acalculator

What is the square’s area?

Type any one measure of a square, such as its side, area or diagonal. The square calculator works out the side and then every other measure from it.

Your numbers

Units
Area (A)
25 in²

A square with side 5 in has a perimeter of 20 in, an area of 25 in² and a diagonal of 7.071 in.

Perimeter (P)
20 in
Diagonal (d)
7.071 in
Inradius (r)
2.5 in
Circumradius (R)
3.536 in

Area (A): 25 in². A square with side 5 in has a perimeter of 20 in, an area of 25 in² and a diagonal of 7.071 in.

How to calculate

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

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.

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

  • 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.

Machine-readable copies: Markdown, JSON.

Worked examples

Each example is checked against the calculator on every build.

  1. Side (s) 196.9 in gives Perimeter (P) 787.4 in, Area (A) 38,750.1 in², Diagonal (d) 278.4 in, Inradius (r) 98.43 in, Circumradius (R) 139.2 in.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. Perimeter (P) 4,724 in gives Side (s) 1,181 in, Area (A) 1,395,000 in², Diagonal (d) 1,670 in.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. Area (A) 75,950.2 in² gives Side (s) 275.6 in, Perimeter (P) 1,102 in, Diagonal (d) 389.7 in.Source: OpenStax, Contemporary Mathematics, §10.6 Area (A = s² for a square). https://openstax.org/books/contemporary-mathematics/pages/10-6-area
  4. Diagonal (d) 393.7 in gives Side (s) 278.4 in, Area (A) 77,500.2 in², Circumradius (R) 196.9 in.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

How it works

For a square with side s:

  • Perimeter: P = 4s.
  • Area: A = s².
  • Diagonal: d = s√2, from the Pythagorean theorem on the right triangle with legs s and s: d² = s² + s².
  • Inradius (the circle that touches the four sides): r = s ÷ 2.
  • Circumradius (the circle through the four corners): R = d ÷ 2 = s√2 ÷ 2.

Type one measure. The page finds the side from it (s = P ÷ 4, s = √A, s = d ÷ √2, s = 2r, or s = 2R ÷ √2), then every other measure from the side. Two typed measures that do not fit one square give no answer.

Rules

  • Every measure must be more than 0.
  • The side can be at most 1,000,000 km (10⁹ m), the perimeter 4 × 10⁹ m, the area 10¹⁸ m², the diagonal 2 × 10⁹ m, the inradius 5 × 10⁸ m and the circumradius 10⁹ m. A value worked out past its limit gives no answer.

Output format. Values are double-precision numbers in the unit shown next to each field. Lengths work in metres and the area in square metres inside the page; the page converts to the shown units.

Sources are OpenStax Contemporary Mathematics (perimeter and area) and Prealgebra 2e (Pythagorean theorem), because Algebra and Trigonometry 2e has no section on squares.

Worked examples by hand

Side 5. P = 4 × 5 = 20. A = 5² = 25. d = 5√2 = 7.0711. r = 5 ÷ 2 = 2.5. R = 7.0711 ÷ 2 = 3.5355.

Perimeter 120 (a square table 30 across). s = 120 ÷ 4 = 30. A = 30² = 900. d = 30√2 = 42.4264.

Area 49. s = √49 = 7. P = 28. d = 7√2 = 9.8995.

Diagonal 10. s = 10 ÷ √2 = 7.0711. A = s² = d² ÷ 2 = 50. R = 10 ÷ 2 = 5.

Other questions people ask

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.