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Segment of a Circle: Definition & Formula

Segment of a Circle: Definition & Formula
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  • 0:02 Definitions
  • 2:07 Area of a Segment
  • 4:07 Examples of Finding…
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Lesson Transcript
Instructor: Beverly Maitland-Frett

Beverly has taught mathematics at the high school level and has a doctorate in teaching and learning.

A circle is an important geometric shape in our world and is very relevant to our everyday experiences. In this lesson, we will explore one aspect of the circle, the segment. In addition, we will also discuss a few other vocabulary words related to the circle.

Definitions

Do you like pizza? Most people do. Some people even like to make their own pizza. Let's image that you just took a circular pizza out of the oven, and you cut it into two equal parts. You would have made two segments. Each half has two parts: a straight line and a curved part. Each of those parts have names: a chord and an arc, respectively.

Before we begin our discussion about segments, let's review some important vocabulary related to the circle. Here are some reminders:

  • The diameter (D) is two times the length of the radius (r); that means D = 2 * r.
  • The radius of a circle is the line segment formed from the center of the circle to any point on the circumference.

As you can see in the diagram, the point O is the center of the circle, and line segment AB is the diameter. There are also three radii: line segments OB, OA, and OC.

Parts of a circle

  • A chord is a line segment whose endpoints lie on the circumference of the circle. Remember that the circumference is the total distance around the circle (all of the crust of your pizza). In other words, the chord is the line formed when one point on the circumference is connected to another point on the circumference (or crust). Note also that the diameter of a circle is the biggest chord.
  • An arc is any curved part on the circumference of the circle. When you have a slice of pizza and eat the crust, you're having the arc.
  • A segment is the section of a circle enclosed by a chord and an arc. Therefore, those halves of the pizza are segments. If you eat one half, you would have eaten a semicircle (half of a circle), which is the biggest segment of a circle.

Since a circle has an infinite number of points on the circumference, there are many possibilities for a chord and, hence, many possibilities for segments.

Segment

When a circle is divided into two segments of different areas, the biggest segment is called the major segment and the smaller segment is called the minor segment.

Major Segment

Area of a Segment

Now, if you order pizza, how does it usually come? Yes, pre-sliced. Those slices look like triangles with a curve, right? Those are sectors. A sector of a circle is the section enclosed by two radii (the two sides of the slice) and an arc (the crust).

We can divide a sector so that it looks like a triangle and a segment.

Sector with Segment

We divided the sector so that a triangle is formed and a segment is formed. If we were to consider the total area of that sector, we could say that the total area of the sector is made up of the area of the triangle and the area of the segment. Some people don't like the crust of the pizza. So when they get a slice, they cut off the crust. However, their full slice includes the crust, plus the part that is left. Therefore, we can say that a sector is made up of the triangle and the segment (the crust).

Using that example, we can find the area of a segment using the formula:

Area of Segment = Area of Sector - Area of Triangle

First, we find the area of the sector, which would be like finding the area of your slice of pizza. Next, we would find the area of the triangle (the part left after you cut off the crust). If we know the area of the full slice of pizza and we know the area of the triangle, then what is left must be the area of the crust. This is the same principle used to find the area of the segment represented by the crust. The formulas for each are as follows:

Areas of a triangle and a sector

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