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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.**

Explore

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. After taking notes about a few coins, Tadeo is shocked about what he discovered. What did he notice?

Discussion

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 $C.$

circle $O,$since it is centered at $O.$

In any given circle, the lengths of any radius and any diameter are constant. They are called

Pop Quiz

Click on the indicated part of the given circle.

Discussion

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

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.

Since $π$ is an irrational number, its decimal part never repeats or terminates. However, its value is often rounded to $3.14$ to make calculations easier. Alternatively, $π$ can be approximated by $722 .$

$π=3.1415926…$

Graphically, $π$ is the number of times that the diameter of the circle fits on top of the circle.

Discussion

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

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 $π.$

$C=2πr$

Example

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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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 $π.$

a The circumference of a circle is calculated by multiplying its diameter by $π.$

$C=πd $

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.
$C=πd$

$π≈3.14$

$C≈3.14d$

Substitute

$d=12$

$C≈3.14(12)$

Multiply

Multiply

$C≈37.68$

RoundDec

Round to $1$ decimal place(s)

$C≈37.7$

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=πd⇒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

$C≈31.4$

Example

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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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 $π.$

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=πd $

Substitute $37$ for $C$ and $3.14$ for $π$ to estimate the diameter of the free throw circle.
$C=πd$

$π≈3.14$

$C≈3.14d$

Substitute

$C=37$

$37≈3.14d$

DivEqn

$LHS/3.14=RHS/3.14$

$3.1437 ≈d$

RearrangeEqn

Rearrange equation

$d≈3.1437 $

CalcQuot

Calculate quotient

$d≈11.783439…$

RoundDec

Round to $1$ decimal place(s)

$d≈11.8$

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

$LHS/6.28=RHS/6.28$

$6.2856.5 ≈r$

RearrangeEqn

Rearrange equation

$r≈6.2856.5 $

CalcQuot

Calculate quotient

$r≈8.996815…$

RoundDec

Round to $1$ decimal place(s)

$r≈9.0$

Pop Quiz

Discussion

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

Example

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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b The area of a circle is $π$ times the radius squared.

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=πr_{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.
$A=πr_{2}$

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)

$A≈12.57$

b The area of the circular mirror can be found by multiplying $π$ by its radius squared.

$A=πr_{2} $

The radius of the mirror is $10$ inches. Substitute $10$ for $r$ into the formula and simplify.
$A=πr_{2}$

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)

$A≈314.16$

Example

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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a The area of a circle is equal to $π$ times the radius squared.

b The diameter of a circle is twice the radius.

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=2r $

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=πr_{2} $

Tadeo's clock has an area of $144π$ square inches. Substitute this value for $A$ and solve the equation for $r.$
$A=πr_{2}$

Substitute

$A=144π$

$144π=πr_{2}$

▼

Solve for $r$

DivEqn

$LHS/π=RHS/π$

$π144π =r_{2}$

CrossCommonFac

Cross out common factors

$π 144π =r_{2}$

SimpQuot

Simplify quotient

$144=r_{2}$

RearrangeEqn

Rearrange equation

$r_{2}=144$

SqrtEqn

$LHS =RHS $

$r_{2} =144 $

SqrtPowToNumber

$a_{2} =a$

$r=144 $

CalcRoot

Calculate root

$r=12$

$h=212 ⇒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=3d $

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=324 ⇒m=8 $

The minute hand is $8$ inches long.
Pop Quiz

Discussion

What is the shape of the space that a car's rear wiper cleans? It looks like half of a circle, right?
## Semicircle

Next, the definition of this figure and the formulas to calculate its area and perimeter are introduced.

Concept

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 |
---|---|

$P=2r+πr$ | $A=21 πr_{2}$ |

Example

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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a Half of the pizza is a semicircle. The perimeter of a semicircle is twice its radius plus $πr.$

b The part of the plate occupied by the pizza is the area of a semicircle. The area of a semicircle is $21 πr_{2}.$

a Start by identifying the shape of half of the pizza. It looks like a semicircle with a radius of $6$ inches.

$P=2r+πr $

Substitute $6$ for $r$ into the formula and simplify. Remember to keep the result in terms of $π.$
$P=2r+πr$

Substitute

$r=6$

$P=2(6)+π(6)$

Multiply

Multiply

$P=12+π(6)$

CommutativePropMult

Commutative Property of Multiplication

$P=12+6π$

b The space occupied by the pizza is its area, which is the area of the semicircle.

$A=21 πr_{2} $

The radius of the pizza is $6$ inches. Substitute $6$ into the area's formula and simplify. Keep the answer in terms of $π.$
$A=21 πr_{2}$

Substitute

$r=6$

$A=21 π(6)_{2}$

CalcPow

Calculate power

$A=21 π(36)$

MoveRightFacToNum

$ca ⋅b=ca⋅b $

$A=2π(36) $

ReduceFrac

$ba =b/2a/2 $

$A=π(18)$

CommutativePropMult

Commutative Property of Multiplication

$A=18π$