Finding the Area of a Sector: Formula & Practice Problems

Finding the Area of a Sector: Formula & Practice Problems
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  • 0:05 Definitions
  • 0:39 Calculating the Area…
  • 1:44 Calculating Area Using Radians
  • 2:42 Area and Known…
  • 3:22 Finding Area: Example
  • 5:48 Lesson Summary
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Lesson Transcript
Instructor: Joseph Vigil
In this lesson, you'll review your knowledge of circle vocabulary and then discover how to calculate the area of a sector of a circle. The lesson covers finding the area of the sector using degrees and radians.

Definitions

What are the parts of a circle?

  • The circle's midpoint is just what it sounds like: the point right in the circle's center.
  • A circle's radius is a line extending from its midpoint to its exterior.
  • A circle's diameter is a line that runs through the circle and passes through its midpoint. Notice that the diameter is really two radii put together; so a circle's radius is always half the length of its diameter, and its diameter is always twice the length of its radius.
  • Lastly, a sector is simply a wedge, or pie piece, of a circle. The sector always originates at the circle's midpoint.

Calculating the Area Using Degrees

Usually, the formula A = pi *r2 is used to calculate a circle's area. However, when finding a portion of the area, that portion can be determined using the sector's angle.

An entire circle encompasses 360 degrees, so the ratio of the sector's angle measurement to 360 degrees is proportional to the fraction of the circle's area being measured. This equation can be written as: sector angle / 360. In other words, if the sector's angle happens to measure 90 degrees, the portion of the circle being measuring is 90 / 360, or 1/4.

Another example: if the sector's angle measures 60 degrees, the fraction would be 60 over 360, or 1/6 of the circle.

So, in order to find the area of a sector, multiply the formula for a circle's area by the portion of the circle that is being calculated. The formula for a sector's area is:

A = (sector angle / 360 ) * (pi *r2)

Calculating Area Using Radians

If dealing with radians rather than degrees to measure the sector angle, the general method of finding the sector's area remains the same. Radians are units used for measuring angles. An entire circle encompasses 2 pi radians, so the fraction of 2 pi is the fraction of the circle's area that is being calculated.

In other words, if the sector's angle happens to measure pi/2 radians, the fraction of the circle being measured is pi/2 divided by 2pi, or 1/4.

If the sector's angle measures pi radians, then measure pi / 2pi, or 1/2, of the circle.

So, to find the area, multiply the circle's area by the fraction of the circle that is being dealt with. Just use radians in place of degrees. The formula for a sector's area in radians is:

A = (sector angle / (2*pi)) * (pi * r2)

Area and Known Portions of a Circle

Sometimes, the portion of a circle is known. In this case, don't divide degrees or radians by any value.

For example, if the known sector is 1/8 of a circle, then simply multiply the formula for the area of a circle by 1/8.

This formula would work with any given portion of a circle. Even if the known sector is 1/160 of a circle, multiply the formula for the area of a circle by 1/160.

So, when given the portion of the circle, the formula for the sector's area is:

A = (fraction of the circle) * (pi * r2)

Finding Area: Example

For this example, imagine an individual has been hired to mow part of a circular field that's 80 feet across. The owner wants a 45 degree sector cleared. But how much area will actually be mowed?

The sector's angle measurement is 45 degrees. Its diameter measures 80 feet. Since the radius is simply half the diameter, this circle's radius measures 40 feet. This is all the information needed to calculate this sector's area. In the examples, approximate pi as 3.14.

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