Free Pre-Algebra
Lesson 33 ! page 1
Lesson 33
Formulas for Circles
What is a Circle?
Everyone knows what a circle looks like.
A sprinkler line rotates around a center
pivot, forming circles of irrigated land.
A band marches around a maypole in
a circular dance. Each band
member’s ribbon is the same length.
The spokes of a bicycle wheel are all
the same length and hold the edge of
the wheel in a circular shape.
Every point on a circle is the same distance from the center.
The parts of a circle are named in the diagram to the right. Any line drawn
from the center to the edge of the circle is called a radius. The distance all
the way across the circle is called the diameter. The diameter consists of two
radii (Latin plural of radius is radii.).
If you know the radius of a circle, you can double it to find the diameter.
2 • radius = diameter
2r = d
Example: The radius of a circle is 14 inches. What is the diameter?
2r = d
( )
2 14 = 28
The diameter is 28 inches.
If you know the diameter of a circle, you can cut it in half to
find the radius.
diameter / 2 = radius
d
1
= r or
d =r
2
2
© 2010 Cheryl Wilcox
Example: The diameter of a circle is 21 inches. What is
the radius?
d
=r
2
21
1
= 10
2
2
The radius is 101/2 inches.
Free Pre-Algebra
Lesson 33 ! page 2
The Circumference of a Circle
The circumference of a circle is the distance around the edge, exactly like the perimeter of a rectangle. The two ideas are
the same, but the circle has a special word for its perimeter.
Just as we used the simpler measurements of length and width to calculate the
perimeter of a rectangle, we’d like to use a simpler measurement such as the
diameter or radius of the circle to calculate the perimeter.
Perhaps you’ve seen this formula before:
C = !d
It reads
Circumference = pi times diameter
The formula contains the Greek letter !, pi. Americans pronounce pi like the dessert,
“pie,” the British pronounce it like the letter “P.” This Greek letter is not a variable. It
is the name of a particular number, just as “four” is the name of a number. The number ! has a special name and symbol
because it is used extensively in mathematics. It helps to have a special symbol for the number because it is impossible to
write accurately in decimal notation.
The formula tells us that the circumference is a multiple of the diameter. Since all circles are the same shape, whether large
or small, the diameter and circumference always have the same relationship: There are ! diameters in one circumference. If
we set the diameter of a circle to 1 unit (1 cm, 1 inch, 1 meter, 1 mile…), then the circumference of the circle will be ! units
long. To figure out the value of !, we could measure around the edge of the circle.
Unwrap the circle and measure it to find the value of !.
1
The diameter of the circle is 1 inch.
2
3
The circumference measures somewhere between 3 1/8 and 3 3/16 inches.
Direct measurement is not really accurate. Is the circle really perfectly round? Is the diameter exactly 1 inch? How can we
be more precise about !?
Methods to caculate ! precisely and mathematically were discovered in antiquity. For example, Archimedes (c. 250 BC)
constructed polygons outside and inside the circle. Since the polygons have straight sides, their perimeters can be
calculated precisely. The circumference of the circle must be between
the perimeter of the outside polygon and the inside polygon. This
helps us narrow down the value of ! as accurately as we have the
patience to calculate.
Modern methods and computers have calculated the value of ! to
trillions of decimal places. You can see the first hundred thousand
decimal places of ! here. Pi is not like the decimals we have encountered so far. We saw in Lesson 30 that we could
convert any fraction to a decimal, and the decimal would either terminate (have a finite number of decimal places) or have a
repeating pattern of digits. The decimal representation of ! neither terminates nor repeats. There is no fraction that exactly
gives the value of !.
© 2010 Cheryl Wilcox
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Lesson 33 ! page 3
Using ! for Calculations
Although we have a special symbol for pi to represent its exact value, to use ! in any practical application we must round it
off and calculate with either a decimal or fraction. Once we substitute a rounded value for !, we have to use the wavy equals
sign ( ! ) that means “approximately equal to” since no matter how many digits we write, the rounded number is not
absolutely accurate.
! " 3.141592653589793238462643383279502884197169399375105820974944592307816406286208998628
Common useful roundings are 3.14 and 3.1416. The fraction
22
1
= 3 = 3.1412857 is sometimes used as a pretty
7
7
good approximation. A calculator with a ! key will use at least as many digits as the display will hold, usually ten.
Finding ! on the Calculator
Scientific calculators usually have a ! key, but the placement varies from model to model. Spend some time looking for the
! key and figuring out how to use it on your calculator.
To use ! on this calculator, press the orange second function
key to access ! written in orange above the “3” key.
This calculator has the ! key available just like the other
numbers. Just press !.
Example: Use the formula C = !d to calculate the circumference of a circle with diameter 85 cm. Compare the
values obtained using three different values for !: a) the fraction approximation 22/7; b) the rounding 3.14; c) the !
key on a scientific calculator.
a)
b)
# 22 &
C = !d = ! 85 " % ( 85
$ 7'
( )
1
267 cm
7
( )
c)
( )
" ( 3.14 ) ( 85)
C = !d = ! 85
266.9 cm
( )
C = !d = ! 85
(
)( )
" 3.141592654 85
267.0353756 cm
We get the same answer only if we round to the nearest whole centimeter. Rounding to the nearest tenth or hundredth
would reveal the round-off error in the different values of !. Moral: Whenever possible, use the ! key on your calculator.
