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CC Geometry H
Aim #3: How can we approximate the area of a circle (also known as a disk)?
Do Now: Complete the chart for the regular hexagons with given side lengths.
r
side
length
r
h
perimeter
r
h
h
radius
height
central angle
(center to vertex)
(center to side)
(between 2 radii)
4
8
5
Relevant Vocabulary of Polygons
• The height, h, of a regular polygon is the length of the segment from the
center of polygon perpendicular to a side of the polygon.
[It is the distance
from the center to a side.]
• s is the length of a side of the regular polygon.
• The radius, r, is the length of a segment joining the center to a vertex.
• The central angle is an angle formed by two consecutive radii of a polygon.
The regular hexagon is inscribed in circle C with radius r.
Find the area of the hexagon, in terms of h and s.
A=6
C
r
h
Area of 1 triangle
=6 (
__ __ )
= (6 s)
s
A=
We can generalize the area formula above, in terms of perimeter, of any regular
polygon Pn. Regular polygon Pn has n sides, each of equal length.
Area(Pn) = Perimeter(Pn) x
hn
2
1. Approximate the area of a disk (circle) of radius 4 using an inscribed regular
hexagon.
2. Approximate the area of a disk of radius 8 using an inscribed regular pentagon.
8
USING INSCRIBED POLYGONS
a) How well does a square approximate the area of a disk? Create a sketch of P4
(a regular polygon with 4 sides; a square) in the following circle. Shade the area
of the disk that is not included in P4.
b) Next, create a sketch of P8 in the following circle.
c) Indicate which inscribed polygon has a greater area.
Area (P4)
Area (P8)
d) Will the area of inscribed regular polygon P16 be greater or less than the area of
P8?
Which is a better approximation of the area of the disk?
The area of an inscribed polygon where n ≥ 3 is _________ than the area of the
disk/circle. Finding the inscribed area would give us a lower limit area
approximation.
USING CIRCUMSCRIBED POLYGONS
We will now approximate the area of a disk using circumscribed polygons.
a. Circumscribe (draw a square on the outside) of the circle
such that each side of the square intersects the circle at
one point. We will use prime notation for each of our outer
polygons. We are sketching P'4.
b. Create a sketch of P'8.
P': Circumscribed polygons
c. Compare the areas of the circumscribed polygons: Area of P'4
Area of P'8
d. Which is a better approximation of the area of the circle: P'4 or P'8?
Explain.
Looking at both polygons, P and P', in the same diagram...
The area of the circle is ____________ than the area of P.
The area of the circle is _____________ then the area of P'.
P'
Area of Pn
P
Area of circle
Area of P'n
From this close-up view of a portion of the
polygons and circle, we see that the area of the
circle is "trapped" between the areas of the outer
and inner polygons.
1. Approximate the area of a disk of radius 2 using an inscribed regular hexagon.
2. Approximate the area of a disk of radius 2 using a circumscribed regular hexagon.
2
3. Based on the areas of the inscribed and circumscribed hexagons, what is an
approximate area of the given disk to the nearest tenth? Take their average.
2
4. What is the area of the disk by the area formula (A = πr ), and how does your
approximation compare?
Let's Sum it Up!
• The area of a regular polygon:
or
Name ______________________
CC Geometry H
Date _______________
HW #3
1a) Name the center, a radius, a height and a central angle of the regular polygon.
T
S
b) Find m≮RCY, m≮RCZ, and m≮ZRC.
R
U
Z
c) If RY = 8, find CZ, to the nearest tenth.
C
V
Y
X
W
2. Find the perimeter and area of a regular octagon with side 15 in. and radius
19.6 in. (answer to the nearest tenth)
in
19.6
15 in
3. The perimeter of a regular nonagon is 18 in. Find the area, to the nearest inch.
4. True or false?
a) The area of a regular n-gon of a fixed radius r increases as n increases.
b) The height of a regular polygon is always less than the radius.
c) The radius of a regular polygon is always less than the side length.
5. Approximate the area of the disk using an inscribed regular pentagon with side
length 12. (nearest whole number).
12
Mixed Review:
1. ΔABC ~ ΔDEF. AB = 8 inches, DE = 10 inches, and the area of ΔABC is 56
square inches. What is the area of ΔDEF?
2. In ΔABC altitude BD is drawn. If AB = 4√5, BD = 8, and AC = 16, is ΔABC a
right triangle? Justify your answer.