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Course: High school geometry (staging) > Unit 9

Lesson 6: Sectors

Deriving the area of a sector

A sector is a fraction of circle defined by two radii. We can find its area by finding the area of the whole circle, then by using the central angle measure (in degrees or radians) to find the fraction of the total area that's inside the sector.

Sectors with angle measures in degrees

A sector is the interior of a circle between two radii.

There are

360 °

in a full circle. The central angle measure

θ

of a sector is between

0 °

and

360 °

, inclusive. Every sector has a fraction of the area of the full circle.

Area of unit fractions of a circle

Problem 1.1

Find the area of the shaded sector.
Enter an exact expression in terms of $π$ ‍.

{Area}_{sector} =

square units

Area of other fractions of a circle

How can we find the area of sectors that make up more complicated fractions of the circle? By definition, each degree is

\frac{1}{360}

of a turn about the full circle.

Problem 2.1

Find the area of the shaded sector.
Enter an exact expression in terms of $π$ ‍.

{Area}_{sector} =

square units

Area of any sector

Describe how you would find the area of a sector if you know the radius $r$ ‍ and the measure $θ$ ‍ of its central angle in degrees.

How did you explain it? Did you use words, like, "First find the area of the whole circle, then multiply it by the fraction of the circle inside the central angle?"

Did you write a formula, like:

A = \frac{θ}{360} π r^{2}

Did you create a proportional relationship, like:

\frac{A}{π r^{2}} = \frac{θ}{360}

In each of the equations, where do we see the area of the full circle? Where do we see the fraction of the circle that the sector takes up?

If you wrote something different, how does it relate to these equations?

Formula for sector area (degrees)

Write a formula for the area $A$ ‍ of a sector with a radius $r$ ‍ and a central angle measure of $θ$ ‍ degrees.
Do not enter the units in the equation.

Sectors with angle measures in radians

We can also measure the central angle of a sector using radians (the number of radius lengths in the sector's arc).

There are

2 π

radians in a full circle because the full circumference of a circle is

2 π

radius lengths long. The central angle measure

θ

of a sector is between

0

and

2 π

, inclusive. Every sector has a fraction of the area of the full circle.

Area of fractions of a circle using radians

Problem 4.1

Find the area of the shaded sector.
Enter an exact expression in terms of $π$ ‍.

{Area}_{sector} =

square units

Formula for sector area (radians)

Write a formula for the area $A$ ‍ of a sector with a radius $r$ ‍ and a central angle measure of $θ$ ‍ radians.
Do not enter the units in the equation.

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