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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

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.

Nickel, Dime, and Quarter
External credits: Wikipedia, Wikipedia, Wikipedia
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.
Random Circles
After taking notes about a few coins, Tadeo is shocked about what he discovered. What did he notice?
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
  • Diameter - A segment whose endpoints are on the circle and that passes through the center. Its length is usually represented algebraically by
  • Circumference - The perimeter of a circle, usually represented algebraically by
The following circle can be referred to as or circle since it is centered at
Parts of a circle
In any given circle, the lengths of any radius and any diameter are constant. They are called the radius and the diameter of the circle, respectively.
Pop Quiz

Identifying the Parts of a Circle

Click on the indicated part of the given circle.

Random Circles
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 is a constant defined as the ratio between the circumference and the diameter of a circle. This ratio is the same for all circles.
A circle with its diameter and circumference labeled
Since is an irrational number, its decimal part never repeats or terminates. However, its value is often rounded to to make calculations easier. Alternatively, can be approximated by

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

Animation unrolling a circle
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

Since the diameter is twice the radius, the circumference of a circle can also be calculated by multiplying by

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.

Detroit Pistons Court
a The diameter of the blue outer circle is feet. What is the circumference of this circle? Round the answer to one decimal place.
b The radius of the red inner circle is feet. What is the circumference of this circle? Round the answer to one decimal place.

Hint

a The circumference of a circle is calculated by multiplying its diameter by Approximate as
b The diameter of a circle is twice its radius. The circumference of a circle equals twice the radius times

Solution

a The circumference of a circle is calculated by multiplying its diameter by
The diameter of the outer circle is feet. The value of can be approximated as Substitute these two values into the formula to find the circumference, then round to one decimal place.

The circumference of the outer circle is about feet long.
Detroit Pistons Court
b This time the diameter of the inner circle is not given but its radius is — feet. Remember, the diameter of a circle is twice its radius.
Substitute this expression into the formula used in Part A.
The circumference of a circle equals twice the radius times Substitute for and for and simplify.

The circumference of the inner circle is about feet long.
Detroit Pistons Court
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 point shot and is fouled. The clock stops at seconds. He prepares to take a free throw. If he makes it, the Pistons will win.

Free throws zone
a The circumference of the free throw circle is about feet long. How long is the free throw line inside the circle? Round the answer to one decimal place.
b The basketball hoop has a circumference of about inches. What is the radius of the hoop? Round the answer to one decimal place.

Hint

a The free throw line is a diameter of the circle. The circumference of a circle equals its diameter times Substitute for and solve the resulting equation for the diameter.
b The circumference is equal to twice the radius times

Solution

a Notice that the free throw line is the diameter of the circle. This circle has a circumference of feet, according to the given information.
Remember that the circumference of a circle is equal to the diameter multiplied by
Substitute for and for to estimate the diameter of the free throw circle.

The diameter of the free throw circle is about feet long. This means that the free throw line is about feet long.
b Start by recalling that the circumference of a circle is twice its radius times
It is given that the circumference of the basketball hoop is about inches. Substitute for and for into the formula. Then, solve it for to determine the radius of the basketball hoop.

The radius of the basketball hoop is about inches.
Pop Quiz

Determining the Dimensions of the Circle

Determine the indicated dimension of the given circle. Round the answer to one decimal place.

Random circles
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

The area of a circle is the product of and the square of its radius.

Circle
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.

Bedroom seen from the top
a The rug has a diameter of feet. How much space does the rug take up on the bedroom floor? Round the answer to two decimal places.
b Tadeo has a circular mirror next to the chest of drawers. The mirror has a radius of inches. What is the area of the mirror? Round the answer to two decimal places.

Hint

a The radius of a circle is half its diameter. The space occupied by the rug refers to its area.
b The area of a circle is 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 by the radius of the rug squared.
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 feet.
The radius of the rug is feet. Next, substitute for into the formula for the area of a circle.
The area of the circular rug is about square feet.
b The area of the circular mirror can be found by multiplying by its radius squared.
The radius of the mirror is inches. Substitute for into the formula and simplify.
The area of the circular mirror is about square inches.
Example

Length of the Clock Hands

Tadeo has a cool basketball hoop-shaped clock on his bedroom wall.

Clock

The area of this clock is square inches.

a The length of the hour hand is half the radius of the clock. How long is the hour hand?
b The length of the minute hand is one-third the diameter of the clock. How long is the minute hand?

Hint

a The area of a circle is equal to 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 be the length of the minute hand and be the radius of the clock. The following equation represents the relation between and
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 times the radius squared.
Tadeo's clock has an area of square inches. Substitute this value for and solve the equation for
Solve for
The clock has a radius of inches. Finally, substitute for into the first equation.
The minute hand is inches long.
b Let be the length of the minute hand and 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.
The diameter of the clock was not given but remember that the diameter of a circle is twice its radius.
The radius of the clock is inches, according to Part A. Therefore, substitute for to find the diameter.
The diameter of Tadeo's clock is inches. Lastly, substitute for into the first equation to calculate the length of the minute hand.
The minute hand is inches long.
Pop Quiz

Determining the Area and Other Dimensions of a Circle

Determine the indicated dimension of the given circle. Round the answer to one decimal place.

Random circles
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?
Car back view and wiper
Next, the definition of this figure and the formulas to calculate its area and perimeter are introduced.
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.

Semicircle

The radius of a semicircle is defined as the distance from the midpoint of the segment to any point of the arc.

Upper semicircle with radius r, where the radius is the distance between the center and the arc.

The perimeter of a semicircle with radius is the length of the segment plus half the circumference of a circle with radius The area of a semicircle with radius is half the area of a circle with radius

Perimeter Area
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.

Half of a pizza
External credits: @macrovector

The radius of the pizza is inches.

a What is the perimeter of the remaining pizza? Write the answer in terms of
b How much space does the remaining pizza take up on the plate? Write the answer in terms of

Hint

a Half of the pizza is a semicircle. The perimeter of a semicircle is twice its radius plus
b The part of the plate occupied by the pizza is the area of a semicircle. The area of a semicircle is

Solution

a Start by identifying the shape of half of the pizza. It looks like a semicircle with a radius of inches.
Half of a pizza
The perimeter of the remaining pizza is the perimeter of a semicircle, which is equal to twice the radius plus
Substitute for into the formula and simplify. Remember to keep the result in terms of
The perimeter of the pizza that is left is inches.
Half of a pizza
b The space occupied by the pizza is its area, which is the area of the semicircle.
Half of a pizza
The area of a semicircle is half the area of a circle.
The radius of the pizza is inches. Substitute into the area's formula and simplify. Keep the answer in terms of
The portion of the plate occupied by the remaining pizza is square inc