16.1 Justifying Circumference and Area of a Circle

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Class
Date
16.1 Justifying Circumference
and Area of a Circle
Essential Question: How can you justify and use the formulas for the circumference and area
of a circle?
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Justifying the Circumference Formula
To find the circumference of a given circle, consider a regular polygon that is inscribed in the
circle. As you increase the number of sides of the polygon, the perimeter of the polygon gets
closer to the circumference of the circle.
Inscribed
pentagon
Inscribed
hexagon
Inscribed
octagon
Let circle O be a circle with center O and radius r. Inscribe a regular n–gon in circle O and
draw radii from O to the vertices of the n-gon.
O
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r
A
1 2
r
x M
B
_
_
_
Let AB be one side of the n-gon. Draw OM, the segment from O to the midpoint of AB.
the SSS Congruence Criterion
A
Then △AOM ≅ △BOM by
B
So, ∠1 ≅ ∠2 by
C
There are n triangles, all congruent to △AOB, that surround point O and fill the n-gon.
.
CPCTC .
360°
180°
____
Therefore, m∠AOB = ____
n and m∠1 =
n .
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
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Since ∠OMA ≅ ∠OMB by CPCTC, and ∠OMA and ∠OMB form a linear pair, these
angles are supplementary and must have measures of 90°. So △AOM and △BOM are
right triangles.
length of oppodite leg _
In △AOM, sin∠1 =
= xr .
length of hypotenuse
__
So, x = r sin∠1 and substituting the expression for m∠1 from above gives
180°
x = r sin ____
n .
 Now express the perimeter of the
The length of
n-gon in terms of x.
―
_
AB is 2x, since M is the midpoint of AB .
n-gon is 2nx .
This means the perimeter of the
Substitute the expression for
The perimeter of the
x in Step D.
n-gon in terms of x is
(
180°
2nr sin ____
n
)
.
( )
180°
n–gon should include the factor n sin ____
n .
What happens to this factor as n gets larger?
 Your expression for the perimeter of the
Use your calculator to do the following.
( )
180
• Enter the expression x sin ___
x as Y 1.
• Go to the Table Setup menu and enter the
values below.
• View a table for the function.
• Use arrow keys to scroll down.
( )
180°
What happens to the value of x sin ____
as x gets larger?
x
The value gets closer to π.
n–gon. What happens to
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 Look at the expression you wrote for the perimeter of the
the value of this expression, as n gets larger?
The expression gets closer to 2πr.
Reflect
1.
When n is very large, does the perimeter of the n-gon ever equal the circumference of
the circle? Why or why not?
No; the perimeter of the n-gon gets very close to the circumference, but is always a bit
less than the circumference.
2.
How does the above argument justify the formula C = 2πr?
When n is very large, the regular n-gon is virtually indistinguishable from the circle and
the expression for the n-gon’s perimeter is virtually indistinguishable from 2πr.
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Explain 1
Applying the Circumference Formula
Example 1 Find the circumference indicated.
 A Ferris wheel has a diameter of 40 feet. What is its
circumference? Use 3.14 for π.
Diameter = 2r
40 = 2r
20 = r
Use the formula
C = 2πr to find the circumference.
C = 2πr
C = 2π(20)
C = 2(3.14)(20)
C ≈ 125.6
The circumference is about 125.6 feet.
 A pottery wheel has a diameter of 2 feet. What is its circumference? Use 3.14 for
The diameter is 2 feet, so the radius in inches is
π.
r = 12 .
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iStockPhoto.com
C = 2πr
C = 2 ∙ 3.14 ∙ 12
C≈
75.36
Reflect
3.
Discussion Suppose you double the radius of a circle. How does the circumference of this larger circle
compare with the circumference of the smaller circle? Explain.
The circumference is also doubled. The new radius is 2r and the new circumference is
2π(2r) = 2(2πr).
Your Turn
4.
The circumference of a tree is 20 feet. What is its 5.
diameter? Round to the nearest tenth of a foot.
Use 3.14 for π.
Since, d = 2r, C = 2πr = πd.
