Inscribed Polygon in a Circle — Definition, Formula & Examples
An inscribed polygon in a circle is a polygon drawn inside a circle so that every one of its corners touches the circle. Each side of the polygon is a chord of that circle.
A polygon is inscribed in a circle if all of its vertices lie on the circle. The circle is then called the circumscribed circle (circumcircle) of the polygon, and its center is equidistant from every vertex. A polygon that admits such a circle is called cyclic.
Key Formula
Where:
- = side length of a regular n-sided polygon inscribed in the circle
- = radius of the circle
- = number of sides of the regular polygon (n ≥ 3)
Worked Example
Problem: A regular hexagon is inscribed in a circle of radius 5 cm. Find the side length and the perimeter of the hexagon.
Step 1: Join the center to all six vertices. The six central angles are equal and together make a full turn.
Step 2: Each side is a chord that subtends a 60° central angle. Use the chord length formula with half of that angle.
Step 3: Multiply the side length by the number of sides to get the perimeter.
Answer: Each side is 5 cm and the perimeter is 30 cm. A regular hexagon's side always equals the radius of its circumscribed circle.
Why It Matters
Archimedes estimated π by inscribing polygons with more and more sides in a circle; their perimeters approach the circumference from below. In geometry courses, inscribed quadrilaterals lead to the theorem that opposite angles of a cyclic quadrilateral are supplementary.
Common Mistakes
Mistake: Assuming every polygon can be inscribed in a circle.
Correction: Every triangle and every regular polygon can be inscribed, but many polygons cannot. A quadrilateral can be inscribed only if its opposite angles add to 180°. A non-square rhombus, for example, cannot be inscribed.