area-of-figures-2014-12-18.notebook
Geometry
March 26, 2015
Area of a Regular
Polygons
Area of Circles
Area of Figures
Area of Sectors
2014-10-26
Area of Regular Polygons
regular polygon is a polygon that has all its sides and
Area of Regular Polygons
Contents
area-of-figures-2014-12-18.notebook
Circumscribing a Regular Polygon
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Circumcenter and Circumradius
The center of a regular polygon is the center of its
circumscribed circle, shown here as "C."
on which lie all the vertices of the polygon.
That circle is called the circumcircle of the polygon.
C
This is called the circumcenter of the polygon.
The radius of a regular
polygon is the radius of the
circumscribed circle shown here as
C
This is called the circumradius.
central angle of a regular polygon is an angle with one
vertex at the circumcenter and two vertices on the circumcircle.
The apothem of a Regular Polygon is the shortest distance
from the center of the polygon to one of its sides.
The sides of the central angle are radii of the circle.
An apothem is perpendicular to a side of the polygon.
The degrees of the central angle
can be found using the formula
An apothem is also the altitude
(height) of the isosceles triangle
formed by the sides of a central
angle and the side of the polygon
that angle subtends.
Central angle =
where n is the number of sides
in the regular polygon.
C
C
area-of-figures-2014-12-18.notebook
March 26, 2015
Area of Regular Polygons
Each triangle has a height equal to the apothem, "a."
number of sides, n: b = P/n.
Area of Regular Polygons
= 1/2
= 1/2(P/n)
There are n triangles in a
regular polygon, so the area
of the polygon is given by:
What is the area of a triangle
whose base is P/n and whose
height is a?
Area of Regular Polygons
Example
Let's find the area of a regular pentagon whose sides have length 7.
But, often we are not given both P and a.
area-of-figures-2014-12-18.notebook
March 26, 2015
Example
Example
Let's find the area of a regular pentagon whose sides have length 7.
Let's find the area of a regular pentagon whose sides have length 7.
For a pentagon, n = 5, so it's
perimeter will be 5 times the
length of a side.
For a pentagon, n = 5, so it's
perimeter will be 5 times the
length of a side.
But, how do we find the
apothem?
But, how do we find the apothem?
Let's look at one of the five
triangles that comprise the
pentagon.
Let's look at one of the five
triangles that comprise the
pentagon.
Example
Example
Let's find the area of a regular pentagon whose sides have length 7.
Let's find the area of a regular pentagon whose sides have length 7.
The two legs of the right triangle
shown are a and 3.5.
The central angle is given by:
m = 360/n = 360º/5 = 72
r
54º
72º
r
54º
Since the s must add to 180º
and the measures of the base
of an isosceles triangle are equal,
those base s must = 54º.
The apothem, a, is the altitude of
this triangle.
θ = opposite / adjacent
tan θ = opp / adj
r
36º r
opp = adj (tan )
a = 3.5 (tan 54º)
54º
54º
a = 3.5 (1.38)
3.5
a = 4.8
area-of-figures-2014-12-18.notebook
Example
Let's find the area of a regular pentagon whose sides have length 7.
March 26, 2015
Example
Fnd the area of a regular octagon whose sides have length 8.
and a = 4.8.
r
54º
72º
r
54º
Area of Regular Polygons
Area of Regular Polygons
Calculate the apothem of the regular polygon shown in the
Calculate the side length of the regular polygon
area-of-figures-2014-12-18.notebook
Area of Regular Polygons
Calculate the perimeter & area of the regular polygon
Area of Regular Polygons
Calculate the perimeter & area of the regular polygon
March 26, 2015
Area of Regular Polygons
Calculate the side length of the regular polygon shown in
Area of Regular Polygons
Calculate the apothem of the regular polygon shown in the
area-of-figures-2014-12-18.notebook
March 26, 2015
Area of Regular Polygons
Area of Regular Polygons
7
8
Calculate the perimeter of the regular
Calculate the area of the regular
Area of a Circle
Interestingly, the formula, A = 1/2Pa, also
leads to the formula for the area of a
circle.
Area of Circles and Sectors
If you let n go to infinity, the regular
polygon approaches the shape of a circle.
The apothem, a, approaches the radius of
the circle, r.
And, the perimeter of the polygon
approaches the circumference of the
circle: 2πr.
Then, A = 1/2Pa approaches
Contents
A = 1/2(2πr)(r)
A = πr2
area-of-figures-2014-12-18.notebook
9 Find the area of a circle that has a radius of 8 in.
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10 Find the area of a circle that has a diameter of 17 in.
A 4π in2
A 8.5π in2
B 8π in2
B 17π in2
C 16π in2
C 72.25π in2
D 64π in2
D 289π in2
11 Find the area of a circle that has a circumference of
18π in.
A 324π in2
Sectors of Circles
A sector of a circle is the portion of the circle enclosed by two
radii and the arc that connects them.
B
B 81π in2
Minor Sector
C 18π in
2
A
D 9π in2
Major Sector
C
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12 Which arc borders the minor sector?
13 Which arc borders the major sector?
C
C
A
A
C
B
B
C
D
D
A
θ
The area of a complete circle is
circle = πr
Similar to the arc length, we have to
find the fraction of the circle in the
sector and multiply this fraction by the
area of the entire circle to find the area
of a sector
In this case, we know that the measure
of our central angle,
C
r
B
D
If the central angle of the sector is given in degrees, that's just the
measure of that angle divided by 360 degrees, yielding:
sector = (θ/360 )(πr )
when
is the central angle of the sector measured in degrees
A
But, if we are also told that the radius of
the circle is 20 cm, we can determine the
C 20 cm
CB.
D
Using our formula, we know
that
and r = 20 cm.
If we substitute these numbers into our formula, it will give us the
π(20)
sector
sector
= 383.97 cm
B
area-of-figures-2014-12-18.notebook
March 26, 2015
14 Find the area of the sector to the nearest tenth.
A
B
ratio and solve it in one step.
A
Area of Sector
Area of Circle
C 20 cm
Central angle
360
C
sector
B
πr
sector
360
360
D
sector
sector
=
360
(πr )
π(20)
cm = 383.97 cm
sector
15 Find the area of the major sector to the nearest tenth.
C
= (θ/360 )(πr )
16 Find the area of the minor sector of the circle. Round
your answer to the nearest hundredth.
8 cm
C
A
5.5 cm
A
T
sector
= (θ/360 )(πr )
T
area-of-figures-2014-12-18.notebook
17 Find the Area of the major sector for the circle. Round
your answer to the nearest thousandth.
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18 What is the central angle for the major sector of the
circle?
C
C
12 cm
15 cm
A
A 120
T
19 Find the area of the major sector. Round to the nearest
thousandth.
20 If a circle is divided into 2 sectors, one major and one
minor, then the sum of the areas of the 2 sectors is equal
to the total area of the circle.
C
False
15 cm
A 120
area-of-figures-2014-12-18.notebook
21 A group of 10 students order pizza. They order 5 12"
pizzas, that contain 8 slices each. If they split the pizzas
equally, how many square inches of pizza does each
student get (to the nearest hundredth)?
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22 You have a circular sprinkler in your yard. The sprinkler
has a radius of 25 ft. How many square feet does the
sprinkler water if it only rotates 120 degrees?
Question 13/25
The remaining slides in this presentation contain questions from the
answer these questions.
Good Luck!
of Contents
area-of-figures-2014-12-18.notebook
March 26, 2015