Inscribing Polygons and Circles

Quarter Wit, Quarter WisdomLast week we looked at regular and irregular polygons.  Today, let’s try to understand how questions involving one figure inscribed in another are done.  The most common example of a figure inscribed in another is a polygon inscribed in a circle or a circle inscribed in a polygon. Let’s see the various ways in which this can be done.

To inscribe a polygon in a circle, the polygon is placed inside the circle so that all the vertices of the polygon lie on the circumference of the circle.

There are a few points about inscribing a polygon in a circle that you need to keep in mind:

–  Every triangle has a circumcircle so all triangles can be inscribed in a circle.

–  All regular polygons can also be inscribed in a circle.

–  Also, all convex quadrilaterals whose opposite angles sum up to 180 degrees can be inscribed in a circle.

There are also a few points about inscribing a circle in a polygon that you need to keep in mind:

–  All triangles have an inscribed circle (called incircle). When a circle is inscribed in a triangle, all sides of the triangle must be tangent to the circle.

–  All regular polygons have an inscribed circle.

–  Most other polygons do not have an inscribed circle

A simple official question will help us see the relevance of these points:

Question: Which of the figures below can be inscribed in a circle?

(A) I only
(B) III only
(C) I & III only
(D) II & III only
(E) I, II & III


I think it will suffice to say that the answer is (C).

Next week, we will look at the relations between the sides of these polygons and the radii of the circles.

Karishma, a Computer Engineer with a keen interest in alternative Mathematical approaches, has mentored students in the continents of Asia, Europe and North America. She teaches the GMAT for Veritas Prep and regularly participates in content development projects such as this blog!