Diagonal of a Rectangle
Calculate the diagonal length of a rectangle using the Pythagorean theorem.
Used in construction, screens, and layout design.
The Formula
d = √(l² + w²)
The diagonal of a rectangle is the straight line connecting opposite corners. It forms a right triangle with the length and width, so the Pythagorean theorem applies.
Variables
| Symbol | Meaning |
|---|---|
| d | Length of the diagonal |
| l | Length of the rectangle |
| w | Width of the rectangle |
Example 1
A TV screen is 48 inches wide and 27 inches tall. What is the diagonal?
d = √(48² + 27²) = √(2304 + 729)
d = √3033
d ≈ 55.1 inches (a "55-inch" TV)
Example 2
A room is 5 m long and 4 m wide. What is the diagonal distance?
d = √(5² + 4²) = √(25 + 16) = √41
d ≈ 6.40 m
When to Use It
Use the diagonal formula when:
- Measuring TV, monitor, or phone screen sizes
- Checking if a rectangular space is square (diagonals should be equal)
- Calculating the longest item that fits in a rectangular box
- Laying out foundations and framing in construction
Key Notes
- A rectangle always has two diagonals of equal length — if measured diagonals differ, the shape is a parallelogram (sides parallel but not 90° corners), not a true rectangle
- TV and monitor sizes are quoted as the diagonal — a "55-inch" display is about 48 in wide × 27 in tall at 16:9 aspect ratio
- For a square (l = w = s), the diagonal simplifies to d = s√2 ≈ 1.414s — about 41% longer than each side
- The 3-4-5 right triangle is a special rectangle diagonal: l=3, w=4 gives d=5 exactly
Key Notes
- Formula: d = √(l² + w²): Direct application of the Pythagorean theorem to the right triangle formed by the two sides and the diagonal. The diagonal is always longer than either side and shorter than their sum.
- Square special case: d = s√2: For a square with side s, both legs are equal and d = √(s² + s²) = s√2 ≈ 1.414s. The ratio of diagonal to side is always √2, which is why the Pythagoreans found the √2 irrational so unsettling — it appears in the most ordinary geometric figure.
- Converse check for squareness: A rectangle is a parallelogram with equal diagonals. A square is a rectangle with perpendicular diagonals. Carpenters check whether a rectangular frame is truly square by measuring both diagonals — they should be equal and cross at right angles.
- 3D extension: space diagonal = √(l² + w² + h²): The longest diagonal in a rectangular prism (box) runs from one corner to the opposite corner. Each successive dimension adds another squared term under the radical. In n dimensions: d = √(x₁² + x₂² + … + xₙ²).
- Applications: Diagonal formulas are used to determine monitor and TV screen sizes (quoted as diagonal), check whether a foundation or frame is square during construction, calculate the minimum aperture for fitting a rectangle through a door, and find vector magnitudes in 2D and 3D space.