Diagonal of Square Formula — d = s√2
Diagonal of Square Formula — d = s√2
TL;DR
The diagonal of square formula is d = s√2, where s is the side length — the diagonal of any square is always its side times the square root of 2. This article derives the formula from the Pythagorean theorem, shows how to find the diagonal from the side, the area, or the perimeter, works six examples from a one-step computation to a real-world problem, and clears up the mistakes that cost the most marks.
The diagonal of square formula sizes TV and monitor screens, picture frames, floor tiles, and any layout where the corner-to-corner span decides whether something fits.
The Formula
For a square with side length s, the diagonal d is:
[ d = s \sqrt{2} ]
Each symbol points to the figure above. s is the side of the square — and because all four sides are equal, you only need one. d is the diagonal — the line joining opposite corners, which is always longer than a side by the fixed factor ( \sqrt{2} \approx 1.414 ).
If you do not know the side directly, two related forms reach the diagonal from other measurements:
- From the area: since ( s = \sqrt{A} ), the diagonal is ( d = \sqrt{2A} ).
- From the perimeter: since ( s = \frac{P}{4} ), the diagonal is ( d = \frac{P}{4} \sqrt{2} ).
How the Diagonal of Square Formula Is Derived
The factor of ( \sqrt{2} ) comes straight out of the Pythagorean theorem. A diagonal cuts the square into two identical right triangles. In each triangle, the two legs are sides of the square (both length s), and the hypotenuse is the diagonal d. Apply the Pythagorean theorem:
[ d^2 = s^2 + s^2 = 2s^2 ]
Taking the square root of both sides:
[ d = s \sqrt{2} ]
That is the whole derivation. The diagonal is a side scaled by ( \sqrt{2} ), because the two equal legs of the inner right triangle force the hypotenuse to that exact multiple.
Why Is the Diagonal s√2 and Not 2s?
A common first guess is that the diagonal is twice the side, but it is not. The diagonal is the straight shortcut, which must be shorter than that detour. The straight-line distance comes out to ( s \sqrt{2} ), longer than one side, but well short of two.
Examples of the Diagonal of Square Formula
Example 1:
A square has a side of 7 cm. Find its diagonal. Apply the formula directly: [ d = 7 \sqrt{2} \approx 9.9 , \text{cm} ]
Example 2:
A square has a diagonal of 10√2 cm. Find its side length. Reverse the formula: [ s = \frac{10\sqrt{2}}{\sqrt{2}} = 10 , \text{cm} ]
Example 3:
A square has a diagonal of 12 cm. Find its side length. The correct way is: [ s = \frac{12}{\sqrt{2}} = 6\sqrt{2} \approx 8.49 , \text{cm} ]
Example 4:
A square has an area of 36 cm². Find its diagonal. First the side, then the diagonal: [ s = \sqrt{36} = 6 , \text{cm} ] [ d = 6 \sqrt{2} \approx 8.49 , \text{cm} ]
Example 5:
A square has a perimeter of 32 m. Find its diagonal. The side is a quarter of the perimeter: [ s = \frac{32}{4} = 8 , \text{m} ] [ d = 8 \sqrt{2} \approx 11.31 , \text{m} ]
Example 6:
A square floor tile must fit through a doorway with a clear opening of 50 cm. The tile has a side of 36 cm. Will it pass through diagonally? The diagonal: [ d = 36 \sqrt{2} \approx 50.9 , \text{cm} ] The diagonal is slightly wider than the doorway.
Key Takeaways
- The diagonal of square formula is ( d = s \sqrt{2} ).
- From the area, use ( d = \sqrt{2A} ); from the perimeter, ( d = \frac{P}{4} \sqrt{2} ).
- The diagonal is about 1.414 times the side — longer than one side but shorter than two.