C = πd
20
_
π =d
C = πd
32
_
π =d
6.37 ≈ d
10.19 ≈ d
So, the diameter is about 6.4 feet.
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The circumference of a circular fountain is 32
feet. What is its diameter? Round to the nearest
tenth of a foot. Use 3.14 for π.
Since, d = 2r, C = 2πr = πd.
So, the diameter is about 10.2 feet.
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Justifying the Area Formula
Explain 2
To find the area of a given circle, consider a regular polygon that is inscribed in the circle.
As you increase the number of sides of the polygon, the area of the polygon gets closer to
the area of the circle.
Inscribed
pentagon
Inscribed
hexagon
Inscribed
octagon
Let circle O be a circle with center O and radius r. Inscribe a regular n–gon in
circle O and draw radii from O to the vertices of the n-gon.
_
_
Let AB be one
_side of the n–gon. Draw OM, the segment from O to the
midpoint of AB.
_
_
We know that OM is perpendicular to AB because triangle AOM is congruent to
triangle BOM.
_
Let the length of OM be h.
O
1
r
r
h
x M x
A
B
Example 2 Justify the formula for the area of a circle.
Thereare
n triangles, all congruent to △AOB, that surround point O and fill the n-gon.
Therefore, the measure of
(
(
)
180° h.
1 (2x)(h) = xh = r sin _
△AOB is _
n
2
Substitute your value for
h above to get △AOB =
180°
180°
_
(_
n )cos( n )
r 2sin
There are n of these triangles, so the area of the n-gon is
.
180°
180°
_
(_
n )cos( n )
nr 2sin
(
) (
.
)
180° cos _
180° . What
n–gon includes the factor n sin _
n
n
happens to this expression as n gets larger?
 Your expression for the area of the
Use your graphing calculator to do the following.
( ) ( )
180° cos _
180° as Y .
• Enter the expression x sin _
1
x
x
• View a table for the function.
• Use arrow keys to scroll down.
What happens to the value of x sin
The value gets closer to π.
180° cos _
180°
(_
x ) ( x ) as x gets larger?
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
)
180°
(_
n )
h = rcos
180° . Write a similar expression for h.
x = r sin _
n
We know that
 The area of
180° .
360° , and the measure of ∠1 is _
∠AOB is _
n
n
for the area of the n–gon. What happens to the value of
 Look at the expression you wrote The
expression gets closer to πr
2
this expression as n gets larger?
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Reflect
6.
When n is very large, does the area of the n-gon ever equal the area of the circle?
Why or why not?
No; the area of the n-gon is always less than the area of the circle.
7.
How does the above argument justify the formula A = πr 2?
When n is very large, the regular n-gon is virtually indistinguishable from the circle and
the expression for the n-gon’s area is virtually indistinguishable from πr 2.
Applying the Area Formula
Explain 3
Example 3 Find the area indicated.
 A rectangular piece of cloth is 3 ft by 6 ft. What is the area of the largest circle that
can be cut from the cloth? Round the nearest square inch.
The diameter of the largest circle is 3 feet, or 36 inches. The radius of the circle is 18 inches.
A = πr 2
2
A = π(18)
A = 324π
A ≈ 1,017.9 in2
So, the area is about 1,018 square inches.
 A slice of a circular pizza measures 9 inches in length. What is the area of the entire pizza?
Use 3.14 for π.
radius
The 9-in. side of the pizza is also the length of the
A = πr 2
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A = π 9
of the circle. So, r =
9 .
2
A = 81 π ≈ 254.34.
255in 2
To the nearest square inch, the area of the pizza is
.
Reflect
8.
Suppose the slice of pizza represents _16 of the whole pizza. Does this affect your
answer to Example 3B? What additional information can you determine with this fact?
No; the pizza is still assumed to be a circle with a radius of 9 in. The area of the slice of
1
the area of the whole pizza, or about 42 square inches.
pizza is _
6
Your Turn
9.
A circular swimming pool has a diameter of 18 feet. To the nearest square foot, what is
the smallest amount of material needed to cover the surface of the pool? Use 3.14 for π.
d = 18 so r = 9; A = πr 2 = (3.14)(9) ≈ 254.34; 254 ft
2
2
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