8. Circles
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Chapter 9
8.
Circles
| Student Learning Objectives: |
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Why
Circles are fundamental in geometry and have diverse real-world applications, from architecture to engineering. This lesson provides insights into understanding key properties of circles, including how to measure their circumference and area. It explains how the number pi relates to circles and explores its significance in defining their boundaries and enclosed regions. Whether for academics or practical use, it helps build an intuitive grasp of circular shapes and their dimensions.
Lesson Settings & Tools
| | 16 Theory slides |
| | 17 Exercises - Grade E - A |
| | Each lesson is meant to take 1-2 classroom sessions |
Circles
Slide of 16
The world is full of circles. They are everywhere. Try to look around right now — it is very likely to find at least one circle. For this reason, it might be useful to learn about them and their dimensions. Grab some paper and a pencil and buckle up, because it is about to begin!
Catch-Up and Review
Here are a few recommended readings before getting started with this lesson.
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Explore
Comparing Circle Dimensions
For his birthday, Tadeo's parents are taking him to a Detroit Pistons game, his favorite NBA team. Tadeo plans to bring some money to buy souvenirs at the stadium. He asks his mother to change the coins in his piggy bank to bills to make it easier to take his money with him.
His mother tells him that the coins all have something in common. She works with Tadeo to measure the distance around the coins and the width of their faces. Tadeo then calculates the ratio of the distance around each coin to its width.
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Discussion
Circle
A circle is the set of all the points in a plane that are equidistant from a given point. There are a few particularly notable features of a circle.
- Center - The given point from which all points of the circle are equidistant. Circles are often named by their center point.
- Radius - A segment that connects the center and any point on the circle. Its length is usually represented algebraically by r.
- Diameter - A segment whose endpoints are on the circle and that passes through the center. Its length is usually represented algebraically by d.
- Circumference - The perimeter of a circle, usually represented algebraically by C.
The following circle can be referred to as ⊙ O, or circle O,
since it is centered at O.
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Pop Quiz
Identifying the Parts of a Circle
Click on the indicated part of the given circle.
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Discussion
The Number Pi
When Tadeo opened his piggy bank and compared the measures of the coins, he found that dividing the circumference by the diameter always resulted in the same number. This fact is true for all circles and is so important that mathematicians gave a unique name to this number.
Concept
Pi
The number pi, usually written with the Greek letter π,
is a constant defined as the ratio between the circumference and the diameter of a circle. This ratio is the same for all circles.
π=3.1415926...
Graphically, π is the number of times that the diameter of the circle fits on top of the circle.
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Discussion
Calculating the Circumference of a Circle
Back in his room, Tadeo wants to measure the circumference of one of his coins. He does not have a tape measure and wonders whether there is another way to find the circumference of a circle. Maybe he could use a ruler? Good news! He can find this information if he knows either the circle's diameter or its radius. Remember, the diameter of a circle is twice the radius.
Rule
Circumference of a Circle
The circumference of a circle is calculated by multiplying its diameter by π.
C=π d
Since the diameter is twice the radius, the circumference of a circle can also be calculated by multiplying 2r by π.
C=2π r
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Example
Basketball Court
Tadeo is finally at the Detroit Pistons stadium with his parents. During the halftime show, he was able to go onto the court for a contest. There, he realized how big the logo in the central circle is.
a The diameter of the blue outer circle is 12 feet. What is the circumference of this circle? Round the answer to one decimal place.
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b The radius of the red inner circle is 5 feet. What is the circumference of this circle? Round the answer to one decimal place.
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Hint
a The circumference of a circle is calculated by multiplying its diameter by π. Approximate π as 3.14.
b The diameter of a circle is twice its radius. The circumference of a circle equals twice the radius times π.
Solution
a The circumference of a circle is calculated by multiplying its diameter by π.
C = π d The diameter of the outer circle is 12 feet. The value of π can be approximated as 3.14. Substitute these two values into the formula to find the circumference, then round to one decimal place.
