Inscribe a Square in a Circle — Definition, Formula & Examples

Inscribing a square in a circle means drawing a square inside a circle so that all four vertices of the square lie exactly on the circle. The circle is then called the circumscribed circle of the square.

A square is inscribed in a circle of radius $r$ when the circle is the circumscircle of the square — that is, the four vertices are concyclic points on the circle, and the center of the circle coincides with the intersection of the square's diagonals. Each diagonal of the inscribed square is a diameter of the circle.

Key Formula

s = r\sqrt{2}

Where:

$s$ = Side length of the inscribed square
$r$ = Radius of the circle

How It Works

Using a compass and straightedge, draw any diameter of the circle. Then construct a second diameter perpendicular to the first by finding the perpendicular bisector of the first diameter. The four points where the two diameters meet the circle are the vertices of the inscribed square. Connect adjacent endpoints to complete the square. This works because two perpendicular diameters divide the circle into four equal arcs of 90° each, guaranteeing a regular quadrilateral.

Worked Example

Problem: A square is inscribed in a circle of radius 5 cm. Find the side length and area of the square.

Find the diagonal: Each diagonal of the inscribed square equals the diameter of the circle.

d = 2r = 2(5) = 10 \text{ cm}

Find the side length: A square's diagonal and side are related by

d = s\sqrt{2}

, so solve for

s

s = \frac{d}{\sqrt{2}} = \frac{10}{\sqrt{2}} = 5\sqrt{2} \approx 7.07 \text{ cm}

Find the area: Square the side length, or use the shortcut

A = \frac{d^2}{2}

A = s^2 = (5\sqrt{2})^2 = 50 \text{ cm}^2

Answer: The inscribed square has a side length of

5\sqrt{2} \approx 7.07

cm and an area of 50 cm².

Why It Matters

This construction appears frequently in high school geometry proofs and compass-and-straightedge exercises. It also shows up in optimization problems — the inscribed square is the largest square that fits inside a given circle, a fact used in engineering and design contexts.

Common Mistakes

Mistake: Confusing the side length with the diameter. Students sometimes set

s = 2r

instead of

s = r\sqrt{2}

Correction: The diameter equals the diagonal of the square, not its side. Use the Pythagorean relationship

d = s\sqrt{2}

to convert between them.