Diagonal of Rectangle Calculator

Calculate the diagonal of a rectangle from its dimensions or area with step-by-step solutions.

Choose Calculation Method

Result

Diagonal
10
units
Length 8
Width 6
Diagonal 10
Area 48
Perimeter 28

Step-by-Step Solution

d = sqrt(l^2 + w^2)

Understanding Rectangle Diagonals

The diagonal of a rectangle is the straight line segment connecting two opposite vertices (corners). Every rectangle has exactly two diagonals, and they are always equal in length. The diagonal divides the rectangle into two congruent right triangles, which is the key to the formula used to calculate it.

The relationship between the diagonal and the sides of a rectangle comes directly from the Pythagorean theorem. Since the diagonal forms the hypotenuse of a right triangle with the length and width as the two legs, we get the formula d = sqrt(l2 + w2).

Diagonal Formulas

From Length & Width

Using the Pythagorean theorem directly with both sides of the rectangle.

d = sqrt(l^2 + w^2)

From Area & One Side

First find the other side from the area, then apply the Pythagorean theorem.

w = A/l, then d = sqrt(l^2 + w^2)

Square (Special Case)

When length equals width, the diagonal formula simplifies.

d = s * sqrt(2)

Diagonal Angle

The angle the diagonal makes with the length can be found using trigonometry.

theta = arctan(w / l)

The Pythagorean Connection

The diagonal of a rectangle is a direct application of the Pythagorean theorem, one of the most fundamental results in geometry. The theorem states that in a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides. Since the diagonal creates a right triangle with the length and width, d2 = l2 + w2.

This is why the classic 3-4-5 right triangle gives us a rectangle with length 4, width 3, and diagonal 5. Other common Pythagorean triples like 5-12-13 and 8-15-17 also correspond to rectangles with integer diagonals.

Properties of Rectangle Diagonals

  • Both diagonals of a rectangle are equal in length.
  • The diagonals bisect each other (divide each other into two equal parts).
  • Each diagonal divides the rectangle into two congruent right triangles.
  • The diagonals of a rectangle do NOT bisect the corner angles (unlike a square).
  • The diagonal is always longer than either side of the rectangle.

Practical Applications

Rectangle diagonal calculations are used in construction to verify that corners are square (the 3-4-5 method), in TV and monitor sizing (screen size is measured diagonally), in architecture for structural bracing, and in everyday tasks like fitting furniture through doorways. Understanding diagonals is essential for anyone working with rectangular spaces and objects.

Worked Example

A room measures 12 feet by 16 feet. What is the diagonal distance across the room? Using the formula: d = sqrt(122 + 162) = sqrt(144 + 256) = sqrt(400) = 20 feet. This is a 3-4-5 triple scaled by 4, so the diagonal is exactly 20 feet.