The diagonal of a rectangle
A diagonal cuts a rectangle into two right triangles, with the length and width as the legs and the diagonal as the hypotenuse. The Pythagorean theorem therefore gives it directly: diagonal = √(length² + width²). You need it to check that a frame or foundation is square, to size a TV or a screen, or to know whether a piece of furniture fits through a doorway.
The formulas
- Diagonal d = √(l² + w²)
- Square: d = side × √2 ≈ side × 1.414
- Angle between the diagonal and the length = arctan(w ÷ l)
- Area = l × w and perimeter = 2 × (l + w)
- The four corners lie on a circle whose diameter is the diagonal.
Worked example: 8 × 6
- Diagonal: √(64 + 36) = √100 = 10.
- Angle with the long side: arctan(6 ÷ 8) = 36.87°.
- Area: 48; perimeter: 28; aspect ratio: 8 ÷ 6 = 1.333, which is 4:3.
The 3-4-5 rule for squaring corners
Builders use this exact triangle to lay out right angles. Measure 3 units along one side and 4 along the other: when the corner is square, the distance between the two marks is exactly 5. For bigger layouts, use multiples such as 6-8-10 or 9-12-15 feet. On a finished rectangle, the two diagonals must be equal; if they differ, the shape is a parallelogram, not a rectangle.
Diagonal as the length changes (width 6)
| Length | Diagonal | Angle |
|---|---|---|
| 3 | 6.71 | 63.43° |
| 4 | 7.21 | 56.31° |
| 6 | 8.49 | 45.00° |
| 8 | 10.00 | 36.87° |
| 12 | 13.42 | 26.57° |
| 16 | 17.09 | 20.56° |
Everyday uses
- Screens: a 16:9 TV with a 55-inch diagonal is 47.9 inches wide and 27 inches tall. The TV size calculator does this conversion for any screen.
- Moving furniture: a door 36 inches wide and 80 inches tall has a diagonal of 87.7 inches, the longest flat panel that can pass when tilted.
- Tiles and sheets: cutting a square tile corner to corner gives two triangles whose long side is the side × 1.414.
- Paper sizes: A-series paper (A4, A3) has an aspect ratio of √2, so halving it keeps the same shape.