Pythagoras' Theorem

1 · Right-angled triangles and the hypotenuse

A right angle is a quarter turn, exactly 90°. It's the corner of a book, a door frame or a sheet of paper. On diagrams we mark it with a small square in the corner instead of an arc.

A right-angled triangle is a triangle with one right angle. Its three sides have names:

• The two sides that make the right angle (they touch the little square) are the shorter sides, sometimes called the legs. We'll call them a and b. It doesn't matter which is which.
• The side opposite the right angle, the one that doesn't touch the little square, is the hypotenuse. We always call it c. It is always the longest side of the triangle.

Triangles are not always drawn sitting flat on their bottom edge, so the hypotenuse can point in any direction. The trick is the same every time: find the little square, then look straight across from it. Try it. Tap the side you think is the hypotenuse (or use the buttons).

2 · Squares on the sides

Here's the big discovery, shown with numbers before any formula. Take a right-angled triangle whose shorter sides are 3 and 4 squares long, and whose hypotenuse is 5. Now build a square on each side, so each side becomes the edge of a square:

• the square on the side of length 3 has area 3 × 3 = 9 little squares;
• the square on the side of length 4 has area 4 × 4 = 16;
• the square on the hypotenuse has area 5 × 5 = 25.

And 9 + 16 = 25. The two small squares exactly make the big one. (If squaring or square roots feel rusty, have a look at the powers and roots page first: this whole topic uses them.)

Is that a coincidence? Drag the three corners (they snap to the grid) and watch the areas. Keep the angle at 90° and the two small areas always add to the big one, even when the triangle is tilted. Pull the angle away from 90° and the sum breaks.

So, for every right-angled triangle, and only for right-angled triangles:

In words: the square on the hypotenuse equals the sum of the squares on the other two sides. This is Pythagoras' theorem. A theorem is a fact that has been proved to be true every time, not just checked on a few examples. It's named after Pythagoras, a Greek thinker from about 2500 years ago, although Babylonian and Chinese mathematicians knew the pattern even earlier.

3 · Why it's always true: a proof you can watch

Checking examples is convincing, but a proof shows it must work for every right-angled triangle. Here's one of the oldest and neatest. Take four identical copies of a right-angled triangle with shorter sides a, b and hypotenuse c, and a big square frame with sides of length a + b. Step through it.

The trick is that nothing was added or taken away. The same frame holds the same four triangles both times, so the empty space must be the same size both times. In the first picture the empty space is one tilted square, c². In the second it is two squares, a² + b². They must be equal. Change the sliders: it works whatever a and b are.

4 · Finding the hypotenuse

Now use it. If you know the two shorter sides, you can find the hypotenuse in four moves: square both, add, then square root (because the sum is c², and you want c).

Example with numbers: shorter sides 6 cm and 8 cm. 6² = 36 and 8² = 64. 36 + 64 = 100, so c² = 100. The square root of 100 is 10, so c = 10 cm. Use the sliders and watch every line of the working.

Rounding sensibly

Most answers are not whole numbers. With shorter sides 5 and 7, c² = 25 + 49 = 74, and √74 = 8.602325… on a calculator. The digits go on forever, so you round:

• Keep the full calculator value until the very last line. Round only the final answer. Rounding halfway through makes the answer drift.
• If the question says how to round, do that. "To 1 d.p." (one decimal place) means one digit after the point: 8.602… → 8.6.
• If it doesn't say, 1 d.p. is usually sensible for lengths in cm or m. Don't give 8 decimal places for a ladder: nobody can measure that accurately.
• Optional, for an exact answer: you can leave it as √74. A square root left like this is called a surd. Some questions ask for an "exact answer"; that means leave the √ in.

5 · Finding a shorter side: subtract!

Sometimes you know the hypotenuse and one shorter side, and need the other shorter side. The formula is the same, but you have to undo it. Start with numbers: the hypotenuse is 13 and one side is 5.

• The big square is 13² = 169. One small square is 5² = 25.
• The two small squares add up to the big one, so the missing small square is what's left: 169 − 25 = 144.
• The missing side is √144 = 12.

In symbols: from a² + b² = c², take b² from both sides to get a² = c² − b². So the rule of thumb is:

This solver does either kind. First decide whether the missing side is the hypotenuse, then enter what you know.