© 2010 Cheryl Wilcox
Free Pre-Algebra
Lesson 33 ! page 4
Since the diameter of a circle is equal to twice the radius, the formula for the circumference can be written either
C = !d or C = 2!r
Use whichever version is appropriate for the information you are given.
Example: Find the circumference of each circle. Round answers to the nearest tenth.
The radius of the circle is 35.8 cm.
(
The diameter of the circle is 145 cm.
)
(
)
C = !d = ! 145 " 455.5309348
C = 2!r = 2! 35.8 " 224.938034
" 224.9 cm
" 455.5 cm
The Area of a Circle
To find a formula for the area of a circle, we’re going to cut it into pieces and rearrange them. The goal is to get something
close to a rectangle, because we already know how to find the area of a rectangle.
These pictures are from the Victoria Department of Education website.
1. Cut a circle into wedges (like pizza
slices):
2. Separate into two sections that you
can slot together:
( )( )
3. The circle has been transformed and
looks like a rectangle. The more slices
you start with, the more like a rectangle
the result will be.
The length of the rectangle is half the
circumference of the circle, so L = !r.
The width is the radius of the circle, so
W = r.
The area of the rectangle is A = LW = !r r = !r . The rectangle is made of all the pieces of the circle, so the area
2
of the rectangle is equal to the area of the circle.
The formula for the area of a circle is A = !r . Notice that the radius units of length are squared and become square
units of area.
2
Example: Find the area of each circle. Round answers to the nearest tenth.
The radius of the circle is 23 cm.
(
A = !r 2 = ! 23 cm
(
)
The diameter of the circle is 78 cm.
)
r = d / 2 = 78 cm / 2 = 39 cm
2
= ! 529 cm " 1661.9 cm
© 2010 Cheryl Wilcox
2
2
(
)
2
(
A = !r 2 = ! 39 cm = ! 1521 cm2
" 4778.4 cm
2
)
Free Pre-Algebra
Lesson 33 ! page 5
The Volume of a Sphere
A sphere is a three-dimensional object, but it is similar to a circle, because all of the
points on the sphere are the same distance from the center. The radius of the sphere
measures the distance from the center to the edge just as in a circle.
Since the sphere is a three-dimensional object, we measure the space it takes up in
cubic units, just as we measured the volume of a box.
The formula for the volume of a sphere can be derived similarly to the way we found the area of a circle formula.
The volume of a sphere is calculated with the formula V =
cubic units of volume when you calculate.
4 3
!r . The cubed value of the length of the radius will result in
3
Example: Find the volume of the sphere. Round your answer to the nearest tenth.
The radius of the sphere is 14 cm.
(
Calculator keystrokes:
)
3
4 3 4
!r = ! 14 cm
3
3
4
= ! 2744 cm3 " 11494.0 cm3
3
V=
(
)
Since the value on the calculator is 11494.04032, to round to
the nearest tenth results in a zero in the tenths place. We
leave the zero in place to show the level of accuracy rather
than just writing 11494 cm3.
Comparing Formulas
The formulas for perimeter, area, and volume for rectangles/boxes use the length, width, (and height) measurements. The
formulas for circles/spheres use the radius measurement.
Perimeter of a Rectangle
P = 2L + 2W
Circumference of a Circle
C = 2!r
Area of a Rectangle
A = LW
Area of a Circle
V = LWH
Volume of a Sphere
A = !r 2
!
© 2010 Cheryl Wilcox
Volume of a Box
V=
4 3
!r
3
Free Pre-Algebra
Lesson 33 ! page 6
Lesson 33: Formulas for Circles
Worksheet
Diameter and Radius
Name _________________________________________
Circumference of a Circle
d = 2r
C = 2!r
d
2
C = !d
r=
1. Fill in the labels for the parts of the circle, using the words
center, radius, diameter, circumference.
Area of a Circle
A = !r 2
Volume of a Sphere
V=
4 3
!r
3
2. Use the measurements given for the radius to calculate
the diameter, circumference, and area of each circle. Round
to the nearest tenth.
a. r = 4.5 cm
diameter
circumference
area
b. r = 45 cm
diameter
How many diameters make up one circumference?
circumference
area
3. The diameter of circle is 51/4 inches.
a. Change 51/4 to decimal form.
4. Which is larger, a cube with sides 10 cm or a sphere with
radius 10 cm?
a. Volume of box
b. What is the circumference of the circle?
c. Approximate with a fraction found on the inch ruler.
© 2010 Cheryl Wilcox
b. Volume of sphere
Free Pre-Algebra
Lesson 33 ! page 7
The arm (radius) of this center pivot irrigation system is 1/4 mile.
5. What area is irrigated? Write the answer
given by your calculator without rounding.
Include units.
