7. Area of a Sector
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Chapter 5
7.
Area of a Sector
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Why
A sector in a circle is like a piece of pie, and understanding how much space it occupies is essential in various fields. The lesson elaborates on how to determine the area of these sectors. Knowing the area of a sector is particularly useful in designing objects like clocks, fans, and protractors. It also has practical applications in sectors like agriculture, where farmers might need to measure the area of a partial circular plot. By delving into the concept, one can grasp the nuances of spaces in circular forms, aiding in both academic pursuits and real-world problem-solving.
Lesson Settings & Tools
| | 8 Theory slides |
| | 11 Exercises - Grade E - A |
| | Each lesson is meant to take 1-2 classroom sessions |
Area of a Sector
Slide of 8
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.
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Challenge
Investigating the Area of a Slice of Pizza
Three friends are sharing a 16-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.
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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.
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 360^(∘).
Area of Sector = θ/360^(∘) * π r^2
From the fact that 2πrad equals 360^(∘), an equivalent formula can be written if the central angle is given in radians.
Area of Sector &= θ/2π * π r^2 &⇓ Area of Sector &= θ/2* r^2
Since the measure of an arc is equal to the measure of its central angle, the arc AB measures θ. Therefore, by substituting mAB for θ, another version of the formula is obtained which can also be written in degrees or radians.
Area of Sector = mAB/360^(∘) * π r^2
or
Area of Sector = mAB/2* r^2
Proof
Consider sector ACB bounded by AC, BC, and AB.
Since a circle measures 360^(∘), this sector represents θ360^(∘) of ⊙ C. Therefore, the ratio of the area of a sector to the area of the whole circle is proportional to θ360^(∘). Area of Sector/Area of Circle = θ/360^(∘)
Recall that the area of a circle is π r^2. By substituting it into the equation and solving for the area of a sector, the desired formula can be obtained.
Area of Sector/π r^2 = θ/360^(∘)
Therefore, the area of a sector of a circle can be found by using the following formula.
Area of Sector = θ/360^(∘) * π r^2
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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 4-meter radius. as shown.
Tiffaniqua knows that ∠ ABC measures 135^(∘).
a Help Tiffaniqua calculate the area of the lawn covered by the sprinkler. If necessary, round the answer to the nearest square meter.
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b What is the area of the lawn that is not watered? If necessary, round the answer to the nearest square meters.
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Hint
a The sprinkler covers a 135-degree sector of a circle with a radius of 4 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 135-degree sector of a circle with radius 4 meters. The area of a sector of a circle can be calculated using the following formula.
Area of Sector = θ/360^(∘) * π r^2 Substituting 135^(∘) for θ and 4 for r into the formula will give the result.
Area of Sector = θ/360^(∘) * π r^2
SubstituteII
θ= 135^(∘), r= 4
Area of Sector = 135^(∘)/360^(∘) * π ( 4)^2
Area of Sector ≈ 19
The area of the lawn covered is about 19 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.
On the diagram, the trapezoid's bases are 5 and 9 meters, and its height is 4 meters. Substitute these values into the formula for the area of a trapezoid.
Area of Trapezoid = 1/2h(b_1+b_2)
SubstituteValues
Substitute values
Area of Trapezoid = 1/2 4( 5+ 9)
▼
Evaluate right-hand side
AddTerms
Add terms
Area of Trapezoid = 1/2 4(14)
Multiply
Multiply
Area of Trapezoid = 1/256
MoveRightFacToNumOne
1/b* a = a/b
Area of Trapezoid = 56/2
CalcQuot
Calculate quotient
Area of Trapezoid =28
This means that the area that Tiffaniqua should water, initially, is 28 square meters. By subtracting the area of the sector from the total area, the lawn area that is not watered can be found. 28-19 = 9 Tiffaniqua needs to water an area of 9 square meters.
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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 66.5 square millimeters.
If the radius of the circle used to draw Pac-Man is 5 millimeters, find the measure of the central angle formed in the colored region. If necessary, round the answer to the nearest degree.
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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. Area of Sector = θ/360^(∘)* π r^2
The area of the sector and the radius are given. Substitute these values into the formula.
Area of Sector = θ/360^(∘)* π r^2
SubstituteValues
Substitute values
66.5 = θ/360^(∘)* π ( 5)^2
▼
Solve for θ
CalcPow
Calculate power
66.5 = θ/360^(∘)* π * 25
Multiply
Multiply
66.5 = θ/360^(∘)* 25π
DivEqn
.LHS /25π .=.RHS /25π .