C = π d
π ≈ 3.14
C ≈ 3.14d
Substitute
d= 12
C ≈ 3.14( 12)
Multiply
Multiply
C ≈ 37.68
RoundDec
Round to 1 decimal place(s)
C ≈ 37.7
The circumference of the outer circle is about 37.7 feet long.
b This time the diameter of the inner circle is not given but its radius is — 5 feet. Remember, the diameter of a circle is twice its radius.
d = 2r Substitute this expression into the formula used in Part A. C = π d ⇒ C = π (2r) The circumference of a circle equals twice the radius times π. Substitute 5 for r and 3.14 for π and simplify.
C = π (2r)
π ≈ 3.14
C = 3.14(2r)
Substitute
r= 5
C ≈ 3.14(2( 5))
Multiply
Multiply
C ≈ 3.14(10)
Multiply
Multiply
C ≈ 31.4
The circumference of the inner circle is about 31.4 feet long.
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Example
A Decisive Free Throw
It is the last seconds of the fourth quarter. The Detroit Pistons are about to lose by two points. Alec Burks manages to score a 2-point shot and is fouled. The clock stops at 0.2 seconds. He prepares to take a free throw. If he makes it, the Pistons will win.
a The circumference of the free throw circle is about 37 feet long. How long is the free throw line inside the circle? Round the answer to one decimal place.
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b The basketball hoop has a circumference of about 56.5 inches. What is the radius of the hoop? Round the answer to one decimal place.
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Hint
a The free throw line is a diameter of the circle. The circumference of a circle equals its diameter times π. Substitute 3.14 for π and solve the resulting equation for the diameter.
b The circumference is equal to twice the radius times π.
Solution
a Notice that the free throw line is the diameter of the circle. This circle has a circumference of 37 feet, according to the given information.
C = 37 Remember that the circumference of a circle is equal to the diameter multiplied by π. C = π d Substitute 37 for C and 3.14 for π to estimate the diameter of the free throw circle.
C = π d
π ≈ 3.14
C ≈ 3.14 d
Substitute
C= 37
37 ≈ 3.14d
DivEqn
.LHS /3.14.=.RHS /3.14.
37/3.14 ≈ d
RearrangeEqn
Rearrange equation
d ≈ 37/3.14
CalcQuot
Calculate quotient
d ≈ 11.783439...
RoundDec
Round to 1 decimal place(s)
d ≈ 11.8
The diameter of the free throw circle is about 11.8 feet long. This means that the free throw line is about 11.8 feet long.
b Start by recalling that the circumference of a circle is twice its radius times π.
C = π (2r) It is given that the circumference of the basketball hoop is about 56.5 inches. Substitute 56.5 for C and 3.14 for π into the formula. Then, solve it for r to determine the radius of the basketball hoop.
C = π (2r)
π ≈ 3.14
C ≈ 3.14(2r)
Substitute
C= 56.5
56.5 ≈ 3.14(2r)
Multiply
Multiply
56.5 ≈ 6.28r
DivEqn
.LHS /6.28.=.RHS /6.28.
56.5/6.28 ≈ r
RearrangeEqn
Rearrange equation
r ≈ 56.5/6.28
CalcQuot
Calculate quotient
r ≈ 8.996815...
RoundDec
Round to 1 decimal place(s)
r ≈ 9.0
The radius of the basketball hoop is about 9 inches.
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Pop Quiz
Determining the Dimensions of the Circle
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Discussion
The Space Inside a Circle
The amount of space inside a two-dimensional figure is known as the area of the figure. The area can usually be calculated if some dimensions of the figure are known. In the particular case of a circle, only its radius is needed.
Rule
Area of a Circle
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Example
Area of a Circular Rug
When they got home from the game, Tadeo's parents gave him another birthday present. They had bought him a circular rug with the Detroit Pistons logo on it for his bedroom. Tadeo was so happy that he ran to his room and placed it on the floor next to his bed.
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Hint
b The area of a circle is π times the radius squared.
Solution
a The space occupied by a plane figure refers to its area. Since the rug is circular, its area can be calculated by multiplying π by the radius of the rug squared.
A = π r^2 The radius of the rug is not given, but its diameter is. The radius of the rug can be found by using the fact that the diameter is twice the radius. The diameter of the rug is 4 feet.
The radius of the rug is 2 feet. Next, substitute 2 for r into the formula for the area of a circle.