6. Convert the area of the field from square miles to acres. Do not round off
until the end of your calculation, then round to the nearest acre.
(1 acre = 0.0015625 square miles)
7. How far would the farmer drive around
the edge of the field? Round to the nearest
tenth.
8. The irrigation arm makes a complete circuit of the field every 3 days.
Use the formula rate =
distance
to find the speed of the end of the
time
irrigation arm in miles per day.
9. Find the volume of a sphere with
radius 1 m. Round to the nearest tenth.
© 2010 Cheryl Wilcox
10. Find the volume of a sphere with
radius 10 m. Round to the nearest
tenth.
11. Find the volume of a sphere with
radius 100 m. Round to the nearest
tenth.
Free Pre-Algebra
Lesson 33 ! page 8
Lesson 33: Formulas for Circles Name
Homework 33A
Name______________________________________
1. Find the perimeter. The grid is square centimeters.
2. Find the perimeter using the measurements given.
3. Find the area of the figure above.
4. Find the area of the figure above.
5. Find the perimeter and area of the triangle.
6. Find the perimeter and area of the triangle.
Perimeter
Perimeter
Area
Area
© 2010 Cheryl Wilcox
Free Pre-Algebra
Lesson 33 ! page 9
Round answers to the nearest tenth.
7. Find the area and perimeter of the rectangle.
8. Find the volume of the box.
Perimeter
Area
9. Find the circumference and area of the circle.
10. Find the volume of the sphere.
11. Find the circumference and area of the circle.
12. Find the volume of the sphere.
13. Convert the decimals to fractions in lowest terms.
14. Convert the fractions to decimals. Use a bar for
repeating decimals. Do not round.
a. 0.3
a. 1/7
b. 2.05
c. 55.125
© 2010 Cheryl Wilcox
b. 1/8
c. 3/16
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Lesson 33 ! page 10
Lesson 33: Formulas for Circles
Homework 33A Answers
1. Find the perimeter. The grid is square centimeters.
2. Find the perimeter using the measurements given.
The perimeter is 28 cm.
3. Find the area of the figure above.
The perimeter is 14.4 m.
4. Find the area of the figure above.
Counting squares, we find
the area is 23 cm2.
The area is 7.68 m2.
5. Find the perimeter and area of the triangle.
6. Find the perimeter and area of the triangle.
Perimeter
Perimeter
P = a + b + c = 221.8 cm
Area
P = a + b + c = 32.4 m
Area
A = bh / 2 = 1785 cm2
© 2010 Cheryl Wilcox
A = bh / 2 = 42 m2
Free Pre-Algebra
Lesson 33 ! page 11
Round answers to the nearest tenth.
7. Find the area and perimeter of the rectangle.
8. Find the volume of the box.
Perimeter
P = 2L + 2W = 26.3 m
V = LWH = 176.085 cm3
Area
! 176.1 cm3
A = LW = 32.4723 m2 ! 32.5 m2
9. Find the circumference and area of the circle.
(
)
10. Find the volume of the sphere.
C = 2!r = 2! 2.2 m " 13.8 m
(
)
2
A = !r 2 = ! 2.2 m " 15.2 m2
11. Find the circumference and area of the circle.
(
)
(
)
C = !d = ! 2.2 m " 6.9 m
2
V=
(
)
3
4 3 4
!r = ! 2.2 m " 44.6 m3
3
3
12. Find the volume of the sphere.
V=
(
)
3
4 3 4
!r = ! 1.1 m " 5.6 m3
3
3
A = !r 2 = ! 1.1 m " 3.8 m2
13. Convert the decimals to fractions in lowest terms.
a. 0.3 =
3
10
b. 2.05 = 2
a. 1/7 = 0.142857
5
1
=2
100
20
c. 55.125 = 55
14. Convert the fractions to decimals. Use a bar for
repeating decimals. Do not round.
125
1
= 55
1000
8
© 2010 Cheryl Wilcox
b. 1/8 = 0.125
c. 3/16 = 0.1875
Free Pre-Algebra
Lesson 33 ! page 12
Lesson 33: Formulas for Circles
Homework 33B
Name_________________________________________
1. Find the perimeter. The grid is square centimeters.
2. Find the perimeter using the measurements given.
3. Find the area of the figure above.
4. Find the area of the figure above.
5. Find the perimeter and area of the triangle.
6. Find the perimeter and area of the triangle.
Perimeter
Perimeter
Area
Area
© 2010 Cheryl Wilcox
Free Pre-Algebra
Lesson 33 ! page 13
Measurements on this page are given in centimeters. Round answers to the nearest tenth.
7. Find the area and perimeter of the rectangle.
8. Find the volume of the box.
Perimeter
Area
9. Find the circumference and area of the circle.
10. Find the volume of the sphere.
11. Find the circumference and area of the circle.
12. Find the volume of the sphere.
13. Convert the decimals to fractions in lowest terms.
14. Convert the fractions to decimals. Use a bar for
repeating decimals. Do not round.
a. 9.4
a.
9
40
b.
40
9
b. 3.1416
c. 0.0015
c. 3
© 2010 Cheryl Wilcox
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