# What is the square’s area?

Finds every measure of a square from any one of them: side, perimeter, area, diagonal, and the radii of the inscribed and circumscribed circles.

- Page: https://www.acalculator.org/math/square-calculator
- JSON spec: https://www.acalculator.org/math/square-calculator.json
- Version: ef590406cd29

## Default answer

Example with the default inputs (Side (s) 5 in): A square with side 5 in has a perimeter of 20 in, an area of 25 in² and a diagonal of 7.071 in.

## Inputs

| Key | Label | Description |
| --- | --- | --- |
| s | Side (s) | The length of one side of the square. |
| p | Perimeter (P) | The distance around the square: four sides. |
| a | Area (A) | The space inside the square: side times side. |
| d | Diagonal (d) | The distance from one corner to the opposite corner. |
| r | Inradius (r) | The radius of the circle that touches all four sides: half the side. |
| cr | Circumradius (R) | The radius of the circle through all four corners: half the diagonal. |

## Outputs

| Key | Label | Description |
| --- | --- | --- |
| s | Side (s) | The length of one side of the square. |
| p | Perimeter (P) | The distance around the square: four sides. |
| a | Area (A) | The space inside the square: side times side. |
| d | Diagonal (d) | The distance from one corner to the opposite corner. |
| r | Inradius (r) | The radius of the circle that touches all four sides: half the side. |
| R | Circumradius (R) | The radius of the circle through all four corners: half the diagonal. |

## Method

P = 4s; A = s²; d = s√2 (Pythagorean theorem: d² = s² + s²); r = s ÷ 2; R = d ÷ 2 = s√2 ÷ 2.

## Assumptions

- All four sides are equal and every corner is a right angle.
- Type one measure; the side follows from it, and every other measure from the side.
- Lengths are in the unit shown next to each field and the area in the area unit shown.

## Worked examples

1. s = 5 gives p = 20, a = 25, d = 7.071068, r = 2.5, cr = 3.535534. Source: OpenStax, Contemporary Mathematics, §10.4 Polygons, Perimeter, and Circumference (P = n·s; a square has n = 4). https://openstax.org/books/contemporary-mathematics/pages/10-4-polygons-perimeter-and-circumference; OpenStax, Contemporary Mathematics, §10.6 Area (A = s² for a square). https://openstax.org/books/contemporary-mathematics/pages/10-6-area; OpenStax, Prealgebra 2e, §9.3 Use Properties of Angles, Triangles, and the Pythagorean Theorem (a² + b² = c²). https://openstax.org/books/prealgebra-2e/pages/9-3-use-properties-of-angles-triangles-and-the-pythagorean-theorem.
2. p = 120 gives s = 30, a = 900, d = 42.426407. Source: OpenStax, Contemporary Mathematics, §10.4 Polygons, Perimeter, and Circumference (P = n·s; a square has n = 4). https://openstax.org/books/contemporary-mathematics/pages/10-4-polygons-perimeter-and-circumference.
3. a = 49 gives s = 7, p = 28, d = 9.899495. Source: OpenStax, Contemporary Mathematics, §10.6 Area (A = s² for a square). https://openstax.org/books/contemporary-mathematics/pages/10-6-area.
4. d = 10 gives s = 7.071068, a = 50, cr = 5. Source: OpenStax, Prealgebra 2e, §9.3 Use Properties of Angles, Triangles, and the Pythagorean Theorem (a² + b² = c²). https://openstax.org/books/prealgebra-2e/pages/9-3-use-properties-of-angles-triangles-and-the-pythagorean-theorem.

## FAQ

### How do I find the area of a square?

Multiply the side by itself: A = s². A square with 5 in sides has an area of 25 in².

### How do I find the side from the area?

Take the square root of the area: s = √A. An area of 49 m² gives a side of 7 m.

### What is the diagonal of a square?

The diagonal splits the square into two right triangles with legs s and s, so d² = s² + s² = 2s² and d = s√2. A 5 in square has a diagonal of 5√2 ≈ 7.0711 in.

### How do I find the side from the diagonal?

Divide the diagonal by √2: s = d ÷ √2. A diagonal of 10 cm gives a side of about 7.0711 cm and an area of d² ÷ 2 = 50 cm².

### What are the inradius and the circumradius?

The inradius is the radius of the largest circle inside the square; it touches all four sides and is s ÷ 2. The circumradius is the radius of the circle through the four corners; it is half the diagonal, s√2 ÷ 2.

### How do I find the perimeter of a square?

Add the four equal sides: P = 4s. A square table 30 in across has a perimeter of 120 in.

## Sources

- OpenStax, Contemporary Mathematics, §10.4 Polygons, Perimeter, and Circumference (P = n·s for a regular polygon; a square table 30 inches across). https://openstax.org/books/contemporary-mathematics/pages/10-4-polygons-perimeter-and-circumference (retrieved 2026-10-02)
- OpenStax, Contemporary Mathematics, §10.6 Area (A = s·s = s² for a square). https://openstax.org/books/contemporary-mathematics/pages/10-6-area (retrieved 2026-10-02)
- OpenStax, Prealgebra 2e, §9.3 Use Properties of Angles, Triangles, and the Pythagorean Theorem (a² + b² = c²). https://openstax.org/books/prealgebra-2e/pages/9-3-use-properties-of-angles-triangles-and-the-pythagorean-theorem (retrieved 2026-10-02)