Perimeter of a Sector of a Circle

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  • 0:00 Perimeter and Sector: Defined
  • 0:50 Calculating the Perimeter
  • 1:34 Finding the Radius
  • 2:29 Finding the Arc Length
  • 3:23 Example Problem
  • 4:57 Lesson Summary
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Lesson Transcript
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.

A sector of a circle is like a slice of pizza or pie. Finding the perimeter, or the distance around the outside edge, can be a little tricky. In this lesson, we will discuss how to find that perimeter and work some examples.

Perimeter and Sector Defined

It's an exciting event: ant races! You've set up a tiny track around the outside edge of a slice of pizza. You've decorated each ant with a different color and lined them up at the starting gate. You lift the tiny gate. They're off! The blue ant completes the path around the pizza slice in just four exciting seconds; but how far was it around the slice?

Perimeter is the distance along the outside edge of any flat object. When the racing ants follow the outside edge of the pizza slice, they are running along the perimeter of the surface. A sector of a circle is a slice of the circle, bordered by two different radius lines and the arc between them. It's like a slice of pizza. In this lesson, we'll figure out how to calculate the perimeter of that sector.

Calculating the Perimeter

Perimeter of sectors is calculated by adding all of the sides together. As part of a filled-in circle, a sector has three different sides that make up its outside edge. Two of them are radii, or segments of lines that extend from the center out to the edge of the circle. The third part is the arc, the curved part.


perimeter of a pizza slice
image of pizza slice perimeter


For example, say you have a piece of pizza that has a radius length of four inches and a curved outside edge of three inches. You can calculate the perimeter by adding the sides together.

perimeter = 2(4) + 3 = 11 inches

But, what if you're missing the arc or the radius? Either one can be calculated using other dimensions. Let's look at some of the ways you can get arc and radius.

Finding the Radius

Finding the radius is often a matter of dividing the diameter, or the distance all the way across the circle and going through the center, by two. For example, if you have a swimming pool with a diameter of 8 feet, then the radius would be half of 8, or 4 feet. Of course, for sector perimeter problems, you need two radius lengths, so you can just use the diameter, which is equal to 2 times the radius, anyway.

If you need to, you can get diameter from the circle perimeter (circumference). Since π is the magic ratio between the diameter and the circumference (perimeter) of a circle, you can use π (which is about 3.14) to get either one from the other. Diameter times π will give you the circumference, and circumference divided by π will give you the diameter. For example, if the perimeter of your circle is 25 feet, then the diameter would be 25 / π, or about 7.96.

Finding the Arc Length

Finding the arc length is usually a matter of using the perimeter of the circle along with the proportion of the sector to the circle. If you know how much of the circle the sector represents, then that will tell you how much of the perimeter is within the sector.

Let's start with Step 1. If you know that the arc is one-fourth of the circle, then you can multiply the perimeter by 1/4 (or divide by 4) to get the arc length. For example, if a sector is 1/4 of a 500 perimeter, then the arc of the sector is 500 / 4, or 125.

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