Area: Inscribed Circle & Square

Learn about squares inscribed in circles and circles inscribed in squares. Understand why their area ratios are fixed and solve exercises finding area differences.

Inscribed Circle & Square

Use the workspace below. Write equations like A = 30 to solve for the area.

Learning Topics

📖 Area Study Guide

1. Inscribed Shapes & Ratios

An inscribed shape is a geometric figure that is drawn inside another figure, so that their boundaries touch.
We will explore two fundamental inscribed setups:
• A square inscribed in a circle (its vertices lie on the circle).
• A circle inscribed in a square (it is tangent to all four sides of the square).

2. Square Inscribed in a Circle

Square inscribed in a circle: Diagonal

d = 2r

, side

a

. Area ratio is exactly 2/pi.

Let a circle have radius

r

. The square inscribed inside has vertices touching the circle boundary.

The diagonal of the square is equal to the diameter of the circle:

d = 2r

.
Using the Pythagorean theorem for side length

a

of the square:

a^2 + a^2 = d^2 \implies 2a^2 = (2r)^2 = 4r^2 \implies a^2 = 2r^2

• Area of the square:

A_{\text{square}} = a^2 = 2r^2

• Area of the circle:

A_{\text{circle}} = \pi r^2

The ratio of the area of the square to the circle is constant:

\frac{A_{\text{square}}}{A_{\text{circle}}} = \frac{2r^2}{\pi r^2} = \frac{2}{\pi} \approx 0.637

Notice that the radius

r^2

cancels out completely! The ratio is always exactly

\frac{2}{\pi}

regardless of the size.

Ratio of Inscribed Square to Circle

\frac{A_{\text{square}}}{A_{\text{circle}}} = \frac{2}{\pi}

3. Circle Inscribed in a Square

Circle inscribed in a square: Side length

a = 2r

. Area ratio is exactly pi/4.

Let a circle of radius

r

be inscribed in a square of side length

a

. The circle fits perfectly inside, tangent to all four sides.

The diameter of the circle is equal to the side length of the square:

d = a = 2r

.
• Area of the square:

A_{\text{square}} = a^2 = (2r)^2 = 4r^2

• Area of the circle:

A_{\text{circle}} = \pi r^2

The ratio of the area of the circle to the square is constant:

\frac{A_{\text{circle}}}{A_{\text{square}}} = \frac{\pi r^2}{4r^2} = \frac{\pi}{4} \approx 0.785

Again, the radius

r^2

cancels out! The ratio is always exactly

\frac{\pi}{4}

regardless of the size.

Ratio of Inscribed Circle to Square

\frac{A_{\text{circle}}}{A_{\text{square}}} = \frac{\pi}{4}

Frequently Asked Questions

What is the area ratio of inscribed circles and squares?

For a square inscribed in a circle, the area ratio of the square to the circle is always exactly 2/π ≈ 0.637. For a circle inscribed in a square, the area ratio of the circle to the square is always exactly π/4 ≈ 0.785. These ratios are constant regardless of the shapes' actual sizes.

How do you calculate the area of a circle?

The area of a circle is calculated as π times the radius squared (A = πr²). Since π (pi) is an irrational number, the area of a circle with a rational radius will always be an irrational number.