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| 16 Theory slides |
| 17 Exercises - Grade E - A |
| Each lesson is meant to take 1-2 classroom sessions |
Here are a few recommended readings before getting started with this lesson.
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.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.
circle O,since it is centered at O.
Click on the indicated part of the given circle.
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.
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.
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.
The circumference of a circle is calculated by multiplying its diameter by π.
C=πd
Since the diameter is twice the radius, the circumference of a circle can also be calculated by multiplying 2r by π.
C=2πr
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.
π≈3.14
d=12
Multiply
Round to 1 decimal place(s)
π≈3.14
r=5
Multiply
Multiply
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.
π≈3.14
C=37
LHS/3.14=RHS/3.14
Rearrange equation
Calculate quotient
Round to 1 decimal place(s)
π≈3.14
C=56.5
Multiply
LHS/6.28=RHS/6.28
Rearrange equation
Calculate quotient
Round to 1 decimal place(s)
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.
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.
r=2
Calculate power
Use a calculator
Round to 2 decimal place(s)
r=10
Calculate power
Use a calculator
Round to 2 decimal place(s)
Tadeo has a cool basketball hoop-shaped clock on his bedroom wall.
The area of this clock is 144π square inches.
A=144π
LHS/π=RHS/π
Cross out common factors
Simplify quotient
Rearrange equation
LHS=RHS
a2=a
Calculate root
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+πr | A=21πr2 |
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.
The radius of the pizza is 6 inches.
r=6
Multiply
Commutative Property of Multiplication
r=6
Calculate power
ca⋅b=ca⋅b
ba=b/2a/2
Commutative Property of Multiplication
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.
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.
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.
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.
For each of the given figures, find the area of the colored region. Write the answers in terms of π.
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.
A radio station upgraded its antenna and now sends out a circular signal that covers an area of about 20096 square miles.
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.
Consider a pair of circles with the same center. The distance between the inner circle and the outer one is 21 inches.
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.