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11.2 Areas of Regular Polygons
Unit IIIH Day 2
Do Now
• What are the two formulas for a 30º-60º-90º
triangle?
▫ hypotenuse = _________
▫ long leg = __________
• If the hypotenuse = x, label the other two sides
in terms of x.
x
Formula for area of equilateral
triangle
• A = ½ bh…
Theorem 11.3 Area of an
equilateral triangle
• The area of an
equilateral triangle is
one fourth the square of
the length of the side
times √3
A = ¼ 3 s2
s
s
s
Ex. 2: Finding the area of an
Equilateral Triangle
• Find the area of an equilateral triangle with 8inch sides.
Vocab
• The center of the polygon and radius of the
polygon are the center and radius of its
circumscribed circle, respectively.
• The apothem is the height of a triangle between
the center and two consecutive vertices of a
polygon.
Hexagon ABCDEF with
center G, radius GA,
and apothem GH
Apothem and Area of Regular Polygons
• A = Area of 1 triangle • # of triangles
Theorem 11.4 Area of a Regular
Polygon
• The area of a regular n-gon with side lengths s is
half the product of the apothem (a) and the
perimeter (P).
▫ So A = ½ a  P, or A = ½ a • n  s.
Vocab.
• A central angle of a regular polygon is an
angle whose vertex is the center and whose sides
contain two consecutive vertices of the polygon.
• You can find the measure of each central angle
of the regular polygon by ________________
▫
360º/
n
= central angle
Do Now
• Fill in the following:
▫ Sum of the interior angles of an n-gon:
▫ Each interior angle of a regular n-gon:
▫ Sum of the exterior angles of an n-gon:
▫ Each exterior angle of a regular n-gon:
▫ Area of a regular n-gon:
▫ Each central angle of a regular n-gon:
Ex. 3: Finding the area of a
regular polygon
• A regular pentagon is inscribed in a circle with
radius 1 unit. Find the area of the pentagon.
Ex. 4
• A regular octagon is inscribed in a circle with
radius 4 centimeters. Find the area of the
octagon.
Ex. 5: regular dodecagon
• The enclosure on the floor
underneath the Foucault
Pendulum at the Houston
Museum of Natural
Sciences in Houston, Texas,
is a regular dodecagon with
side length of about 4.3 feet
and a radius of about 8.3
feet. What is the floor area
of the enclosure?
Foucault Pendulum, Houston
Museum of Natural Sciences
Ex. 6
• What is the area of a regular nonagon with a side
length of 6 in.?
Ex. 7
• What is the area of a regular pentagon with an
apothem length 10 cm.?
Closure
• You are given the radius of a regular polygon.
Describe how you would find its area.