EQN-04-13

Inscribed & Circumscribed Figures

Relate a circle to a square or triangle drawn inside or around it using one shared length.

~6 min read•Difficulty 3/5

When one shape sits inside another, find the single length they share, then express both areas in terms of it.

The circle's diameter equals the square's side.

Square side s \to circle radius s /2 \to circle area π (s /2)^{2} = π s^{2} /4

The square's diagonal equals the circle's diameter.

Circle radius r \to diameter 2 r = square diagonal \to square side = 2 r / 2 = r 2

square area = (r 2)^{2} = 2 r^{2}

The gap between the two shapes is (outer area $-$ inner area). For a circle inscribed in a square, the four corners total $s^{2} - π s^{2} /4.$

The trap is using the side when the figure gives the diameter, or vice versa — identify the shared length first.

Diagram

A circle inscribed in a square: its diameter equals the side.

Worked examples

Example 1 (difficulty 2/5)

A circle is inscribed in a square of side 10. What is the radius of the circle?

◎ Predict the answer first:

Example 2 (difficulty 3/5)

A square is inscribed in a circle of radius 4. What is the area of the square?

◎ Predict the answer first:

Setting the inscribed square’s SIDE equal to the diameter

Why: Side feels like the natural match.

Fix: It is the DIAGONAL of the inscribed square that equals the diameter.

Using the square’s side as the inscribed circle’s radius

Why: Both are prominent lengths.

Fix: The inscribed circle’s diameter (not radius) equals the side.