10. Circles

Lesson

A polygon is said to be inscribed in a circle if all of the vertices of the polygon lie on the circumference of the circle. By this definition, the sides of the polygon are also chords of the circle, and the angles are inscribed angles.

Using only a compass, straight edge, pencil, and piece of paper, construct each of the following regular polygons. Then answer the questions at the bottom of the page.

Follow along with the video below to construct your own equilateral triangle inscribed in a circle.

Follow along with the video below to construct your own regular hexagon inscribed in a circle.

Follow along with the video below to construct your own square inscribed in a circle.

1. How can you be certain that the sides of each polygon are all the same length?

2. Why do you think the steps for constructing a regular hexagon and an equilateral triangle start off the same?

3. Do you think it is possible to construct other regular polygons inscribed in a circle (for example, a regular octagon)? If so, try a construction and explain your steps. If not, explain why not.

4. Do you think that there are any types of regular polygons that cannot be inscribed in a circle? Explain your reasoning.

Apply geometric theorems to verify properties of circles.

Use chords, tangents, and secants to find missing arc measures or missing segment measures.