66.5/25 π = θ/360^(∘)
UseCalc
Use a calculator
0.846704 ... = θ/360^(∘)
MultEqn
LHS * 360^(∘)=RHS* 360^(∘)
304.813547 ... ^(∘) = θ
RearrangeEqn
Rearrange equation
θ = 304.813547 ... ^(∘)
RoundInt
Round to nearest integer
θ ≈ 305^(∘)
The corresponding central angle is about 305^(∘).
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Example
Comparing the Radius of Two Circles
The diagram below models the motion of two gears S and L. Gear S has a radius of 2 inches.
a Find the radius of the larger gear.
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b Find the area of the sector of Gear L formed when Gear S completes two revolutions. Write the answer in terms of π.
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Hint
a How can the measure of the central angle formed in Gear L 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 144^(∘). Since the measure of an arc is the same as its corresponding central angle, the measure of the colored arc of ⊙ L is also 144^(∘).
Furthermore, the length of the arc is equal to the circumference of ⊙ S. Recall that the circumference is given by the formula C= 2π r. Substituting r= 2 into the formula will give the circumference C_S of ⊙ S. C_S & = 2 π ( 2) & = 4 π The length of the arc of ⊙ L is, therefore, 4π inches. Now that the measure and length of the arc is known, the radius R of ⊙ L can be found. To do so, substitute the values into the formula for the arc length.
Arc length/Circumference of ⊙ L= Arc measure/360^(∘)
SubstituteValues
Substitute values
4π/2π R= 144^(∘)/360^(∘)
▼
Solve for R
ReduceFrac
a/b=.a /2π./.b /2π.
2/R= 144^(∘)/360^(∘)
UseCalc
Use a calculator
2/R= 0.4
MultEqn
LHS * R=RHS* R
2 = 0.4 R
DivEqn
.LHS /0.4.=.RHS /0.4.
5 = R
RearrangeEqn
Rearrange equation
R= 5
The radius of the larger gear is 5 inches.
b As can be seen on the diagram, when Gear S makes two complete revolutions, the central angle measures 288^(∘).
Note that the arc created due to revolutions also measures 288^(∘). In the previous part, the radius of ⊙ L is found as 5 inches. Using the formula for the area of a sector, the sector of ⊙ L can be calculated.
Area of Sector = θ/360^(∘)* π R^2
SubstituteII
θ= 288^(∘), R= 5
Area of Sector = 288^(∘)/360^(∘)* π ( 5)^2
Area of Sector = 20 π
The area of the sector of ⊙ L is 20π square inches.
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Example
Using Areas Of Sectors to Solve Problems
In his free time, Dylan enjoys making decorative figures by hand. He has 5 identical sectors and brings these sectors together as shown.
Dylan knows that the area of each sector is 18 square millimeters.
a Find the perimeter of the star-shaped figure. If necessary, round the answer to one decimal place.
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b Find the perimeter of the figure. If necessary, round the answer to one decimal place.
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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.
The value of the radius r can be found using the formula for the area of a sector. Substitute 18 for the area of the sector A, and 108^(∘) for θ into the formula.
Since 10 identical radii form the figure, the perimeter is 10r. r = 4.4 ⇔ 10r = 44 The perimeter of the star-shaped figure is 44 millimeters.
b The perimeter of the figure created by Dylan is the sum of 5 identical arc lengths.
In Part A, the radii of the sectors were found to be about 4.4 millimeters. The measure of the arc is 108^(∘) because the corresponding central angle measures 108^(∘). Now, the Arc Length Formula can be used.
Arc Length = θ/360^(∘)* 2 π r
SubstituteII
θ= 108^(∘), r= 4.4
Arc Length = 108^(∘)/360^(∘)* 2 π ( 4.4)
Arc Length ≈ 8.3
There are 5 of these arc. 5 * 8.3 ≈ 41.5 Therefore, the figure has a perimeter of approximately 41.5 millimeters.
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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 6 meters. The diagram shows a bird's eye view of Mark's house.
If the measure of arc MN is 100^(∘), what is the area of the region that is illuminated outside of the courtyard area? If necessary, round the answer to two decimal places.
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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 MP, NP, and MN is a sector of ⊙ P. 
The region bounded by MN and MN is called segment of the circle P. To find the area of the segment, the area of the triangle MPN should be subtracted from the area of the sector MPN.
