{{ tocSubheader }}
{{ 'ml-label-loading-course' | message }}
An error ocurred, try again later!
Chapter {{ article.chapter.number }}
{{ article.number }}.
{{ article.displayTitle }}
{{ article.intro.summary }}
Show less Show more expand_more Lesson Settings & Tools
| | {{ 'ml-lesson-number-slides' | message : article.intro.bblockCount }} |
| | {{ 'ml-lesson-number-exercises' | message : article.intro.exerciseCount }} |
| | {{ 'ml-lesson-time-estimation' | message }} |
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.
Error loading image.
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. 
In the diagram, sector is created by and $\overgroup{AB}.$
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 $m\overgroup{AB}$ for another version of the formula is obtained which can also be written in degrees or radians.
$\text{Area of Sector} = \dfrac{m\overgroup{AB}}{360^\circ} \cdot \pi r^2$
or
$\text{Area of Sector} = \dfrac{m\overgroup{AB}}{2}\cdot r^2$
Proof
Consider sector bounded by and $\overgroup{AB}.$
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.
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.
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":["x"],"constants":["PI"]},"rendered":false},"text":""},"formTextBefore":null,"formTextAfter":null,"answer":{"text":["19"]}}
b What is the area of the lawn that is not watered? If necessary, round the answer to the nearest square meters.
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":["x"],"constants":["PI"]},"rendered":false},"text":""},"formTextBefore":null,"formTextAfter":null,"answer":{"text":["9"]}}
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.
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.
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.
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":["x"],"constants":["PI"]},"rendered":false},"text":""},"formTextBefore":null,"formTextAfter":null,"answer":{"text":["305"]}}
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.
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. 
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.
The radius of the larger gear is inches.
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.
The area of the sector of is square inches.
a Find the radius of the larger gear.
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":["x"],"constants":["PI"]},"rendered":false},"text":""},"formTextBefore":null,"formTextAfter":null,"answer":{"text":["5"]}}
b Find the area of the sector of Gear formed when Gear completes two revolutions. Write the answer in terms of
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":false,"useShortLog":false,"variables":["x"],"constants":["PI"]},"rendered":false},"text":""},"formTextBefore":null,"formTextAfter":null,"answer":{"text":["20 \\pi"]}}
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
b As can be seen on the diagram, when Gear makes two complete revolutions, the central angle measures
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.
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.
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":["x"],"constants":["PI"]},"rendered":false},"text":""},"formTextBefore":null,"formTextAfter":null,"answer":{"text":["44"]}}
b Find the perimeter of the figure. If necessary, round the answer to one decimal place.
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":["x"],"constants":["PI"]},"rendered":false},"text":""},"formTextBefore":null,"formTextAfter":null,"answer":{"text":["41.5"]}}
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.
b The perimeter of the figure created by Dylan is the sum of identical arc lengths.
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.
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":["x"],"constants":["PI"]},"rendered":false},"text":""},"formTextBefore":null,"formTextAfter":null,"answer":{"text":["13.69"]}}
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 $\overgroup{MN}$ is a sector of
Edit Lesson
Loading content