6 · Pythagorean triples and checking for a right angle

A few right-angled triangles have three whole-number sides. Such a set of three whole numbers is a Pythagorean triple. The famous one is 3, 4, 5: 9 + 16 = 25. Others worth knowing: 5, 12, 13 (25 + 144 = 169), 8, 15, 17 (64 + 225 = 289) and 7, 24, 25 (49 + 576 = 625).

Multiply every number in a triple by the same amount and you get another triple, because you've just made a bigger copy of the same shape: 3, 4, 5 × 2 gives 6, 8, 10; × 3 gives 9, 12, 15. Spotting a triple can save you the whole calculation.

Pythagoras' theorem also works backwards (this is called the converse): if the three sides of a triangle fit a² + b² = c², then the triangle must have a right angle, opposite the longest side. So you can test whether a triangle is right-angled using only its lengths:

1. Pick out the longest side. That's the only side that could be c.
2. Square the two shorter sides and add them.
3. Square the longest side. Same answer? Right-angled. Different? Not right-angled.

Builders use this with a tape measure to check corners are square: mark 3 m along one wall and 4 m along the other, and the diagonal should be exactly 5 m. Type any three lengths, in any order, or tap a set to try.

Notice what happens when it fails. If a² + b² is bigger than c², the angle opposite c is less than 90° (acute): the corner is pinched closed. If a² + b² is smaller than c², that angle is more than 90° (obtuse): the corner has been pushed open. You saw both in section 2.

7 · Real-world problems

In real problems nobody hands you a labelled triangle. You have to spot it. Look for a right angle hiding in the situation (a wall meets the ground, the corner of a rectangle, "north" and "east"), sketch the triangle, label the hypotenuse c, then decide: add or subtract?

The ladder against a wall

Walls stand at right angles to the ground. A ladder leaning on a wall makes a right-angled triangle: the wall and the ground are the shorter sides, and the ladder itself is the hypotenuse (it's opposite the corner where the wall meets the ground). To find how high it reaches, you know the hypotenuse, so subtract.

Diagonals of rectangles and TV screens

Every corner of a rectangle is a right angle, so a diagonal cuts it into two right-angled triangles. The width and height are the shorter sides and the diagonal is the hypotenuse: add. TV and phone screens are sold by their diagonal, measured in inches. A "55-inch TV" is about 48 inches wide and 27 inches tall.

8 · Distance between two points

How far apart are the points (−2, −3) and (4, 5)? You can't count squares along a slanting line, but you can draw a right-angled triangle under it: go across from one point and up to the other. The across part and the up part are the shorter sides, and they are easy to count. The straight-line distance is the hypotenuse.

• Across: from x = −2 to x = 4 is 4 − (−2) = 6.
• Up: from y = −3 to y = 5 is 5 − (−3) = 8.
• Distance² = 6² + 8² = 36 + 64 = 100, so the distance is √100 = 10.

Drag either point. Negative coordinates are fine: a distance across or up is always positive, and squaring would make a negative positive anyway.

Worked examples

Quick check

What you've learned

• A right-angled triangle has one 90° angle. The side opposite it is the hypotenuse, c, and it is always the longest side. The other two are the shorter sides a and b. Label c first.
• Pythagoras' theorem: a² + b² = c². The squares on the two shorter sides add up to the square on the hypotenuse. The rearrangement proof shows why: the same four triangles in the same frame leave c² uncovered one way and a² + b² the other.
• Finding the hypotenuse: square, add, square root.
• Finding a shorter side: square, subtract (biggest minus smaller), square root. The answer must come out shorter than the hypotenuse.
• Don't forget the final square root, and don't add before squaring: (a + b)² is not a² + b².
• Round only at the end, usually to 1 d.p.; an exact answer can be left as a surd like √74.
• Pythagorean triples such as 3, 4, 5 and 5, 12, 13 (and their multiples) are whole-number right-angled triangles.
• The converse: if a² + b² = c² for the sides, the triangle is right-angled. If not, it isn't (acute if a² + b² is bigger, obtuse if smaller).
• Real problems hide a right angle: ladders and walls, rectangle diagonals, and the across-and-up triangle between two coordinate points.