The Squares Calculator finds the area, perimeter and diagonal of a square from a single side length, since all four sides of a square are equal by definition. Enter one side, and the tool returns all three derived measurements with the substituted arithmetic shown, or enter the area or diagonal instead to work backward to the side length.
A square is the simplest quadrilateral to calculate precisely because one measurement fixes everything else: all four sides match, all four angles are right angles, and both diagonals are equal in length and bisect each other at 90 degrees.
Calculate the area of a square
Area counts the unit squares that cover the interior, which for a square is simply the side length multiplied by itself.
A = side²Worked example: a square with a side of 6 units.
1. Square the side length. 6² = 6 × 6 = 36 square units.
Calculate the perimeter of a square
Perimeter is the total distance around the figure, walking all four equal edges.
P = 4 × sideWorked example: the same square with a side of 6 units.
1. Multiply the side by 4. 4 × 6 = 24 units.
Calculate the diagonal of a square
The diagonal connects two opposite corners and equals the hypotenuse of a right triangle formed by two adjacent sides. Because both legs of that triangle are equal (the square's side), the diagonal formula simplifies from the general Pythagorean theorem.
diagonal = side × √2Worked example: the same square with a side of 6 units.
1. Multiply the side by the square root of 2. 6 × 1.41421 ≈ 8.49 units.
The diagonal is always longer than any single side, and specifically about 41 percent longer, since √2 ≈ 1.414.
Find the side length from area
When the area is known instead of the side, reverse the area formula by taking the square root.
side = √AWorked example: a square with an area of 49 square units.
1. Take the square root of the area. √49 = 7 units.
Find the side length from the diagonal
When the diagonal is known instead of the side, divide by √2.
side = diagonal / √2Worked example: a square with a diagonal of 10 units.
1. Divide by the square root of 2. 10 / 1.41421 ≈ 7.07 units.
Compare squares and general rectangles
A square is a special case of a rectangle where length and width happen to be equal. Every formula that applies to a rectangle, area as length times width and perimeter as twice the sum of the two sides, still works for a square once both dimensions are set equal to the single side length.
What makes the square calculations simpler is that only one measurement needs to be entered rather than two independent ones.
| Shape | Area formula | Perimeter formula |
|---|---|---|
| Square | side² | 4 × side |
| Rectangle | length × width | 2(length + width) |
Work through a second full example
A square with a side of 10 units shows all three formulas together. 1. Area. 10² = 100 square units. 2. Perimeter. 4 × 10 = 40 units. 3. Diagonal. 10 × 1.41421 ≈ 14.14 units.
Doubling the side from 6 to 10 (roughly 1.67 times) increases the area from 36 to 100, a factor of about 2.78, since area scales with the square of the side rather than in direct proportion to it.
The perimeter and diagonal, by contrast, both scale directly with the side: perimeter grows from 24 to 40 and diagonal from about 8.49 to 14.14, matching the same 10/6 ratio as the side length itself. That contrast between squared and linear scaling is the same reason doubling a room's dimensions more than doubles the flooring needed to cover it.
Check the relationship between area and diagonal directly
The diagonal can also be found straight from the area, without going through the side length as an intermediate step, since diagonal = √(2 × area). For the 100-square-unit example above, that gives √200 ≈ 14.14, matching the diagonal found from the side length of 10.
This shortcut is useful whenever the area is already known and only the diagonal is needed, skipping the square-root-of-area step that would otherwise come first.
Frequently asked questions
How do you find the area of a square?
Multiply the side length by itself: A = side². A square with a 6-unit side has an area of 36 square units.
How do you find the perimeter of a square?
Multiply the side length by 4: P = 4 × side. A square with a 6-unit side has a perimeter of 24 units.
How do you find the diagonal of a square?
Multiply the side length by the square root of 2: diagonal = side × √2. A square with a 6-unit side has a diagonal of about 8.49 units.
How do you find the side length from the area?
Take the square root of the area: side = √A. A square with an area of 49 square units has a side length of 7 units.
How do you find the side length from the diagonal?
Divide the diagonal by the square root of 2: side = diagonal / √2. A square with a diagonal of 10 units has a side length of about 7.07 units.
Is a square a type of rectangle?
Yes. A square is a rectangle with all four sides equal, so every rectangle formula applies to a square once length and width are set to the same value.
Why is the diagonal always longer than the side?
Because the diagonal is the hypotenuse of a right triangle whose two legs are the square's sides, and a hypotenuse is always longer than either leg alone. The exact ratio is √2, about 1.414 times the side length.
Summary
A square's area, perimeter and diagonal all follow directly from its single side length: area is side squared, perimeter is 4 times the side, and the diagonal is the side times √2. A square with a 6-unit side has an area of 36, a perimeter of 24, and a diagonal of about 8.49 units.
The same relationships work in reverse to find the side length from a known area or diagonal.