Inscribed and circumscribed polygons

Two ways to combine a polygon and a circle

Every triangle works

Every triangle can be inscribed in a circle and has an inscribed circle:

• the circumscribed circle is centered at the circumcenter;
• the inscribed circle is centered at the incenter.

Quadrilaterals: two tests

Not every quadrilateral can be inscribed in a circle or circumscribed about one.

Such a quadrilateral is called cyclic. The reason: two opposite angles are inscribed angles whose intercepted arcs together make the whole circle, \(360°\). Half of \(360°\) is \(180°\).

This follows from the fact that two tangent segments from the same point are congruent: each side is made of two tangent segments, and each tangent segment appears in exactly one side of each opposite pair.

Regular polygons

• The radius of a regular polygon goes from the center to a vertex.
• The apothem goes from the center perpendicular to a side; it's the radius of the inscribed circle.
• The central angle is formed by two radii to consecutive vertices and measures \(\frac{360°}{n}\).

A regular octagon has central angles of \(\frac{360°}{8} = 45°\).

Practice