Transcript Area Problems - My Teacher Site

Example 1
Explain how you
could find the area
of the regular
hexagon shown.
Regular Inscribed Polygon
The diagram shows a
regular polygon
inscribed in a circle.
– Center of circle =
center of the
polygon
– Radius of circle =
radius of the
polygon
Regular Inscribed Polygon
The apothem of the
polygon is the
distance from the
center to any side of
the polygon.
– Apothem = height
of isosceles
triangle with 2 radii
as legs
Regular Inscribed Polygon
A central angle of a
polygon is an angle
formed by two
consecutive radii.
– Measure of central
angle = 360 
n
Areas of Regular Polygons
Perimeter and Area of Similar Figures
Objective:
1. To find the area of a regular n-gon
2. To describe the effects on perimeter and
area when dimensions are changed
proportionally
Example 2
1. Identify the center,
a radius, an
apothem, and a
central angle of the
polygon.
2. Find m<XPY,
m<XPQ, m<PXQ.
Example 3
Assume a regular
n-gon has a side
length of s and an
apothem of a. Find
a formula for the
area of the regular
n-gon.
Area of a Regular Polygon
The area of a regular n-gon with side length s
is half the product of the apothem a and
the perimeter P.
Regular 3-gon
What is the measure
of each central
angle in an
equilateral triangle?
What is the measure
of the angle formed
by the apothem and
the radius of the
triangle?
Regular 4-gon
What is the measure
of each central
angle in a square?
What is the
measure of the
angle formed by the
apothem and the
radius of a square?
Regular 5-gon
What is the measure
of each central
angle in a regular
pentagon? What is
the measure of the
angle formed by the
apothem and the
radius of the
pentagon?
Regular 6-gon
What is the measure
of each central
angle in a regular
hexagon? What is
the measure of the
angle formed by the
apothem and the
radius of the
hexagon?
Example 4
Find the area of each regular polygon.
Summary
Example 5
Find the area of each regular polygon.
1. A =
2. A =
3. A =
Example 6
Find the area of each regular polygon.
1. A =
2. A =
Example 7
Find a formula for the
area of a regular
hexagon in terms of
s, the side length.
Example 8
The perimeter of a regular hexagon is 48 cm.
What is the area of the hexagon?
Example 9
Find the area of the
shaded region.
Example 10
Rectangle ABCD ~ PQRS
with a scale factor of 3:4.
Find the perimeter and
area of rectangle PQRS.
A
9
Q
P
R
S
D
6
B
C
Perimeter of Similar Polygons
If two polygons are similar with the lengths of
corresponding sides in the ratio of a:b, then
the ratio of their perimeters is a:b.
Perimeter
of Polygon
I
Perimeter
of Polygon
II

a
b
Area of Similar Polygons
If two polygons are similar with the lengths of
corresponding sides in the ratio of a:b, then
the ratio of their areas is a2:b2.
Example 11
In the diagram ΔABC ~ ΔDEF. Find the
indicated ratio.
1. Ratio (red to blue) of the perimeters
2. Ratio (red to blue) of the areas
Example 12
Stuart is installing the
same carpet in a
bedroom and den.
The floors of the rooms
are similar. The carpet
for the bedroom costs
$117. Carpet is sold
by the square foot.
How much does it cost
to carpet the den?
Example 13
The polygons below are similar. Find the
values of x and y.