Area of a Square Calculator
Enter any one measurement of a square - side length, diagonal, perimeter, circumscribed circle radius, or inscribed circle radius - and this calculator works out all the others instantly. Switch between metric and imperial units, and see each step of the math.
Formula
Worked example
A square with side 5 cm: Area = 5^2 = 25 cm^2. Perimeter = 4 x 5 = 20 cm. Diagonal = 5 x sqrt(2) = 7.071 cm. Circumradius = 7.071 / 2 = 3.536 cm. Inradius = 5 / 2 = 2.5 cm.
What is the area of a square?
The area of a square is the amount of two-dimensional space enclosed within its four equal sides. Because all four sides of a square are the same length, the area formula is simply the side length multiplied by itself: A = a x a, or a squared. For example, a square with a side of 5 centimetres has an area of 25 square centimetres. The unit of area is always the square of whatever length unit you use: square centimetres (cm²), square metres (m²), square inches (in²), and so on.
Solving from diagonal, perimeter, circumradius, or inradius
You do not need the side length to find the area. If you know the diagonal (the distance from one corner to the opposite corner), the side is d divided by the square root of 2, so the area is d² / 2. If you know the perimeter, each side is P / 4 and the area is (P / 4)². The circumradius R is the radius of the circle that passes through all four corners. Because the circumradius equals half the diagonal, the side equals R x sqrt(2) and the area is 2R². The inradius r is the radius of the largest circle that fits inside the square - it equals half the side length, so the side is 2r and the area is 4r². All five entry points are supported in this calculator: choose the one you know from the "Solve from" dropdown.
Diagonal and circle relationships
Every square has two circles naturally associated with it. The circumscribed circle (circumcircle) passes through all four corners and has a radius equal to half the diagonal. The inscribed circle (incircle) touches all four sides at their midpoints and has a radius equal to half the side length. The diagonal of a square is always the side multiplied by the square root of 2 (approximately 1.41421). This follows from the Pythagorean theorem: because the diagonal forms a right triangle with two sides, d² = a² + a² = 2a², so d = a x sqrt(2). The diagonal is also useful for confirming whether a shape is a true square: if the two diagonals are equal in length and bisect each other at right angles, the quadrilateral is a square.
How area scales with side length
Area grows as the square of the side length, so small increases in side produce large increases in area. Doubling the side quadruples the area. Tripling the side multiplies the area by nine. Halving the side reduces the area to one quarter. This quadratic scaling is visible in the chart below the result: the curve rises steeply as the side grows. It matters in practice - a square room that is 4 m on each side has 16 m² of floor space, but a room that is only 3 m on each side has just 9 m², about 44 percent less despite the sides being only 25 percent shorter.
Common square sizes and their areas
| Side (cm) | Area (cm²) | Perimeter (cm) | Diagonal (cm) |
|---|---|---|---|
| 1 | 1 | 4 | 1.414 |
| 2 | 4 | 8 | 2.828 |
| 5 | 25 | 20 | 7.071 |
| 10 | 100 | 40 | 14.142 |
| 15 | 225 | 60 | 21.213 |
| 20 | 400 | 80 | 28.284 |
| 25 | 625 | 100 | 35.355 |
| 30 | 900 | 120 | 42.426 |
| 50 | 2500 | 200 | 70.711 |
| 100 | 10000 | 400 | 141.421 |
Side lengths in centimetres with corresponding area, perimeter, and diagonal.
Frequently asked questions
What is the formula for the area of a square?
The area of a square is the side length multiplied by itself: A = a x a, or a squared. For example, a square with sides of 6 cm has an area of 36 cm². The result is always in square units matching the length unit you used.
How do I find the area if I only know the diagonal?
Use the formula A = d² / 2, where d is the diagonal length. Because the diagonal of a square equals the side times sqrt(2), squaring it gives 2a², and dividing by 2 recovers a². For example, a square with a diagonal of 10 cm has an area of 100 / 2 = 50 cm².
How do I find the area from the perimeter?
Divide the perimeter by 4 to get the side length, then square it. For a perimeter of 20 cm: side = 20 / 4 = 5 cm, area = 5² = 25 cm². The formula is A = (P / 4)².
What is the difference between circumradius and inradius for a square?
The circumradius (R) is the radius of the circle that passes through all four corners of the square. It equals half the diagonal: R = a x sqrt(2) / 2. The inradius (r) is the radius of the circle that fits perfectly inside the square, touching all four sides at their midpoints. It equals half the side: r = a / 2. For any square, R is always larger than r by a factor of sqrt(2).
How does the area change when I double the side length?
When you double the side length, the area increases by a factor of four (2² = 4). More generally, if you multiply the side by any factor k, the area multiplies by k². This quadratic relationship means area grows much faster than side length: a side three times longer produces nine times the area.
Can I use this calculator for a room or land area measured in metres or feet?
Yes. The calculator works with any length unit - just be consistent. If your side is in metres, your area will be in square metres. If it is in feet, the area will be in square feet. For non-square rooms, use a rectangle area calculator instead.