A = π r^2
Substitute
r= 2
A = π ( 2)^2
CalcPow
Calculate power
A = π(4)
UseCalc
Use a calculator
A = 12.566370...
RoundDec
Round to 2 decimal place(s)
A ≈ 12.57
The area of the circular rug is about 12.57 square feet.
b The area of the circular mirror can be found by multiplying π by its radius squared.
A = π r^2 The radius of the mirror is 10 inches. Substitute 10 for r into the formula and simplify.
A = π r^2
Substitute
r= 10
A = π ( 10)^2
CalcPow
Calculate power
A = π(100)
UseCalc
Use a calculator
A = 314.159265...
RoundDec
Round to 2 decimal place(s)
A ≈ 314.16
The area of the circular mirror is about 314.16 square inches.
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Example
Length of the Clock Hands
Tadeo has a cool basketball hoop-shaped clock on his bedroom wall.
The area of this clock is 144π square inches.
a The length of the hour hand is half the radius of the clock. How long is the hour hand?
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b The length of the minute hand is one-third the diameter of the clock. How long is the minute hand?
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Hint
a The area of a circle is equal to π times the radius squared.
b The diameter of a circle is twice the radius.
Solution
a According to the given information, the length of the hour hand is half the radius of the clock. Let h be the length of the minute hand and r be the radius of the clock. The following equation represents the relation between h and r.
h = r/2 The radius of the clock is not given but its area. The radius can be calculated by using the fact that the area of a circle is π times the radius squared. A = π r^2 Tadeo's clock has an area of 144π square inches. Substitute this value for A and solve the equation for r.
A = π r^2
Substitute
A= 144π
144π = π r^2
▼
Solve for r
DivEqn
.LHS /π.=.RHS /π.
144π/π = r^2
CrossCommonFac
Cross out common factors
144π/π = r^2
SimpQuot
Simplify quotient
144 = r^2
RearrangeEqn
Rearrange equation
r^2 = 144
SqrtEqn
sqrt(LHS)=sqrt(RHS)
sqrt(r^2) = sqrt(144)
SqrtPowToNumber
sqrt(a^2)=a
r = sqrt(144)
CalcRoot
Calculate root
r = 12
The clock has a radius of 12 inches. Finally, substitute 12 for r into the first equation. h = 12/2 ⇒ h = 6 The minute hand is 6 inches long.
b Let m be the length of the minute hand and d be the diameter of the clock. It is given that the length of the minute hand is one-third of the diameter of the clock.
m = d/3 The diameter of the clock was not given but remember that the diameter of a circle is twice its radius. d = 2r The radius of the clock is 12 inches, according to Part A. Therefore, substitute 12 for r to find the diameter. d = 2( 12) ⇒ d = 24 The diameter of Tadeo's clock is 24 inches. Lastly, substitute 24 for d into the first equation to calculate the length of the minute hand. m = 24/3 ⇒ m = 8 The minute hand is 8 inches long.
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Pop Quiz
Determining the Area and Other Dimensions of a Circle
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Discussion
Half of a Circle
What is the shape of the space that a car's rear wiper cleans? It looks like half of a circle, right?
Concept
Semicircle
A semicircle is half of a circle. It is a two-dimensional figure obtained when a circle is cut into two halves. Its shape consists of an arc and a segment.
The radius of a semicircle is defined as the distance from the midpoint of the segment to any point of the arc.
The perimeter of a semicircle with radius r is the length of the segment plus half the circumference of a circle with radius r. The area of a semicircle with radius r is half the area of a circle with radius r.
| Perimeter | Area |
|---|---|
| P=2r + π r | A=1/2π r^2 |
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Example
Half of a Pizza
Tadeo's parents bought pizza to celebrate both Tadeo's birthday and the Pistons' victory. It took Tadeo a while to get to the table because he was playing a video game. When he arrived, his parents had already eaten half of the supreme pizza.
External credits: @macrovector
The radius of the pizza is 6 inches.
a What is the perimeter of the remaining pizza? Write the answer in terms of π.
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b How much space does the remaining pizza take up on the plate? Write the answer in terms of π.
