11.1­11.2 Theorems w­quizzes.notebook
March 12, 2012
11.1 Notes and Theorems
?
Apr 12­8:22 AM
Regular Interior
10 * 180
1800
1800/12
150
Apr 14­9:35 AM
1
11.1­11.2 Theorems w­quizzes.notebook
March 12, 2012
Regular Exterior & Central Angle
Regular Interior
72
120
720
60
36
10 * 180
1800
1800/12
150
30
Apr 14­9:35 AM
11.2
Apr 12­12:14 PM
2
11.1­11.2 Theorems w­quizzes.notebook
March 12, 2012
Quiz 11.1
115o
1. Find the value of x.
96o
64o
xo
2. The measure of each interior angle of a regular polygon is 144o. How many sides does the polygon have?
3. Find the value of x. 76o
52o
xo
Apr 12­12:16 PM
11.2 Notes and Theorems
Geometric Shapes
Apr 12­12:33 PM
3
11.1­11.2 Theorems w­quizzes.notebook
March 12, 2012
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
Apr 12­2:41 PM
11.2 Notes and Theorems
Find the area of an equilateral triangle with 12 foot
sides.
Solution:
Use s = 12 in the Equilateral Δ Area Theorem.
A = ¼√3 s2 = ¼√3 (122)
= ¼√3(144)
= ¼(144)√3
= 36√3
Apr 12­2:41 PM
4
11.1­11.2 Theorems w­quizzes.notebook
March 12, 2012
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
Solution:
C
Area of regular polygon
A = ½aP
1) Find the measurements of ΔDBC.
a) Find the measure of central ∠ABC:
∠ABC = 1/8 • 360o = 45o
b) Isosceles ΔABC, the altitude bisects base AC
∠DBC = 22.5o.
c) Use trig ratios to find the legs:
BD BD
DC DC
cos 22.5o = = and sin 22.5o = =
BC 2
BC 2
2) Determine the apothem (or the altitude of inner triangle).
3) Find the Perimeter of the octagon.
Apr 12­2:41 PM
11.2 Notes and Theorems
A regular octagon is incsribed in a circle with radius 2 units.
Find the area (A) of the octagon.
B
A
Solution:
Area of regular polygon
A = ½aP
D
C
2) Determine the apothem (a): a = BD = 2 cos 22.5o
3) Find the Perimeter (P):
P = 8(AC) = 8(2 DC) = 8(2 • 2 sin 22.5o) = 32 sin 22.5o
∴
The area of the octagon is A = ½aP
A = ½(2 cos 22.5o)(32 sin 22.5o) ≈ 11.3 square units
Apr 12­2:41 PM
5
11.1­11.2 Theorems w­quizzes.notebook
March 12, 2012
Reg Poly Quiz
A regular octagon is inscribed in a circle with
radius
3 units. Find the area (A) of the octagon.
B
A
D
C
Apr 12­2:41 PM
Reg Poly Quiz
A regular octagon is inscribed in a circle with
radius
3 units. Find the area (A) of the octagon.
B
A
Solution:
D
C
Area of regular polygon
A = ½aP
1) Find the measurements of ΔDBC.
a) Find the measure of central ∠ABC:
∠ABC = 1/8 • 360o = 45o
b) Isosceles ΔABC, the altitude bisects base AC
∠DBC = 22.5o.
c) Use trig ratios to find the legs:
BD BD
DC DC
cos 22.5o = = and sin 22.5o = =
BC 3
BC 3
2) Determine the apothem (or the altitude of inner triangle).
3) Find the Perimeter of the octagon.
Apr 12­2:41 PM
6
11.1­11.2 Theorems w­quizzes.notebook
March 12, 2012
Reg Poly Quiz
A regular octagon is inscribed in a circle with
radius
3 units. Find the area (A) of the octagon.
B
A
Solution:
Area of regular polygon
D
A = ½aP
2) Determine the apothem (a): C
a = BD = 3 cos 22.5o
a = 2.7716
3) Find the Perimeter (P): = 18.3688
P = 3 sin 22.5o = ½ side
2(3 sin 22.5o) = one whole side AC
8(AC) = 8(2(3 sin 22.5o)) = all 8 sides or Perimeter
∴
The area of the octagon is A = ½aP
A = ½(3 cos 22.5o)(48 sin 22.5o) ≈ 25.46 square units
A = ½(2.7716)(18.3688) ≈ 25.46 u2
Apr 12­2:41 PM
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
T
B
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)
or you c
= 24√3
ould u
30­60­9
0
A = ¼
√3s 2 se: Apr 12­2:41 PM
7
11.1­11.2 Theorems w­quizzes.notebook
March 12, 2012
Mar 22­5:19 PM
Internal / External Angle Quiz
1)
2)
3)
4)
5)
Mar 22­5:19 PM
8
11.1­11.2 Theorems w­quizzes.notebook
March 12, 2012
Mar 22­5:20 PM
9