11.1 Notes and Theorems
?
11.2
Quiz 11.1
1. Find the value of x.
o
115
o
96
64o
x
o
2. The
measure of each interior angle of a regular polygon is
o
144 . How many sides does the polygon have?
3. Find the value of x.
o
76
o
52
x
o
11.2 Notes and Theorems
Geometric Shapes
Circumscribed Circle
Inscribed Polygon
Inscribed Circle
Circumscribed Polygon
S
A
B
T
11.2 Notes and Theorems
Find the area of an equilateral triangle with 4 foot sides.
Solution:
Use s = 4 in the Equilateral Δ Area Theorem.
A = ¼√3 s2
= ¼√3 (42)
= ¼√3 (16)
= ¼ (16) √3
= 4√3
11.2 Notes and Theorems
A regular octagon is inscribed in a circle with radius 2 units.
Find the area (A) of the octagon.
B
A
D
C
Solution:
Area of regular polygon
1)
A = ½aP
Find the measurements of ΔDBC.
a) Find the measure
of central
ABC:
1
o
o
ABC = /8 · 360 = 45
b) Isosceles ΔABC,
the altitude bisects base AC
o
DBC = 22.5 .
c) Use trig ratios to find the legs:
cos 22.5o =
BD
BC
=
BD
2
and sin 22.5o =
DC = DC
BC
2
11.2 Notes and Theorems
A regular octagon is inscribed in a circle with radius 2 units.
Find the area (A) of the octagon.
B
A
D
C
Solution:
Area of regular polygon
A = ½aP
2)
Determine the apothem
(a):
o
a = BD = 2 cos 22.5
3) Find the Perimeter (P):
·
o
o
P = 8(AC) = 8(2 DC) = 8(2 2 sin 22.5 ) = 32 sin 22.5
The area of the octagon is A = ½aP
o
o
A = ½(2 cos 22.5 )(32 sin 22.5 ) ≈ 11.3 square units
11.2 Notes and Theorems
Find the perimeter (P) and area (A) of a regular hexagon with side
length of 4 cm and radius 4 cm.
S
A
B
T
Solution:
1)
Find the perimeter:
Hexagon = 6 sides with 4 cm length
P = 6(4) = 24 cm.
A = ½aP
2)
Determine the apothem (a): consider Δ SBT.
BT = ½(BA) = ½(4) = 2cm
Use the Pythagorean Theorem to find the apothem ST.
a = √42 - 22 = 2√3 cm.
∴ The area of the hexagon is A = ½aP
A = ½(2√3)(24)
= 24√3