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Hint
a Half of the pizza is a semicircle. The perimeter of a semicircle is twice its radius plus π r.
b The part of the plate occupied by the pizza is the area of a semicircle. The area of a semicircle is 12π r^2.
Solution
a Start by identifying the shape of half of the pizza. It looks like a semicircle with a radius of 6 inches.
The perimeter of the remaining pizza is the perimeter of a semicircle, which is equal to twice the radius plus π r. P = 2r + π r Substitute 6 for r into the formula and simplify. Remember to keep the result in terms of π.
P = 2r + π r
Substitute
r= 6
P = 2( 6) + π ( 6)
Multiply
Multiply
P = 12 + π(6)
CommutativePropMult
Commutative Property of Multiplication
P = 12 + 6π
The perimeter of the pizza that is left is 12+6π inches.
b The space occupied by the pizza is its area, which is the area of the semicircle.
The area of a semicircle is half the area of a circle. A = 1/2π r^2 The radius of the pizza is 6 inches. Substitute 6 into the area's formula and simplify. Keep the answer in terms of π.
A = 1/2π r^2
Substitute
r= 6
A = 1/2π ( 6)^2
CalcPow
Calculate power
A = 1/2π (36)
MoveRightFacToNum
a/c* b = a* b/c
A = π (36)/2
ReduceFrac
a/b=.a /2./.b /2.
A = π (18)
CommutativePropMult
Commutative Property of Multiplication
A = 18π
The portion of the plate occupied by the remaining pizza is 18π square inches.
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Closure
Measuring Wheel
While watching the sports news with his dad, Tadeo saw an interesting object referees use to measure how far an athlete throws a javelin. Tadeo said that it looks like a unicycle.
His dad explained to Tadeo how the object works. It uses two main pieces of information, the circumference of the wheel and the number of turns the wheel makes on the path, to measure the path. For example, if the wheel has a radius of 0.4 meters, then it has a circumference of 0.8π meters. This means that every revolution of the wheel covers 0.8π meters, or about 2.5 meters.
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Circles
Exercise 2.1
The diameter of a circle is doubled. How does it affect the circumference of the circle?
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Let's consider a circle with diameter d.
The circumference of a circle is found by multiplying its diameter by π. C_1 = π d Let's double the diameter of the circle.
We can find an equation for the circumference of the larger circle. C_2 = π (2d) Notice that if we factor out 2, the factor at the right-hand side becomes C_1. Let's do it!
This equation tells us that the circumference of the larger circle is twice the circumference of the first one. This means that if the diameter of a circle is doubled, its circumference also doubles.
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Solution
The two most popular pizzas in Vincenzo's pizzeria are the La Ragazza and the Mamma Mia. The La Ragazza is a 12-inch-diameter pizza that costs $8.99. The Mamma Mia is 14-inches across and costs $10.99.
External credits: @macrovector
Which pizza is the better deal? Use 3.14 for π.
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We can decide which pizza is the better deal by determining which one costs less per square inch.
| Diameter | Cost | |
|---|---|---|
| La Ragazza | 12 inches | $ 8.99 |
| Mamma Mia | 14 inches | $ 10.99 |
First, let's find out exactly how much pizza comes with each option by finding their areas. Remember, the area of a circle is π times the radius squared. A = π r^2 Let's find the area of the La Ragazza. The diameter of this pizza is 12 inches, which means that it has a radius of 6 inches. We will use 3.14 for π.
The area of La Ragazza is 113.04 square inches. Next, let's find the area of the Mamma Mia pizza. It has a diameter of 14 inches, so its radius is 7 inches.
The Mamma Mia has an area of 153.86 square inches. Let's update our table.
| Diameter | Cost | Amount of Pizza (in^2) | |
|---|---|---|---|
| La Ragazza | 12 inches | $ 8.99 | 113.04 |
| Mamma Mia | 14 inches | $ 10.99 | 153.86 |
Now let's find the cost per square inch of pizza. We can find this unit rate by dividing the cost of each pizza by its area. We will round each answer to two decimal places.
| Diameter | Cost | Amount of Pizza (in^2) | Cost per Square Inch ($) | |
|---|---|---|---|---|
| La Ragazza | 12 inches | $ 8.99 | 113.04 | 8.99/113.04 = 0.079529... ≈ 0.08 |
| Mamma Mia | 14 inches | $ 10.99 | 153.86 | 10.99/153.86 = 0.071428... ≈ 0.07 |
We can see that the La Ragazza gives us 1 square inch of pizza for 8 cents, but the Mamma Mia gives us the same amount of pizza for 7 cents. This means that the Mamma Mia is the better deal.
