Sec' 10 3 Area of Regular Polygons

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Sec. 10 3 Area of Regular Polygons

• Objective
• 1) To find the area of a regular polygon.

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• A Area of 1 triangle of triangles
• ( ½ apothem side length s) of sides
• ½ apothem of sides side length s
• ½ apothem perimeter of a polygon
• This approach can be used to find the area of any
  regular polygon.

F
A
H
a
E
G
B
D
C
Hexagon ABCDEF with center G, radius GA, and
apothem GH
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Regular Polygon All sides ? All ?s ?
Vertex ?s
Regular Pentagon inscribed inside of a circle.
Center of circle and the reg. pentagon.
Radius of circle and the reg. pentagon.
5 ? isosceles ?s
Apothem - ? distance from the center to a side.
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Find the missing ? measures.
of sides
Vertex ?
360
n
m?1 360/n 360/5 72
1
2
Apothem bisects vertex ? of the Is. ? formed by
the radii.
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m?2 m?3 90 180 36 m?3 90 180 m?3 54
m?2 ½ m?1 m?2 ½ (72) m?2 36
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Finding the area of a reg. polygon?

• Suppose reg. n-agon w/ side s
• The radii divides the figure into n ? isos ?s
  each w/ area ½ as

s
Apothem Side
p ns

• Area of n-gon n ½ as
• Perimeter of n-gon ns

Perimeter of reg. Polygon
Length of sides
of sides
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The area formula for any reg. Polygon
A ½ ap
Apothem
Area of any reg. Polygon
Perimeter of reg. polygon
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Example 2 find the area of a regular decagon
with a 12.3in apothem and an 8in side.
p ns (10)(8cm) 80cm
A ½ ap ½ (12.3in)(80in) 492in2
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Example 3 Find the apothem of a reg. hexagon
with sides of 10mm.
m?1 360/6 60 m?2 ½ (60) 30 m?3
10mm
1
60
2
3
a v3 short side a v3 (5mm) a 5v3 8.66
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30-60-90 ? shortcut
a
Could you find the area of it now??
60
5mm
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Example 4 Find the area of a regular pentagon
with 7.2ft sides and a 6.1ft radius.
p ns (5)(7.2ft) 36ft
A ½ ap ½ (4.92ft)(36ft) 88.6ft2
6.1ft
a2 b2 c2 a2 3.62 6.12 a2 12.96
37.21 a2 24.25 a 4.92
6.1ft
a
3.6 ft
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What is the difference between the radius and the
apothem?
The radius is longer because it is from the
center all the way to the edge of the circle but
the apothem is just from the center to the middle
of the side
What is the formula for the area of a regular
polygon?
A ½asn
A ½aP