# Pythagorean theorem: the 3-4-5 trick, ladders, TV sizes and how to check a right angle

> a² + b² = c² is the most useful line in school geometry. This guide shows what it really says, how carpenters use a 3-4-5 rope to square a room, why a '55-inch' TV is not 55 inches wide and how to test whether any triangle has a right angle.

Published 2026-10-09 by Toolcloak. 4 minute read. Original: https://toolcloak.pages.dev/articles/pythagorean-theorem-3-4-5-triples-and-how-to-check-a-right-angle
Related free tool: [Pythagorean Theorem Calculator](https://toolcloak.pages.dev/tools/pythagorean-theorem-calculator)

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.

> **In short**
> - c is always the side **opposite the right angle** and always the longest. The two shorter sides are called legs.
> - To find c: add the squares of the legs, then take the square root. To find a leg: subtract, then take the square root.
> - The **3-4-5** triangle (and its multiples 6-8-10, 9-12-15) gives a right angle with nothing but a tape measure.
> - The rule only works for right triangles. If a² + b² is not equal to c², the angle is not 90 degrees.

## 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.

**The squares on the sides of a 3-4-5 triangle (areas)**

|  | Square on leg a (3 × 3) | Square on leg b (4 × 4) |
| --- | --- | --- |
| Two legs together | 9 | 16 |
| Hypotenuse (5 × 5) | 25 |

*Both bars are the same length: 25. That equality is the whole theorem, drawn as bars.*

## Three steps to the missing side

**Finding the hypotenuse of a triangle with legs 5 and 12**

1. **Square each leg.** 5² = 25 and 12² = 144.
2. **Add them.** 25 + 144 = 169.
3. **Take 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.*

Try it: [Pythagorean Theorem Calculator](https://toolcloak.pages.dev/tools/pythagorean-theorem-calculator). Enter any two sides and get the third, with the working shown step by step.

## 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.

**Squaring a corner: by eye or with 3-4-5**

| By eye | 3-4-5 check |
| --- | --- |
| Maybe 88° | Exact 90° |
| Looks fine from one side | Mark 3 m and 4 m from the corner |
| A 2° error becomes about 5 cm over a 3 m wall | The diagonal between marks must be 5 m |
| Doors and tiles then refuse to line up | 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.

**Actual width of a 16:9 TV for each advertised diagonal (inches)**

|  | Value |
| --- | --- |
| 32" | 27.9 in |
| 43" | 37.5 in |
| 50" | 43.6 in |
| 55" | 47.9 in |
| 65" | 56.7 in |
| 75" | 65.4 in |
| 85" | 74.1 in |

*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.*

**A 55-inch 16:9 TV in numbers**

|  | Value | Note |
| --- | --- | --- |
| Diagonal | 55 in | The number on the box |
| Width | 47.9 in | About 122 cm |
| Height | 27.0 in | About 68 cm |
| Screen area | 1,293 sq in | Roughly 1.2 sq m |

*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.

**Distance for a 3-east, 4-north trip**

|  | Blocks |
| --- | --- |
| Along the streets | 7 blocks |
| Straight line | 5 blocks |

*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.

**Compare c² with a² + b²**

| Box | What it means |
| --- | --- |
| c² = a² + b² | Exactly equal: a right triangle (90°). Example: 6, 8, 10. |
| c² < a² + b² | Smaller: all angles are sharp, an acute triangle. Example: 6, 7, 8. |
| c² > a² + b² | Larger: one angle is wider than 90°, an obtuse triangle. Example: 4, 5, 8. |
| Not a triangle | If the two short sides add up to less than the long one, no triangle exists. Example: 2, 3, 9. |

*For 6, 7, 8: 36 + 49 = 85, which is more than 64, so the triangle is acute.*

> **Common mistakes:** Using a leg as c, forgetting to take the square root at the end, adding the sides instead of their squares (3 + 4 is 7, not 5), and applying the rule to a triangle that has no right angle. Always check that c is the longest side.

## 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.*

## Frequently asked questions

### 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.