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Solution
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We can see that the given figure looks like a ring. It is created by an outer circle with a radius of 4 and an inner circle with a radius of 3.
We can find the area of the colored region by subtracting the area of the inner circle from the area of the outer one. Let's start by finding the area of the inner circle. Remember, the area of a circle equals π times the radius squared.
The inner circle has an area of 9π square units. Let's now find the area of the outer circle by following a similar procedure. The radius of this circle is 4.
Finally, let's find the area of the ring by subtracting the area of the inner circle from the area of the outer circle.
The area of the colored region is 7π square units.
The given figure looks kind of like an earring, right? We can identify three circles in the figure, two colored and one white.
The area of the colored region is the area of the medium circle subtracted from the area of the largest circle, plus the area of the smallest circle. A_(Colored) = A_(C_1) - A_(C_2) + A_(C_3) We need to know the radius of each of these circles to calculate their areas. We can see that the diameter of C_3 is 2 units. This means that C_3 has a radius of 1 unit. We can also see that the diameter of C_3 is the radius of C_2, so C_2 has a radius of 2 units. We already found two out of three radii! r_1 = ? r_2 = 2 r_3 = 1 We can also see that the diameter of C_2 is the radius of C_1. This means that the radius of C_1 is 4 units.
Now we can calculate the areas of the circles. Let's start with the area of C_1.
Let's follow a similar procedure to calculate the areas of the remaining two circles. We can summarize the information in a table.
| Circle | Radius | A=π r^2 | Simplify |
|---|---|---|---|
| C_1 | r=4 | A_(C_1)=π (4)^2 | A_(C_1) = 16π |
| C_2 | r=2 | A_(C_2)=π (2)^2 | A_(C_2) = 4π |
| C_3 | r=1 | A_(C_3)=π (1)^2 | A_(C_3) = π |
Finally, we are ready to calculate the area of the colored region!
The area of the colored region is 13π square units.
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A radio station upgraded its antenna and now sends out a circular signal that covers an area of about 20 096 square miles.
What is the maximum distance from the antenna that a person can tune in to the radio station? Use 3.14 for π.
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Let's start by making a graph that illustrates the given information. The antenna sends a circular signal that covers 20 096 square miles.
The maximum distance from the antenna at which the radio station can be tuned in to is the radius of the circle formed by the signal. Therefore, let's use the formula for the area of a circle to find the radius of the circle formed by the signal. A = π r^2 Let's substitute 20 096 for A and 3.14 for π into the formula. Then, we will solve the resulting equation for r.
The radius of the circle formed by the signal is 80 miles. This means that a person can tune a radio to the station at most 80 miles away from the antenna.
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Consider a pair of circles with the same center. The distance between the inner circle and the outer one is 21 inches.
How many inches longer is the circumference of the outer circle than the circumference of the inner circle? Use 227 for π.
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We want to find the difference between the circumferences of the two circles. Let's start by finding the circumference of the outer circle. Remember, the circumference of a circle is twice the radius multiplied by π. C = 2π r We do not know the radius of either circle, but we do know that the distance between the circles is 21 inches.
This means that the radius of the outer circle is 21 inches greater than the radius of the inner circle. If r_1 is the radius of the inner circle, then we can write the following for the radius r of the outer circle. r = r_1 + 21 Next, let's substitute this expression into the formula for the circumference of the outer circle.
Notice that the first term in the right-hand side is the circumference of the inner circle, C_\text{Inner} = 2\pi r_1. This means that we can write the circumference of the outer circle in terms of the circumference of the inner circle. C_(Outer) = C_(Inner) + 42π This equation tells us that the circumference of the outer circle is 42π inches longer than the circumference of the inner circle. Let's substitute 227 for π to find the exact value.
The circumference of the outer circle is 132 inches longer than the circumference of the inner circle.
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Circles
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