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Circles With and Without Coordinates

Area of a Sector

This lesson will explore the regions bounded by two radii of a circle and their intercepted arc. Additionally, the formula for the area of such regions will be derived considering the part-whole relationships.

Catch-Up and Review

Here is some recommended reading before getting started with this lesson.


Challenge

Investigating the Area of a Slice of Pizza

Three friends are sharing a inch pizza equally. Emily is becoming a nutritionist and is curious about how many calories are in a slice. To find out, she will calculate the area of one slice.

Help Emily to find the area of one slice. How can the measure of a central angle be used to find the area?

Discussion

Investigating the Area of Sectors of a Circle

Concept

Sector of a Circle

A sector of a circle is a portion of the circle enclosed by two radii and their intercepted arc.
Sector of a Circle
In the diagram, sector is created by and

Rule

Area of a Sector of a Circle

The area of a sector of a circle is calculated by multiplying the circle's area by the ratio of the measure of the central angle to

From the fact that equals an equivalent formula can be written if the central angle is given in radians.

Since the measure of an arc is equal to the measure of its central angle, the arc measures Therefore, by substituting for another version of the formula is obtained which can also be written in degrees or radians.


or

Proof

Consider sector bounded by and

sector of a circle

Since a circle measures this sector represents of Therefore, the ratio of the area of a sector to the area of the whole circle is proportional to

Recall that the area of a circle is By substituting it into the equation and solving for the area of a sector, the desired formula can be obtained. Therefore, the area of a sector of a circle can be found by using the following formula.

Example

Using Sectors of Circles to Solve Problems

Like the pizza problem, numerous real-life problems can be modeled by sectors of a circle.

Tiffaniqua has a trapezoid-shaped yard whose side lengths are shown on the diagram. To water the lawn, she sets up a water sprinkler that can water the grass within a meter radius. as shown.

Area of Sector

Tiffaniqua knows that measures

a Help Tiffaniqua calculate the area of the lawn covered by the sprinkler. If necessary, round the answer to the nearest square meter.
b What is the area of the lawn that is not watered? If necessary, round the answer to the nearest square meters.

Hint

a The sprinkler covers a degree sector of a circle with a radius of meters.
b The area of a trapezoid is half the product of the height and the sum of the two bases.

Solution

a The sprinkler covers a degree sector of a circle with radius meters. The area of a sector of a circle can be calculated using the following formula.
Substituting for and for into the formula will give the result.
Evaluate right-hand side
The area of the lawn covered is about square meters.
b To find the area of the lawn that is not watered, the area found in Part A should be subtracted from the area of the trapezoid.
Area of Sector
On the diagram, the trapezoid's bases are and meters, and its height is meters. Substitute these values into the formula for the area of a trapezoid.
Evaluate right-hand side
This means that the area that Tiffaniqua should water, initially, is square meters. By subtracting the area of the sector from the total area, the lawn area that is not watered can be found. Tiffaniqua needs to water an area of square meters.

Example

Calculating the Central Angle of Pac-Man's Mouth

When the area of a sector is given, the measure of the corresponding central angle can be calculated

Consider a two-dimensional image of Pac-Man. The area covered by Pac-Man is about square millimeters.

Pac-Man
If the radius of the circle used to draw Pac-Man is millimeters, find the measure of the central angle formed in the colored region. If necessary, round the answer to the nearest degree.

Hint

Pac-Man is essentially a sector of a circle. Use the formula for the area of a sector of a circle to find the measure of the angle.

Solution

To find the measure of the corresponding central angle, the formula for the area of a sector will be used. The area of the sector and the radius are given. Substitute these values into the formula.
Solve for
The corresponding central angle is about

Example

Comparing the Radius of Two Circles

The diagram below models the motion of two gears and Gear has a radius of inches.
Gears
a Find the radius of the larger gear.
b Find the area of the sector of Gear formed when Gear completes two revolutions. Write the answer in terms of

Hint

a How can the measure of the central angle formed in Gear be used?
b Use the formula for the area of a sector of a circle.

Solution

a When the smaller gear completes a single revolution, the central angle measured in the larger gear becomes Since the measure of an arc is the same as its corresponding central angle, the measure of the colored arc of is also
Furthermore, the length of the arc is equal to the circumference of Recall that the circumference is given by the formula Substituting into the formula will give the circumference of The length of the arc of is, therefore, inches. Now that the measure and length of the arc is known, the radius of can be found. To do so, substitute the values into the formula for the arc length.
Solve for
The radius of the larger gear is inches.
b As can be seen on the diagram, when Gear makes two complete revolutions, the central angle measures
Note that the arc created due to revolutions also measures In the previous part, the radius of is found as inches. Using the formula for the area of a sector, the sector of can be calculated.
Evaluate right-hand side
The area of the sector of is square inches.

Example

Using Areas Of Sectors to Solve Problems

In his free time, Dylan enjoys making decorative figures by hand. He has identical sectors and brings these sectors together as shown.

Sectors

Dylan knows that the area of each sector is square millimeters.

a Find the perimeter of the star-shaped figure. If necessary, round the answer to one decimal place.
b Find the perimeter of the figure. If necessary, round the answer to one decimal place.

Hint

a Notice that each side of the star-shaped figure is a radius.
b Start by finding the corresponding arc length of a sector.

Solution

a Notice that the sides of the star-shaped figure are the radii of the sectors.
Sectors
The value of the radius can be found using the formula for the area of a sector. Substitute for the area of the sector and for into the formula.
Solve for
Since identical radii form the figure, the perimeter is The perimeter of the star-shaped figure is millimeters.
b The perimeter of the figure created by Dylan is the sum of identical arc lengths.
Sectors
In Part A, the radii of the sectors were found to be about millimeters. The measure of the arc is because the corresponding central angle measures Now, the Arc Length Formula can be used.
Evaluate right-hand side
There are of these arc. Therefore, the figure has a perimeter of approximately millimeters.

Closure

Solving Real Life Problems Using Sectors of Circles

Mark set up a lamp in his courtyard. He uses a light bulb that illuminates a circular area with a radius of meters. The diagram shows a bird's eye view of Mark's house.

If the measure of arc is what is the area of the region that is illuminated outside of the courtyard area? If necessary, round the answer to two decimal places.

Hint

The area of a triangle is half the product of the lengths of any two sides and the sine of the included angle.

Solution

From the diagram, it can be seen that the region bounded by and is a sector of

The region bounded by and is called segment of the circle To find the area of the segment, the area of the triangle should be subtracted from the area of the sector The area of is half the product of the lengths of any two sides and the sine of the included angle. Since and are radii of and are meters. Moreover, the included angle measures because it intercepts a arc. Substitute these values.
Evaluate right-hand side
Next, the area of the sector will be calculated.
Evaluate right-hand side
Finally, the area of the region is the difference of and


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