Circumscribed Polygon — Definition, Formula & Examples

A circumscribed polygon is a polygon that surrounds a circle, with each of its sides touching (tangent to) the circle at exactly one point. The circle inside is called the inscribed circle, or incircle.

A polygon is circumscribed about a circle if and only if every side of the polygon is tangent to the circle. Equivalently, the circle is the incircle of the polygon, and the circle's center is equidistant from all sides of the polygon, with that distance equal to the inradius.

Key Formula

A = \frac{1}{2} \cdot p \cdot r

Where:

$A$ = Area of the circumscribed polygon
$p$ = Perimeter of the polygon
$r$ = Radius of the inscribed circle (inradius)

Worked Example

Problem: A regular hexagon is circumscribed about a circle with radius 5 cm. Find the area of the hexagon.

Find the side length: For a regular hexagon circumscribed about a circle of radius r, the apothem equals r. The apothem of a regular hexagon with side length s is s√3/2. Set this equal to 5.

\frac{s\sqrt{3}}{2} = 5 \implies s = \frac{10}{\sqrt{3}} = \frac{10\sqrt{3}}{3} \approx 5.774 \text{ cm}

Find the perimeter: A hexagon has 6 sides, so multiply the side length by 6.

p = 6 \cdot \frac{10\sqrt{3}}{3} = 20\sqrt{3} \approx 34.64 \text{ cm}

Apply the area formula: Use A = (1/2) · p · r with r = 5.

A = \frac{1}{2} \cdot 20\sqrt{3} \cdot 5 = 50\sqrt{3} \approx 86.6 \text{ cm}^2

Answer: The area of the circumscribed hexagon is

50\sqrt{3} \approx 86.6

cm².

Why It Matters

Circumscribed polygons appear in geometry proofs and competition problems involving tangent lines and incircles. Engineers and architects use them when designing structures that must enclose circular features, such as bolt patterns around pipes or frames around circular windows.

Common Mistakes

Mistake: Confusing circumscribed with inscribed. Students sometimes think a circumscribed polygon is inside the circle.

Correction: A circumscribed polygon goes around the circle (sides tangent to the circle). An inscribed polygon sits inside the circle (vertices on the circle). Think: "circum" means "around."