7. Area of a Sector
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Chapter 5
7.
Area of a Sector
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.
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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.
Challenge
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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.
Discussion
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Investigating the Area of Sectors of a Circle
Concept
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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 ACB is created by AC, BC, and AB.
Rule
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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
Example
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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
SubstituteII
θ= 135^(∘), r= 4
Area of Sector = 135^(∘)/360^(∘) * π ( 4)^2
Area of Sector ≈ 19
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 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
Example
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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.
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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.
The corresponding central angle is about 305^(∘).
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^(∘)
Example
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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. 
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.
The radius of the larger gear is 5 inches.
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.
The area of the sector of ⊙ L is 20π square 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^(∘).
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
b As can be seen on the diagram, when Gear S makes two complete revolutions, the central angle measures 288^(∘).
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
Example
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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.
b The perimeter of the figure created by Dylan is the sum of 5 identical arc lengths.
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
Closure
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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.
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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.
A_(triangle) = 1/2* MP * NP * sin(θ)
SubstituteValues
Substitute values
A_(triangle) = 1/2* 6 * 6 * sin( 100^(∘))
A_(triangle) ≈ 17.73
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
A_(segment) = A_(sector)-A_(triangle)
SubstituteValues
Substitute values
A_(segment) ≈ 31.42 - 17.73
SubTerm
Subtract term
A_(segment) ≈ 13.69
Area of a Sector
Exercise 2.1
Davontay is ordering pizzas from La Pizza Surgelata for seven friends and himself. The restaurant sells three different sizes of pizza — small, medium, and large — each for a different price.
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From the given information, we know that three people who share a medium pizza are satisfied. That means one portion from the medium pizza with a central angle of 120^(∘) is enough to meet their expectations.
If we calculate the area of this portion of the pizza, we can determine how many square inches of pizza each person would want. Notice that the diameter of a medium pizza is 12 inches which means the radius is 6 inches.
Davontay has to make sure his friends get at least 12π square inches of pizza each. If we multiply this by 8, we get the total amount of pizza Davontay should order for the party. 12π( 8)=96π
Area of Small and Large Pizzas
Next, we will determine the area of each type of pizza. Remember, the pizzeria is out of medium pizzas. A small and a large pizza have a radius of r= 5 and r= 7 respectively.
| Pizza | A=π r^2 | Evaluate |
|---|---|---|
| Small | A=π( 5)^2 | A=25π |
| Large | A=π ( 7)^2 | A=49π |
Let's list different orders that give at least the desired amount of pizza Davontay's group needs to fill their appetite.
| Order | Total Area | Evaluate |
|---|---|---|
| 4 small | 4(25π) | 100π |
| 2 small, 1 large | 2(25π)+49π | 99π |
| 2 large | 2(49π) | 98π |
Each order's total area is greater than 96π. Therefore, each order fulfills the group's appetite.
Which Pizza Order is the Cheapest?
While the orders meet the needs of their appetite, we want to choose the order that costs the least amount of money. Let's calculate each orders total cost.
| Order | Total Cost | Evaluate |
|---|---|---|
| 4 small | 4($5.99) | $23.96 |
| 2 small, 1 large | 2($5.99)+$9.99 | $21.97 |
| 2 large | 2($9.99) | $19.98 |
As we can now see, Davontay should choose to order 2 large pizzas.
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A group of friends is planning to build a game house. Those long game days will require them to have food nearby. In that case, they decided that outside the house they will have a garden that consists of a square, a rectangle, and a quarter circle as marked in the diagram.
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In order to calculate the area of the three parts, we need to know the value of x.
Calculating x
In the exercise, it is mentioned that the garden consists of a square, a rectangle, and a quarter of a circle. Notice that the radii that creates the quarter circle, also makes up a side of the square, as well as a side of the rectangle.
Since the radii have the same measure, we can equate these expressions and solve for x.
Calculating Areas
When we know that x=30, we can calculate the dimensions of the garden. 3(30)-50&=40 feet 30+10&=40 feet 2(30)&=60 feet Let's add this to the diagram. We will also include the area of the square and rectangle.
We also need to find the area of the quarter circle. This is a sector of a circle with a central angle with a measure of θ=90^(∘). The area of the sector of a circle can be determined with the following formula. A=θ/360^(∘)* π r^2 Let's substitute the angle measure and the radius of the quarter circle and evaluate.
Let's add the area of the quarter circle to the diagram.
Area of the Garden
Finally, by adding all of the individual areas, we can determine the garden's total area. As instructed in the exercise, we will round to the nearest square foot. 40^2+400π+(60)(40)≈ 5257 feet^2
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What is the area of the shaded region? Round the answer to two decimal places.
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The shaded area equals the area of the sector minus the area of the triangle formed by the radii, as illustrated in the following diagram.
With the given information in mind, let's consider the characteristics of the following diagram.
The radius of the circle is 4 centimeters. The marked arc has a measure of 60^(∘), which means the corresponding central angle also has the same measure. Additionally, since vertical angles are congruent, the central angle that relates to our sector also has a measure of 60^(∘).
Area of the Sector
The area of a sector with a central angle θ can be calculated using the following formula. A=θ/360^(∘)* π r^2 Let's substitute the radius and measure of the central angle into this equation and solve for A.
Area of the Triangle
To calculate the area of the triangle, let's consider what we know and what we can derive from the given characteristics of the diagram. As the side lengths are equal, we are able to recognize that this is an isosceles triangle with a vertex angle of 60^(∘). Furthermore, according to the Base Angles Theorem, the base angles are congruent.
Using the Interior Angles Theorem, we can determine the measure of the base angles. m∠ x+m∠ x+60^(∘)=180^(∘) ⇓ m∠ x=60^(∘) Since all angles have a measure of 60^(∘), we know that we have an equilateral triangle.
If we draw the altitude of such a triangle, we get a 30-60-90 triangle. The leg opposite the 30^(∘)-angle will have half the length of the hypotenuse. For our triangle it will have the length 2.
In 30-60-90 triangles, the longer leg is sqrt(3) times greater than the shorter leg. Here it has a length of 2sqrt(3).
We can now calculate the area of the triangle.
Blue Area
It is now time to find the area of the blue sector by subtracting the area of the triangle from the area of the sector. 8π/3- 4sqrt(3)≈ 1.45 cm^2 Bravo.
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The following sectors have the same area.
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We know that the sectors have the same area. Therefore, if we calculate the area of the sector with the smaller central angle, we can then use the result to find the value of r.
We can now use the same formula for the sector with the greater central angle. Let's substitute the area we just calculated together with the central angle of the sector and solve for r.
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Area of a Sector
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