Theorem

The largest circle that fits inside a square

Geometry • Circle • Inscribed Figures

Formula

r_{m a x} = \frac{a}{2}

The largest circle contained in a square of side

a

is the inscribed circle, which touches all four sides: its diameter equals the side, so its radius is

\frac{a}{2}

and its area is

\frac{π a ^{2}}{4}

, which is

\frac{π}{4}

of the square. No circle inside the square can be wider than the side, and the inscribed circle attains that width.

How to use it

1Inside a rectangle with sides $a$ and $b$ the same argument gives a largest circle of diameter $min (a, b)$ .
2In three dimensions it fixes the base of the largest cone or cylinder standing in a cube of edge $a$ : base radius $\frac{a}{2}$ , and the height is then the full edge.
3The fraction $\frac{π}{4} \approx 0.785$ is a quick check on any 'circle cut from a square' answer.

Worked example

A circular disc of the greatest possible size is cut from a square sheet of side

14

cm. What is its area? (Use

π = \frac{22}{7}

Step-by-step

1
$r = \frac{14}{2} = 7$ cm
Why: The largest circle has diameter equal to the side of the square.
2
$π r^{2} = \frac{22}{7} \times 7^{2} = 154$ cm $^{2}$

Answer

154

^{2}

Watch out

This is the INSCRIBED circle, not the circumscribed one: the circle through the four corners has diameter

a 2

and does not fit inside the square at all. Taking the diagonal as the diameter is the standard error.

Builds on

Area and circumference of a circle Perimeter and area of a square

Also known as

incircle of a square, largest circle in a square, inscribed circle in a square, biggest circle cut from a square sheet, circle inscribed in a square, largest circle in a rectangle

Appears in past papers

Formula

Worked example

Appears in past papers

2019 (I)

2018 (II)

2018 (I)