Calculate square area, perimeter, diagonal, inradius, and circumradius from side length with step-by-step formulas.
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A square is a regular quadrilateral with four equal sides and four right angles (90°). It is a special case of a rectangle, rhombus, and parallelogram.
A = a^2, \quad P = 4a, \quad d = a\sqrt{2}, \quad r = \frac{a}{2}, \quad R = \frac{a}{\sqrt{2}}Area is side squared, perimeter is 4 times side, and the diagonal is side times square root of 2.
Scenario: Calculate properties of a square with side a = 8 cm
Area: A = 8² = 64 cm²
Perimeter: P = 4 × 8 = 32 cm
Diagonal: d = 8√2 ≈ 11.3137 cm
Inradius: r = 8 / 2 = 4 cm
All squares are rhombuses (four equal sides), but a square also requires four 90-degree right angles.
By the Pythagorean theorem, d = √(a² + a²) = √(2a²) = a√2 ≈ 1.4142 × a.
The inradius is the radius of the circle inscribed inside the square, which equals half the side length: r = a / 2.
The circumradius is the radius of the circumscribed circle touching all four vertices, which equals half the diagonal: R = d / 2 = a / √2.
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