How to Draw Circumscribed & Inscribed Circles

Instructor: Michael Quist

Michael has taught college-level mathematics and sociology; high school math, history, science, and speech/drama; and has a doctorate in education.

Drawing accurate circumscribed or inscribed circles can be a tricky process. In this lesson, we'll explore the process for drawing perfect inscribed and circumscribed circle for any polygon where it's possible.

What are Circumscribed and Inscribed Circles?

Drawing a circle is pretty easy. Grab your compass, a round object, or tie a string or rope to a fixed point, and you can scribe a fairly good circle. It gets a little more complicated, however, if you need that circle to fit neatly inside another figure (inscribed) or outside it (circumscribed). In this lesson, we'll discuss how to do both.

A circumscribed circle surrounds a polygon, touching every vertex (corner). All triangles can be circumscribed by a circle, as can all regular (all sides are the same length) polygons. An inscribed circle is inside the polygon, touching each side at exactly one point. When a circle is correctly inscribed, each side that it touches will be tangent to the circle, which means they just touch it, sort of like a ball sitting on a hard surface.

Drawing a Circumscribed Circle

There are two things you need to know to draw any circle. You need to know where to put the center, and you need to know how long the radius should be. Once you have those two, you can adjust your compass, set your center point, and draw your circle. Let's explore first of all how to do that when you want your circle to go around the outside of the object (circumscribed).

First all, can it be done? Here's the trick. You can only circumscribe a circle around a polygon (where it touches all of the corners) if the perpendicular bisectors, lines drawn at a 90° angle from the exact center of each side, meet at one point. If they don't, then no circle you draw will be correctly circumscribed. If they do, then you now have the center of our circle. Let's try one.

Irregular quadrilateral
circumscribe step 1

Draw an irregular quadrilateral like ABCD shown. If you want to make sure you'll be able to circumscribe a circle around it, draw the circle first, then draw the quadrilateral inside the circle, with all of the corners touching the outside edge of the circle. That way you know it will work for the exercise.

Draw perpendicular bisectors
circumscribe step 2

Okay, time to remember that geometry practice. Remember how to construct a perpendicular bisector?

  1. Set your compass to some length a little longer than half the length of the polygon's side that you're trying to bisect, and set the point of the compass in one end of the side.
  2. Make a small, light arc on both sides of the line.
  3. Set the point of the compass in the other end of the side, and make the two marks again.
  4. Where the arcs cross are center points. Connect those dots, and you have a perpendicular bisector.

Constructing a perpendicular bisector
perpendicular bisector

Okay, the hard part is done. The radius of your circle will be the distance from the center you located to any corner on the quadrilateral. Set your radius, put the point of your compass in the center, and draw the circle around the quadrilateral. If you have a suitable polygon, this will work every time. Also, notice that since all of the bisectors will intersect at only one point, you only actually have to do two of them, and then use the point where they cross! However, drawing all of the perpendicular bisectors will let you know if you have a suitable polygon. Remember, all triangles and regular (all sides the same length) polygons can be circumscribed with a circle.

Circle circumscribed around irregular quadrilateral
circumscribed circle

Inscribing a Circle

Inscribing a circle within a polygon is similar, but now you're going to use angle bisectors (a line that divides an angle into two equal smaller angles). For example, take a look at triangle ABC shown.

Irregular triangle

The first thing you want to do is draw an angle bisector from at least two corners to the center. Where the two bisectors meet will be the center of the circle.

Angle bisectors
bisecting the angles

Okay, back to that great geometry practice. Remember how to bisect an angle?

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