Last 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

**Solution**:

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!*