the horizontal and vertical gaps are the legs, and the distance between the points is the hypotenuse.

Worked examples

Find the hypotenuse

A right triangle has legs 9 and 12. Find the hypotenuse.

  1. 92 + 122 = 81 + 144 = 225 Square each leg and add.
  2. c = √225 = 15 Take the root of the total.
  3. Check: 15 > 12 and 15 < 21 ✓ Between the longest leg and the sum — plausible. Also 3 × (3-4-5).

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Find a leg

A ladder 17 ft long leans against a wall, its foot 8 ft out. How high up the wall does it reach?

  1. The ladder is the hypotenuse: c = 17. It is opposite the right angle between wall and ground.
  2. a2 = 172 - 82 = 289 - 64 = 225 Subtract, because you are finding a leg.
  3. a = √225 = 15 An 8-15-17 triple.

15 feet up the wall.

Distance on a grid

Find the distance between (2, 3) and (7, 15).

  1. Horizontal gap: 7 - 2 = 5 One leg of a right triangle.
  2. Vertical gap: 15 - 3 = 12 The other leg.
  3. 52 + 122 = 25 + 144 = 169 Apply the theorem to the two gaps.
  4. √169 = 13 A 5-12-13 triple.

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Words to know

Leg
One of the two sides forming the right angle.
Hypotenuse
The side opposite the right angle — always the longest.
Pythagorean triple
Three whole numbers that fit: 3-4-5, 5-12-13, 8-15-17, 7-24-25.
Converse
If a2 + b2 = c2, then the triangle must be right-angled.

Mistakes to watch for

Treating a leg as the hypotenuse. c is always opposite the right angle and always the longest side. Put the longest length in the wrong place and the arithmetic gives a leg that is longer than the hypotenuse — impossible.

Adding when you should subtract. If you know the hypotenuse and one leg, the missing leg comes from c2 - a2. Adding gives an answer bigger than the hypotenuse, which the picture immediately disproves.

Forgetting the final square root. 9 + 16 = 25 is c2, not c. Stopping there gives an answer five times too large and still looks plausible.

Using it on a triangle with no right angle. The whole proof rests on that corner. Without a right angle the relationship simply does not hold.

The theorem has more known proofs than any other in mathematics — one 1927 collection gathered 370 of them.

One was published in 1876 by James A. Garfield, five years before he became President of the United States. His proof uses a trapezium and is genuinely elegant.

Babylonian tablets list Pythagorean triples from around 1800 BC, a thousand years before Pythagoras. What the Greeks added was the proof that it always holds.

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