Lesson 2: Exploring Circles

Let’s explore circles.

2.1: How Do You Figure?

Here are two figures.

Two figures on a grid labeled A and B. Figure A is a square with horizontal distance 6 units. Figure B is a curved figure with horizontal distance of approximately 4 units and vertical distance of approximately 9.5 units.

Figure C looks more like Figure A than like Figure B. Sketch what Figure C might look like. Explain your reasoning.

2.2: Sorting Round Objects

Your teacher will give you some pictures of different objects.

  1. How could you sort these pictures into two groups? Be prepared to share your reasoning.
  2. Work with your partner to sort the pictures into the categories that your class has agreed on. Pause here so your teacher can review your work.
  3. What are some characteristics that all circles have in common?
  4. Put the circular objects in order from smallest to largest.
  5. Select one of the pictures of a circular object. What are some ways you could measure the actual size of your circle?

2.3: Measuring Circles

Priya, Han, and Mai each measured one of the circular objects from earlier.

  • Priya says that the bike wheel is 24 inches.
  • Han says that the yo-yo trick is 24 inches.
  • Mai says that the glow necklace is 24 inches.
  1. Do you think that all these circles are the same size?
  2. What part of the circle did each person measure? Explain your reasoning.

2.4: Drawing Circles

Spend some time familiarizing yourself with the tools that are available in this applet.

GeoGebra Applet Y6dfR3GW

Draw and label each circle. 

  • Circle A, with a diameter of 6 cm.
  • Circle B, with a radius of 5 cm.
  • Circle C, with a radius that is equal to Circle A’s diameter.
  • Circle D, with a diameter that is equal to Circle B’s radius. Pause here so your teacher can review your work.
  • Use a compass to recreate one of these designs.


A circle consists of all of the points that are the same distance away from a particular point called the center of the circle.

A segment that connects the center with any point on the circle is called a radius. For example, segments $QG$, $QH$, $QI$, and $QJ$ are all radii of circle 2. (We say one radius and two radii.) The length of any radius is always the same for a given circle. For this reason, people also refer to this distance as the radius of the circle.

Two circles labeled circle 1 and circle 2. Circle 1 has a center at P and the points A, C, F, B, D, and E lie on the circle. Diameters A B, C D, and E F are drawn.  Circle 2 has a center at Q and the points G, H, I and J lie on the circle. Line segments Q G, Q H, Q I, and Q J are drawn.

A segment that connects two opposite points on a circle (passing through the circle’s center) is called a diameter. For example, segments $AB$, $CD$, and $EF$ are all diameters of circle 1. All diameters in a given circle have the same length because they are composed of two radii. For this reason, people also refer to the length of such a segment as the diameter of the circle.

The circumference of a circle is the distance around it. If a circle was made of a piece of string and we cut it and straightened it out, the circumference would be the length of that string. A circle always encloses a circular region. The region enclosed by circle 2 is shaded, but the region enclosed by circle 1 is not. When we refer to the area of a circle, we mean the area of the enclosed circular region.

Practice Problems ▶




The distance from the center of a circle to any point on the circle. Also the corresponding line segment from the center to a point on the circle.



A line segment that has endpoints on a circle and passes through the center is called a diameter of the circle. The length of this segment is also called the diameter.



The circumference of a circle is the distance around the circle. If you imagine the circle as a piece of string, it is the length of the string. If the circle has radius $r$ then the circumference is $2\pi r$.



A circle of radius $r$ with center $O$ is the set of all points that are a distance $r$ units from $O$. 

To draw a circle of radius 3 and center $O$, use a compass to draw all the points at a distance 3 from $O$.