Square Calculator
Calculate square area, perimeter, diagonal, and inscribed/circumscribed circle properties. Enter any measurement and get all values instantly.
Formula:A = a² | P = 4a | d = a√2
Square Properties
Area
100.0000 cm^2
Side10.0000 cm
Perimeter40.0000 cm
Diagonal14.1421 cm
Square Input
Square Visualization
Showing inscribed circle (inside) and circumscribed circle (outside)
Complete Results
Side Length (a)
10.0000 cm
Area
100.0000 cm^2
A = a^2
Perimeter
40.0000 cm
P = 4a
Diagonal
14.1421 cm
d = a x sqrt(2)
Circle Properties
Inscribed Circle Radius
5.0000 cm
r = a/2
Circumscribed Circle Radius
7.0711 cm
R = d/2 = (a x sqrt(2))/2
Inscribed Circle Area
78.5398 cm^2
78.5% of square
Circumscribed Circle Area
157.0796 cm^2
63.7% filled by square
Square Reference Table
| Side | Area | Perimeter | Diagonal |
|---|---|---|---|
| 1 | 1 | 4 | 1.414 |
| 2 | 4 | 8 | 2.828 |
| 5 | 25 | 20 | 7.071 |
| 10 | 100 | 40 | 14.142 |
| 20 | 400 | 80 | 28.284 |
| 50 | 2500 | 200 | 70.711 |
| 100 | 10000 | 400 | 141.421 |
Square Formulas
Basic Formulas
- Area: A = a^2
- Perimeter: P = 4a
- Diagonal: d = a x sqrt(2)
Finding Side From
- From Area: a = sqrt(A)
- From Perimeter: a = P/4
- From Diagonal: a = d/sqrt(2)
Circle Formulas
- Inscribed radius: r = a/2
- Circumscribed radius: R = (a x sqrt(2))/2 = d/2
- Ratio R/r = sqrt(2) = 1.414...
Key relationship: The diagonal of a square is always sqrt(2) times (about 1.414x) the side length. This comes from the Pythagorean theorem: d^2 = a^2 + a^2 = 2a^2.
Square Properties
Area
100.0000 cm^2
Side10.0000 cm
Perimeter40.0000 cm
Diagonal14.1421 cm
?How to Calculate Square Properties
Square formulas: Area = side x side = side squared. Perimeter = 4 x side. Diagonal = side x sqrt(2), approximately 1.414 x side. Inscribed circle radius = side / 2. Circumscribed circle radius = (side x sqrt(2)) / 2 = diagonal / 2.
What is a Square?
A square is a regular quadrilateral with four equal sides and four right angles (90 degrees each). It is both a special rectangle (with equal sides) and a special rhombus (with right angles). The diagonals of a square are equal, bisect each other at right angles, and each diagonal divides the square into two congruent isosceles right triangles.
Key Facts About Squares
- Area = side squared = a^2
- Perimeter = 4 x side = 4a
- Diagonal = a x sqrt(2), approximately 1.414a
- All four sides are equal in length
- All four angles are 90 degrees (right angles)
- Inscribed circle radius = a/2 (half the side)
- Circumscribed circle radius = a x sqrt(2)/2 = diagonal/2
- A square is a special rectangle and a special rhombus
- Diagonals bisect each other at 90 degrees
- Diagonals divide the square into 4 congruent right triangles
Quick Answer
Square formulas: Area = side x side = side squared. Perimeter = 4 x side. Diagonal = side x sqrt(2), approximately 1.414 x side. Inscribed circle radius = side / 2. Circumscribed circle radius = (side x sqrt(2)) / 2 = diagonal / 2.
Frequently Asked Questions
The area (A) of a square is calculated by squaring the side length: A = side x side = a^2. For example, a square with side 5 has area = 25 square units.
The diagonal (d) of a square equals the side multiplied by the square root of 2: d = a x sqrt(2) = a x 1.414... This comes from the Pythagorean theorem applied to the right triangle formed by two sides and the diagonal.
The inscribed circle (incircle) is the largest circle that fits inside the square, touching all four sides. Its radius equals half the side length: r = a/2. The circle's area is pi x (a/2)^2.
The circumscribed circle (circumcircle) is the smallest circle that completely contains the square, passing through all four corners. Its radius equals half the diagonal: R = (a x sqrt(2))/2 = d/2.
To find the side from the area, take the square root: side = sqrt(Area). For example, if area = 36, then side = sqrt(36) = 6.
A square's diagonals are equal in length, bisect each other at right angles (90 degrees), and divide the square into four congruent right isosceles triangles. Each diagonal = side x sqrt(2).
Last updated: 2025-01-15
How this works
Formulas follow standard definitions from the NIST Digital Library of Mathematical Functions and classical textbook derivations. Calculations run entirely in your browser. Where a closed-form solution exists, it is used; where an iterative or numerical method is required, the implementation is named on the page.
Sources
- [1]NIST Digital Library of Mathematical FunctionsAcademicdlmf.nist.govAccessed Apr 21, 2026
Joseph OrdunaFounder & Software Engineer
Full-stack software engineer specializing in embedded systems, web architecture, and AI/ML. Founder of Practical Web Tools. Built the gesture-controlled drone IP acquired by KD Interactive (Aura Drone, sold on Amazon).
Square Properties
Area
100.0000 cm^2
Side10.0000 cm
Perimeter40.0000 cm
Diagonal14.1421 cm