A_(segment) = A_(sector) - A_(triangle)
The area of △ MPN is half the product of the lengths of any two sides and the sine of the included angle. A_(triangle) = 1/2* MP * NP * sin(θ)
Since MP and NP are radii of ⊙ P, MP and NP are 6 meters. Moreover, the included angle MPN measures 100^(∘) because it intercepts a 100^(∘) arc. Substitute these values.
A_(triangle) = 1/2* MP * NP * sin(θ)
SubstituteValues
Substitute values
A_(triangle) = 1/2* 6 * 6 * sin( 100^(∘))
A_(triangle) ≈ 17.73
Next, the area of the sector MNP will be calculated.
A_(sector) = θ/360^(∘)* π r^2
SubstituteII
r= 6, θ= 100^(∘)
A_(sector) = 100^(∘)/360^(∘)* π ( 6)^2
▼
Evaluate right-hand side
CalcPow
Calculate power
A_(sector)= 100^(∘)/360^(∘)* π (36)
MoveRightFacToNum
a/c* b = a* b/c
A_(sector) = 100^(∘) * 36/360^(∘)* π
Multiply
Multiply
A_(sector)= 3600^(∘)/360^(∘) * π
CalcQuot
Calculate quotient
A_(sector)= 10 * π
UseCalc
Use a calculator
A_(sector) = 31.415926 ...
RoundDec
Round to 2 decimal place(s)
A_(sector) ≈ 31.42
Finally, the area of the region A_R is the difference of A_S and A_T.
A_(segment) = A_(sector)-A_(triangle)
SubstituteValues
Substitute values
A_(segment) ≈ 31.42 - 17.73
SubTerm
Subtract term
A_(segment) ≈ 13.69
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Area of a Sector
Exercise 3.1
The diagram shows three circles that are congruent and have a radius of 1 centimeter.
What percentage of the square does the shaded area with the diagonal lines occupy? Round the answer to the nearest whole percent.
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To determine the percentage that the striped area occupies of the rectangle, we ought to find values for the rectangle's area A_R and the striped area A_(Strpd). The rectangle's area is the product of its width and length. A_R=(4)(2+sqrt(3)) cm^2 ⇕ A_R= 14.92820... cm^2 The striped area is the rectangle's area minus the sum of the area of the three circles and the area of the region between the circles. Therefore, we will also need to find the areas of the circles and the area between them. We know that each circle has a radius of 1 centimeter. We can determine the circles' combined area by calculating the area of one circle A_C and multiplying it by 3. 3A_C=3(π(1)^2) ⇔ 3A_C= 3π cm^2
Area Between the Circles
To determine the area of the region between the circles, we will draw radii to the points of tangency between the circles. Since these radii meet at right angles, they form straight line segments. Additionally, the radii are all congruent which means these segments form an equilateral triangle. In such a triangle all angles are congruent with a measure of 60^(∘).
Further examining the diagram, we notice that the area of the region between the circles is the difference between the triangle's area and the area of three sectors, each with a central angle of 60^(∘).
As stated, we need to find the area of a sector A_S in order to find the area of the region between the circles. To do so, we can use the following formula.
A = θ/360^(∘)* π r^2
Let's substitute θ for 60^(∘) and r for the value 1. Notice that we have three identical sectors. To account for the value of all three sectors, we will multiply both sides of the formula by 3.
Continuing our quest to obtain the area of the region between the circles, we need to find the area of the triangle. By drawing the altitude of such an equilateral triangle, we see that this is a 30-60-90 triangle. Therefore, the leg opposite the 60^(∘)-angle is sqrt(3) times greater than the leg opposite the 30^(∘)-angle. Additionally, the smaller leg is half the length of the hypotenuse.
We have now collected enough measurements to calculate the area of the triangle A_T.
Calculating the Area Between the Circles
To find the area of the region between the circles, we will subtract the area of the three sectors 3A_s from the area of the triangle A_T.
Percentage of Striped Area within Rectangle's Area
Now we have everything we need to determine the striped area A_(strpd). It equals the rectangle's area subtracted by the sum of the circles and the area of the region between the circles. We have found each of those variables. Let's calculate the striped area.
Alas, by dividing the area of the striped area by the area of the rectangle, we can find the percentage that the striped area makes up of the rectangle's area. 5.34217.../14.92820...≈ 0.36 = 36 %
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Area of a Sector
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