Square Diagonal Calculator: Quick & Accurate Tool

Understanding the Diagonal of a Square

A diagonal is a straight line connecting two opposite corners (vertices) of a polygon. Every square possesses two diagonals that share these key properties:

• They are identical in length.
• They intersect at right angles while bisecting each other.
• They bisect the square's interior angles.

Each diagonal splits the square into two congruent isosceles right triangles. The legs of these triangles are the square's sides, and the hypotenuse is the diagonal.

The Essential Diagonal Formula

To determine a square's diagonal length (d), multiply the length of one side (s) by the square root of 2.

d = s × √2

For a practical example, consider a square with a side length of 5 inches. Applying the formula gives:

d = 5 × √2 ≈ 7.071 inches

Derivation of the Formula

This formula originates from the Pythagorean theorem. The diagonal forms the hypotenuse (d) of a right triangle whose legs are both equal to the side length (s).

s² + s² = d²
2s² = d²
d = √(2s²) = s√2

Alternative Calculation Methods

If the side length is unknown, you can calculate the diagonal using other known values:

• Using Area (A): d = √(2 × A)
• Using Perimeter (P): d = (P / 4) × √2

Frequently Asked Questions

About the Pythagorean Theorem

The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. This principle is fundamental to deriving the square diagonal formula. Back to derivation.