Right triangles everywhere
Whenever you know how far along and how far up, and want the straight-line distance, there is a right triangle in the question. The Pythagorean theorem finds its third side from the other two, and run backwards it checks whether a corner is really square. This page collects the places it turns up, so you can spot the triangle before anyone draws it.
Why does one rule about triangles matter so much?
Some distances are hard to measure directly: straight across a pond, corner to corner of a field, the length of a ramp that has not been built yet. The distance along and the distance up are usually easy, because they run beside walls, floors and fences. A right triangle joins the two, and the theorem turns them into the distance you wanted.
What the theorem really says is about area: the squares on the two legs of a right triangle together cover exactly as much as the square on the hypotenuse. It works only for a right angle, and it works in reverse — if the three sides fit, the angle is exactly 90°. That second use is how builders make a square corner without a giant set square.
The rule is far older than Pythagoras, the Greek thinker it is named after, who lived around 500 BCE. A Babylonian clay tablet from about 1800 BCE, now called Plimpton 322, lists whole-number sides of right triangles. Hundreds of proofs have been published since, one of them by James Garfield in 1876, five years before he became president of the United States.
Where it turns up
- Carpentry — framers check that a deck or wall corner is square by marking 3 units along one side and 4 along the other, then making the diagonal exactly 5 — often scaled up to 6, 8 and 10 feet
- TV screens — a 55-inch TV is measured corner to corner; its screen is about 48 inches wide and 27 inches tall, and the square root of 48² + 27² is just over 55
- Baseball — the bases are 90 feet apart and form a square, so the catcher's throw from home plate to second base is the square's diagonal, about 127 feet
- Ladders — a common safety rule puts the foot of a ladder 1 unit out from the wall for every 4 units up, so a ladder reaching 4 m up a wall needs to be √17 m long, about 4.1 m
- 1. The words, first
- 2. Finding a missing side
- 3. Checking a square corner
- 4. Screen sizes
- 5. Ramps and shortcuts
- 6. The diagonal of a box face
The words, first
The idea: Hypotenuse is the one to be sure of. Every use of the theorem starts by finding it.
| Word | What it means |
|---|---|
| Right triangle | A triangle with a 90° angle, a square corner. |
| Legs, a and b | The two sides that form the right angle. |
| Hypotenuse, c | The side across from the right angle. It is always the longest side. |
| Square, square root | 5² = 5 × 5 = 25, and √25 = 5. |
| Pythagorean theorem | For a right triangle, a² + b² = c². |
| Pythagorean triple | Three whole numbers that fit the theorem, such as 3, 4, 5 or 5, 12, 13. |
| Diagonal | A straight line from one corner of a shape to the opposite corner. |
| Rise, run | How far a ramp goes up, and how far it goes along the ground. |
Finding a missing side
The idea: To find the hypotenuse, add the squares of the legs, then take the square root. To find a leg, subtract the square of the known leg from the square of the hypotenuse, then take the square root.
hypotenuse: c² = a² + b² · leg: a² = c² − b²
Worked example: the hypotenuse. The legs are 7 cm and 24 cm. 7² + 24² = 49 + 576 = 625, and √625 = 25, so the hypotenuse is 25 cm.
Worked example: a leg. The hypotenuse is 17 m and one leg is 8 m. 17² − 8² = 289 − 64 = 225, and √225 = 15, so the other leg is 15 m.
When the root is not whole. Legs of 5 and 7: 5² + 7² = 25 + 49 = 74. √74 is between 8 and 9, because 74 is between 64 and 81, and closer to 9. A calculator gives 8.60. An answer under 7 or over 5 + 7 = 12 would be impossible: the hypotenuse is longer than either leg, and shorter than both together.
Find c first. Before any arithmetic, find the right angle and label the side across from it c. Getting that wrong loses more marks than any arithmetic slip.
Checking a square corner
The idea: Run the theorem backwards. If the squares of the two shorter sides add to the square of the longest, the angle between the shorter sides is exactly 90°. Builders use whole-number triples so there is nothing to work out on site.
| Triple | Check |
|---|---|
| 3, 4, 5 | 9 + 16 = 25 |
| 6, 8, 10 | 36 + 64 = 100 |
| 5, 12, 13 | 25 + 144 = 169 |
| 8, 15, 17 | 64 + 225 = 289 |
Multiples work too. 6, 8, 10 is 3, 4, 5 doubled. Scaling a triangle up changes its size but not its angles.
Worked example. You want a square corner on a garden bed. Mark 60 cm along one board and 80 cm along the other, then measure between the marks.
- What it should be: 60² + 80² = 3600 + 6400 = 10 000, and √10 000 = 100, so 100 cm.
- If it measures 104 cm, the corner is open wider than 90°. Push the boards in until it reads 100.
- If it measures 97 cm, the corner is squeezed smaller than 90°. Open it up.
Screen sizes
The idea: A screen is sold by its diagonal. The width and the height are the legs, and the size is the hypotenuse.
Worked example. A TV screen is 48 inches wide and 27 inches tall. What size is it sold as?
- Square and add: 48² + 27² = 2304 + 729 = 3033.
- Estimate the root: 55² = 3025 and 56² = 3136, so √3033 is just over 55.
- So it is sold as a 55-inch TV. A calculator gives 55.07.
Will it fit? A cabinet opening is 50 inches wide. The TV is called 55-inch, but the diagonal is not the width: the screen is 48 inches wide, so it fits if the frame adds less than 2 inches.
Tablets too. A tablet screen 8 inches by 6 inches is sold as 10-inch: 8² + 6² = 64 + 36 = 100, and √100 = 10.
Ramps and shortcuts
The idea: A ramp is the hypotenuse of its rise and its run. A shortcut across a rectangle is its diagonal, which is always shorter than going along two sides.
Worked example: a ramp. A wheelchair ramp has to rise 1 m to a doorway, and it starts 12 m out along the ground. How long is it?
- Square and add the rise and the run: 1² + 12² = 1 + 144 = 145.
- Take the square root. √145 is just over 12, since 12² = 144. A calculator gives 12.04, so the ramp is about 12.04 m long — only 4 cm longer than its run.
A gentle ramp is barely longer than the ground it covers, so the hard part of planning one is finding room for the run.
Worked example: a shortcut. A park is 80 m by 60 m. Along two sides is 80 + 60 = 140 m. Across, 80² + 60² = 6400 + 3600 = 10 000, and √10 000 = 100, so 100 m. The diagonal saves 140 − 100 = 40 m — which is why a worn path cuts across the corner of so many school fields.
The diagonal of a box face
The idea: Every face of a box is a rectangle, and a diagonal cuts it into two right triangles. The diagonal is the longest straight line that fits flat on that face.
Worked example. Will a 45 cm ruler lie flat on the bottom of a box that is 40 cm long and 30 cm wide?
- Along the length? No: 45 cm is longer than 40 cm.
- Corner to corner: 40² + 30² = 1600 + 900 = 2500, and √2500 = 50, so the diagonal is 50 cm.
- So yes, if it goes in diagonally, with 5 cm to spare.
Which face matters. If the box is 20 cm tall, its end face is 30 cm by 20 cm, and 900 + 400 = 1300, so that diagonal is √1300, about 36 cm: too short for the ruler. Even the long side face, 40 cm by 20 cm, falls just short: 1600 + 400 = 2000, and √2000 is about 44.7 cm.