In this article
- What the formula actually says
- Three steps to the missing side
- Pythagorean triples: when all three sides are whole numbers
- The carpenter's trick: squaring a corner with a rope
- Diagonal of a rectangle: from rooms to screens
- Ladders: how far from the wall?
- Walking blocks versus flying straight
- Is this triangle a right triangle?
- Quick values worth remembering
In a triangle with one right angle (90 degrees), the square of the longest side equals the sum of the squares of the other two. Written as a formula: a² + b² = c², where c is the longest side, called the hypotenuse. It works for every right triangle, from a rooftop to a phone screen.
What the formula actually says
Draw a square on each side of a right triangle. The area of the big square on the hypotenuse is exactly the sum of the areas of the two smaller squares. For the 3-4-5 triangle that is 9 + 16 = 25. Nothing is approximate here; it is an exact fact of flat geometry.
Both bars are the same length: 25. That equality is the whole theorem, drawn as bars.
Three steps to the missing side
- 1Square each leg5² = 25 and 12² = 144.
- 2Add them25 + 144 = 169.
- 3Take the square root√169 = 13. The hypotenuse is 13.
To find a leg instead, subtract: if c = 13 and a = 5, then b² = 169 − 25 = 144, so b = 12.
Pythagorean triples: when all three sides are whole numbers
Most right triangles have a messy hypotenuse. Legs of 1 and 1 give √2 = 1.4142… forever. A few special sets of whole numbers fit perfectly. They are called Pythagorean triples, and every multiple of a triple is another triple.
| Triple | Check | Handy multiples |
|---|---|---|
| 3, 4, 5 | 9 + 16 = 25 | 6-8-10, 9-12-15, 30-40-50 |
| 5, 12, 13 | 25 + 144 = 169 | 10-24-26, 15-36-39 |
| 8, 15, 17 | 64 + 225 = 289 | 16-30-34 |
| 7, 24, 25 | 49 + 576 = 625 | 14-48-50 |
| 20, 21, 29 | 400 + 441 = 841 | 40-42-58 |
| 9, 40, 41 | 81 + 1,600 = 1,681 | 18-80-82 |
If your two legs and hypotenuse match one of these (or a multiple), you can skip the square root completely.
The carpenter's trick: squaring a corner with a rope
Builders marking a foundation, gardeners laying a bed and tailors cutting cloth all need a true 90-degree corner. Measure 3 units along one edge and 4 units along the other. If the straight distance between those two marks is exactly 5 units, the corner is square. For big jobs use 6-8-10 or 9-12-15 so small errors matter less.
- Looks fine from one side
- A 2° error becomes about 5 cm over a 3 m wall
- Doors and tiles then refuse to line up
- Mark 3 m and 4 m from the corner
- The diagonal between marks must be 5 m
- Adjust until the tape agrees
The check works because a triangle with sides 3, 4, 5 can only be built with a right angle between the short sides.
Diagonal of a rectangle: from rooms to screens
A rectangle cut along its diagonal makes two right triangles, so width² + height² = diagonal². A 12 ft by 16 ft room has a diagonal of √(144 + 256) = √400 = 20 ft, another 3-4-5 in disguise. Screens are sold by their diagonal, which is why the number on the box is bigger than the width.
For 16:9, width = diagonal × 16 ÷ √337 and height = diagonal × 9 ÷ √337. A 55-inch TV is about 47.9 in wide and 27.0 in tall.
Doubling the diagonal does not double the screen: it makes the area four times larger, because area grows with the square.
Ladders: how far from the wall?
A common safety rule is the 4-to-1 rule: for every 4 units of height you climb, put the foot of the ladder 1 unit away from the wall. That makes an angle of about 76 degrees. Pythagoras gives the numbers: with a 4 m ladder, the base sits about 0.97 m from the wall and the top reaches about 3.88 m.
| Ladder length | Foot from wall | Height reached | Angle |
|---|---|---|---|
| 3 m | 0.73 m | 2.91 m | about 76° |
| 4 m | 0.97 m | 3.88 m | about 76° |
| 5 m | 1.21 m | 4.85 m | about 76° |
| 6 m | 1.46 m | 5.82 m | about 76° |
Every row is the same triangle scaled up: base = length ÷ √17 and height = 4 × length ÷ √17. Follow the maker's label on your ladder first; the 4-to-1 rule is a guide, not a replacement for it.
Walking blocks versus flying straight
On a street grid you cannot cut the corner. Going 3 blocks east and 4 blocks north means walking 7 blocks, but a bird flies 5. The gap between the two is the saving from the diagonal.
A drone or a cable can take the straight route. A taxi cannot. The straight line is always the shortest possible path.
Is this triangle a right triangle?
The theorem also works backwards. Take the longest side as c and compare c² with a² + b². The result tells you the kind of triangle.
For 6, 7, 8: 36 + 49 = 85, which is more than 64, so the triangle is acute.
Quick values worth remembering
| Legs | Hypotenuse | Note |
|---|---|---|
| 1, 1 | √2 ≈ 1.4142 | Diagonal of a unit square |
| 1, 2 | √5 ≈ 2.2361 | Not a whole number |
| 2, 3 | √13 ≈ 3.6056 | Not a whole number |
| 5, 5 | ≈ 7.0711 | 5 × √2 |
| 10, 10 | ≈ 14.1421 | 10 × √2 |
Any square's diagonal is its side times about 1.4142.
Questions people ask
What is the Pythagorean theorem?
In a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides: a² + b² = c². It lets you find any one side when you know the other two.
What is the hypotenuse?
The hypotenuse is the side opposite the right angle. It is always the longest side of a right triangle.
What is a 3-4-5 triangle used for?
It is a quick way to build a perfect right angle. Measure 3 along one edge and 4 along the other; if the distance between the marks is 5, the corner is square.
Does the Pythagorean theorem work for any triangle?
No. It holds only for right triangles. For other triangles you need the law of cosines, which reduces to Pythagoras when the angle is 90 degrees.
How do I find a missing leg?
Subtract instead of adding: b = √(c² − a²). For c = 13 and a = 5, b = √(169 − 25) = 12.
Why is a 55-inch TV not 55 inches wide?
Screen size is measured along the diagonal. For a 16:9 screen, the width is about 87 percent of the diagonal, so a 55-inch TV is roughly 47.9 inches wide.