Diagonal of a Square Calculator
Convert between a square's side and its corner-to-corner diagonal in either direction. Enter one value in any unit — the diagonal is always the side times √2 ≈ 1.414 — and get side, diagonal, area, and perimeter together.
Example: with What do you know? Side length (a) · Value 10 → Diagonal d = a√2: 14.142 units.
- Side (a)10 units
- Area (a²)100 square units
- Perimeter (4a)40 units
Computed by the calculator below using its default values. Change any input to see your own numbers.
d = a√2 ≈ 1.414a, from the Pythagorean theorem. A baseball diamond's 90-ft sides put second base 127.28 ft from home plate.
The √2 shortcut
A square's diagonal cuts it into two 45-45-90 right triangles, so the Pythagorean theorem gives d = √(a² + a²) = a√2 ≈ 1.414a. The reverse is just as clean: side = diagonal ÷ √2, and since area is a², you can go straight from diagonal to area with A = d²/2 — no intermediate step needed.
The number shows up anywhere squares get measured corner to corner. A baseball diamond is a 90-ft square, so the throw from home to second base is 90√2 ≈ 127.28 ft. Framers check a square layout by taping the diagonal and comparing against side × 1.414; a 10-ft square garden bed should tape 14 ft 1.7 in corner to corner, and any shortfall means the corners are out of square.
How it’s calculated
d = a√2 with √2 ≈ 1.4142136; reverse: a = d/√2 = d × 0.7071068. Area = a² (equivalently d²/2); perimeter = 4a. Results stay in whatever unit you enter, with area in that unit squared.
True squares only — four equal sides and 90° corners. For unequal sides use the rectangle diagonal formula √(l² + w²).
Square diagonals worth knowing
| Square | Side | Diagonal |
|---|---|---|
| Unit square | 1 | 1.414 |
| Floor tile | 12 in | 16.97 in |
| Garden bed | 4 ft | 5.657 ft |
| Baseball diamond | 90 ft | 127.28 ft (home to second) |
| City block (approx.) | 100 m | 141.42 m |
Computed with d = a√2; baseball infield is a 90-ft square per MLB rules.
Common mistakes
- Doubling the side to estimate the diagonal — the true factor is 1.414, so doubling overshoots by 41%.
- Multiplying by √2 when converting diagonal to side; going that direction you divide (or multiply by 0.7071).
- Computing area from the diagonal as d² instead of d²/2 — that doubles the real area.
- Applying a√2 to a rectangle that is only roughly square; even a 10 × 11 rectangle needs the full √(l² + w²).
Frequently asked questions
What is the diagonal of a square formula?
d = a√2 ≈ 1.414 × side. It comes from the Pythagorean theorem, because the diagonal is the hypotenuse of a right triangle whose legs are two sides of the square.
How do I find the side from the diagonal?
Divide the diagonal by √2, or equivalently multiply by 0.7071. A 20-unit diagonal means sides of 14.142 units and an area of exactly 200 — d²/2 is a handy one-step check.
How far is the throw from home plate to second base?
A baseball diamond is a 90-ft square, so the diagonal is 90√2 ≈ 127.28 ft — about 42.4 yards, noticeably longer than the 90-ft basepaths themselves.
Why is the diagonal exactly √2 times the side?
Both legs of the right triangle are the same length a, so d² = a² + a² = 2a², and d = a√2. The 45-45-90 triangle ratio 1 : 1 : √2 is the same fact stated as a rule.