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Free Pre-Algebra Lesson 33 ! page 1

© 2010 Cheryl Wilcox

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

2= r or

1

2d = r

Example: The diameter of a circle is 21 inches. What is the radius?

d

2= r

21

2= 10

1

2

The radius is 101/2 inches.



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 !.

The diameter of the circle is 1 inch. 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 !.

1 32



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

7= 3

1

7= 3.1412857 is sometimes used as a pretty

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)

C = !d = ! 85( ) "22

7

#

$%&

'(85( )

2671

7 cm

b)

C = !d = ! 85( )" 3.14( ) 85( )266.9 cm

c)

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.



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.

C = 2!r = 2! 35.8( ) " 224.938034

" 224.9 cm

The diameter of the circle is 145 cm.

C = !d = ! 145( ) " 455.5309348

" 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

2. The rectangle is made of all the pieces of the circle, so the area

of the rectangle is equal to the area of the circle.

The formula for the area of a circle is A = !r2

. Notice that the radius units of length are squared and become square

units of area.

Example: Find the area of each circle. Round answers to the nearest tenth.

The radius of the circle is 23 cm.

A = !r2= ! 23 cm( )

2

= ! 529 cm2( ) " 1661.9 cm

2

The diameter of the circle is 78 cm.

r = d / 2 = 78 cm / 2 = 39 cm

A = !r2= ! 39 cm( )

2

= ! 1521 cm2( )

" 4778.4 cm2



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 =4

3!r

3. The cubed value of the length of the radius will result in

cubic units of volume when you calculate. Example: Find the volume of the sphere. Round your answer to the nearest tenth.

The radius of the sphere is 14 cm.

V =4

3!r

3=

4

3! 14 cm( )

3

=4

3! 2744 cm

3( ) " 11494.0 cm3

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.

Calculator keystrokes:

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

Area of a Rectangle

A =LW

Volume of a Box

V =LWH

Circumference of a Circle

C = 2!r

Area of a Circle

A = !r2

Volume of a Sphere

V =4

3!r

3

!



Lesson 33: Formulas for Circles

Worksheet Name _________________________________________

Diameter and Radius

d = 2r

r =d

2

Circumference of a Circle

C = 2!r

C = !d

Area of a Circle

A = !r2

Volume of a Sphere

V =4

3!r

3

1. Fill in the labels for the parts of the circle, using the words center, radius, diameter, circumference.

How many diameters make up one circumference?

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

circumference

area

3. The diameter of circle is 51/4 inches.

a. Change 51/4 to decimal form.

b. What is the circumference of the circle?

c. Approximate with a fraction found on the inch ruler.

4. Which is larger, a cube with sides 10 cm or a sphere with radius 10 cm?

a. Volume of box

b. Volume of sphere



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

time to find the speed of the end of the

irrigation arm in miles per day.

9. Find the volume of a sphere with radius 1 m. Round to the nearest tenth.





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.


5. Find the perimeter and area of the triangle.

Perimeter

Area


Perimeter

Area



Round answers to the nearest tenth.

7. Find the area and perimeter of the rectangle.

Perimeter

Area

8. Find the volume of the box.

9. Find the circumference and area of the circle.

10. Find the volume of the sphere.



13. Convert the decimals to fractions in lowest terms.

a. 0.3

b. 2.05

c. 55.125

14. Convert the fractions to decimals. Use a bar for repeating decimals. Do not round.

a. 1/7

b. 1/8

c. 3/16




Homework 33A Answers


The perimeter is 28 cm.


The perimeter is 14.4 m.


Counting squares, we find

the area is 23 cm2.


The area is 7.68 m2.


Perimeter

P = a +b +c = 221.8 cm

Area

A = bh / 2 = 1785 cm2


Perimeter

P = a +b +c = 32.4 m

Area

A = bh / 2 = 42 m2



Round answers to the nearest tenth.


Perimeter

P = 2L + 2W = 26.3 m

Area

A =LW = 32.4723 m2! 32.5 m

2


V =LWH = 176.085 cm3

! 176.1 cm3


C = 2!r = 2! 2.2 m( ) " 13.8 m

A = !r

2= ! 2.2 m( )

2

" 15.2 m2


V =4

3!r

3=

4

3! 2.2 m( )

3

" 44.6 m3


C = !d = ! 2.2 m( ) " 6.9 m

A = !r

2= ! 1.1 m( )

2

" 3.8 m2


V =4

3!r

3=

4

3! 1.1 m( )

3

" 5.6 m3


a. 0.3

=3

10

b. 2.05

= 25

100= 2

1

20

c. 55.125

= 55125

1000= 55

1

8

14. Convert the fractions to decimals. Use a bar for

repeating decimals. Do not round.

a. 1/7 = 0.142857

b. 1/8 = 0.125

c. 3/16 = 0.1875




Homework 33B Name_________________________________________






Perimeter

Area


Perimeter

Area



Measurements on this page are given in centimeters. Round answers to the nearest tenth.


Perimeter

Area







a. 9.4

b. 3.1416

c. 0.0015

14. Convert the fractions to decimals. Use a bar for repeating decimals. Do not round.

a.

9

40

b.

40

9

c.

316

113

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