# Geometry and the Circle Lesson

A circle is an important shape in the field of geometry. Let’s look at the definition of a circle and its parts. We will also examine the relationship between the circle and the plane.

A **circle** is a shape with all points the same distance from its center. A circle is named by its center. Thus, the circle to the right is called circle A since its center is at point A. Some real world examples of a circle are a wheel, a dinner plate and (the surface of) a coin.

The distance across a circle through the center is called the **diameter**. A real-world example of diameter is a 9-inch plate.

The **radius** of a circle is the distance from the center of a circle to any point on the circle. If you place two radii end-to-end in a circle, you would have the same length as one diameter. Thus, the diameter of a circle is twice as long as the radius.

We can look at a pizza pie to find real-world examples of diameter and radius. Look at the pizza to the right which has been sliced into 8 equal parts through its center. A radius is formed by making a straight cut from the center to a point on the circle. A straight cut made from a point on the circle, continuing through its center to another point on the circle, is a diameter. As you can see, a circle has many different radii and diameters, each passing through its center.

A **chord** is a line segment that joins two points on a curve. In geometry, a chord is often used to describe a line segment joining two endpoints that lie on a circle. The circle to the right contains **chord AB**. If this circle was a pizza pie, you could cut off a piece of pizza along chord AB. By cutting along chord AB, you are cutting off a segment of pizza that includes this chord.

It turns out that a diameter of a circle is the longest chord of that circle since it passes through the center. A diameter satisfies the definition of a chord, however, a chord is not necessarily a diameter. This is because every diameter passes through the center of a circle, but some chords do not pass through the center. Thus, it can be stated, **every diameter is a chord, but not every chord is a diameter**.

Let’s revisit the definition of a circle. A **circle** is the set of points that are equidistant from a special point in the plane. The special point is the center. In the circle to the right, the center is point A. Thus we have circle A.

A **plane** is a flat surface that extends without end in all directions. In the diagram to the right, Plane P contains points A, B and C.

Can you think of some real world objects that satisfy the definition of a plane? At this level of mathematics, that is difficult to do. Intuitively, a plane may be visualized as a flat infinite sheet of paper. The top of your desk and a chalkboard are objects which can be used to represent a plane, although they do not satisfy the definition above.

A circle divides the **plane** into three parts:

- the points INSIDE the circle
- the points OUTSIDE the circle
- and the points ON the circle

You can see an interactive demonstration of this by placing your mouse over the three items below.

A circle divides a plane into three parts:

- the points
**INSIDE**the circle - the points
**OUTSIDE**the circle - and the points
**ON**the circle

Example 1:

Name the center of this circle.

Answer: Point B

Example 2:

Name two chords on this circle that are not diameters.

Answer: DE and FG

Example 3:

Name all radii on this circle.

Answer: BA, BC, BD and BG

Example 4:

What are AC and DG?

Answer: AC and DG are diameters.

Example 5:

If DG is 5 inches long, then how long is DB?

Solution: The diameter of a circle is twice as long as the radius.

5 inches **รท** 2 = 2.5 inches

Answer: The length of DB is 2.5 inches

Summary: A circle is a shape with all points the same distance from its center. A circle is named by its center. The parts of a circle include a radius, diameter and a chord. All diameters are chords, but not all chords are diameters. A plane is a flat surface that extends without end in all directions. A circle divides the plane into three parts: The points inside the circle, the points outside the circle and the points on the circle.

**1. Which of the following is a chord, but not a diameter?**

**2. Which of the following is a radius?**

**3. Name the center of this circle.**

**4. What is PR (or PQR)?**

**5. If PQ is 3 cm long, then how long is PR?**