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Geometry

Learners calculate the area of a circle, the radius of a circle, and the measure of a circle’s central angle.

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After completing this tutorial, you will be able to complete the following:

- Calculate the area of a sector in a circle when the central angle is given in degrees.
- Calculate the area of a sector in a circle when the central angle is given in radians.
- Calculate the area of a sector, radius or central angle of a circle when two of them are given.

The area of the circle

The area of a circle can be thought of as the number of square units inside the circle.

The area of a circle can be found using the formula below:

Here ? (pi) is the ratio of the circumference of any circle to its diameter, and it can be approximated as 3.14. And r is the radius of the circle measured in some unit of length (for example, centimeters). The resulting area is then measured in units of area corresponding to the unit of length chosen (that is to say, the area of a square with side length 1 cm).

The area of a sector is only a part of the area of an entire circle.

A sector is defined as the region enclosed by two radii of a circle and their subtended arc; its area is a portion of the circle's entire area. Using proportional thinking, we see that the area of the circle is to the area of a sector as the central angle of the circle, , is to the angle between the radii defining the sector, .

For example, if the measure of the central angle of the sector is , then the area of the sector is of the area of the circle, or , since is one-fourth of (which represents the total number of degrees in a circle).

Calculating the Area of a Sector: When the central angle is in degrees:

To find the area of a sector of a circle of radius of 4 centimeters and central angle measure of :

Step 1: Find the area of the circle.

The area of the circle is equal to the radius squared times pi . Using this formula, and approximating , the area of the circle is .

Step 2: Use the proportional relationship.

Or since the area of a sector is just a portion of the circle's entire area, multiply the area of the circle by the ratio of the central angle of the sector to 360 (degrees in a circle):

Calculating the Area of a Sector: When the central angle is in radians:

To find the area of the sector of a circle of radius 2 centimeters and central angle measure of radians.

Step 1: Find the area of the circle

The area of the circle is equal to the radius square times . Using this formula, and approximating , the area of the circle is

Step 2: Use the proportional relationship.

Calculating the Area of a Sector: When the central angle is in radians (simplified version):

If we look at the general formula that we developed for the area of a sector, we'll see that we can simplify it when the central angle is in radians:

NOTE: These formulas can also be used to find the radius of the circle, or the central angle when given the other information.

For example, consider the following problem:

If the area of a sector of a circle with radius of 3 cm is , then what is the measure of the central angle?

First, substitute the known information into the formula

To solve for ?, multiply each side of the equation by 2, and then divide by 9:

Approximate Time | 25 Minutes |

Pre-requisite Concepts | Calculate the area of a circle; understand the concept of pi and the concept of the central angle of a circle; solve simple proportions |

Course | Geometry |

Type of Tutorial | Procedural Development |

Key Vocabulary | area, area of a sector, central angle |