Diagonal Of Rectangle Calculator

How to Calculate the Diagonal of a Rectangle

The diagonal of a rectangle is the straight-line segment connecting two opposite (non-adjacent) corners. Because every interior angle of a rectangle measures exactly 90°, any diagonal divides the shape into two congruent right triangles. This geometric property makes the Pythagorean theorem the natural and precise tool for computing diagonal length.

The Formula

The standard diagonal of a rectangle formula is:

d = √(l² + w²)

• d — diagonal length, expressed in the same unit as l and w
• l — length of the rectangle (the longer side)
• w — width of the rectangle (the shorter side)

Derivation from the Pythagorean Theorem

Drawing a diagonal inside a rectangle with length l and width w produces a right triangle whose two legs equal l and w and whose hypotenuse is the diagonal d. Substituting into the Pythagorean theorem (a² + b² = c²) gives d² = l² + w², and taking the positive square root of both sides yields d = √(l² + w²). This derivation is grounded in classical Euclidean geometry, as presented in Northern Kentucky University's Transition to College Mathematics and supported by mathematical reference material published by NIST in its Formulas and Tables series.

Step-by-Step Calculation

1. Identify the length (l): Measure or record the longer side of the rectangle.
2. Identify the width (w): Measure or record the shorter side of the rectangle.
3. Square both values: Compute l² and w² separately.
4. Add the squares: Find the sum l² + w².
5. Extract the square root: Calculate √(l² + w²) to obtain the diagonal d.

Worked Examples

Example 1 — Classic 6-8-10 Pythagorean Triple

A rectangle has length 8 m and width 6 m. Applying the formula: d = √(8² + 6²) = √(64 + 36) = √100 = 10 m. Because 6, 8, and 10 form a Pythagorean triple, the result is a whole number — a useful check when precise integer answers are expected.

Example 2 — US Letter Paper

A standard letter-size sheet measures 11 in × 8.5 in. d = √(11² + 8.5²) = √(121 + 72.25) = √193.25 ≈ 13.90 inches. This diagonal is relevant when designing custom frames or fitting printed content into angled layouts.

Example 3 — Television Screen

A 4K TV panel measures 47.9 in wide and 27.0 in tall. d = √(47.9² + 27.0²) = √(2294.41 + 729.00) = √3023.41 ≈ 54.99 in. Manufacturers round this to the advertised 55-inch screen size, confirming that all TV sizes represent diagonal measurements of the rectangular display panel.

Real-World Applications

• Consumer electronics: Every television, monitor, tablet, and smartphone screen size is defined as the diagonal of its rectangular display.
• Construction and carpentry: Builders verify that corners are perfectly square by measuring both diagonals; equal diagonals confirm a true rectangle.
• Interior design: Determining whether furniture or appliances can pass diagonally through a doorway requires the diagonal formula.
• Photography: Camera sensor crop factors derive from the ratio of the full-frame sensor diagonal (43.3 mm for 35 mm film) to the crop sensor's diagonal.
• Flooring and tiling: Diagonal tile layouts and border spacing calculations depend on accurate diagonal measurements of individual tiles.

Units and the Special Case of a Square

Both l and w must share the same unit before applying the formula; the result d will also be in that unit. For a square where l = w, the formula simplifies to d = l√2 ≈ 1.4142 × l. A 10 ft × 10 ft room, for instance, has a diagonal of approximately 14.14 ft.

Verification and Reasonableness Checks

After computing the diagonal, verify that your answer makes sense: the diagonal must always be longer than both the length and width, but shorter than their sum. For instance, in a 5 × 12 rectangle, the diagonal of 13 is indeed greater than both 5 and 12, yet less than 5 + 12 = 17. This simple sanity check catches calculation errors before they propagate downstream.