11.1 areas of triangles and parallelogramspapgeometry.weebly.com › uploads › 2 › 2 › 8 › 6...
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Warm-up
Find the area of the shaded region:
About 66.86 sq cm
20 cm
10
22
L E B
A D M
LAMB is a parallelogram
ADE is an equilateral triangle
About 168.25 sq m
Area of quadrilateral is 784 m²
A.5C
Trig work
8. 13.6
9. 6.4
10. 5.0
11. 19.3
12. 8
13. 7.3
14. 29
15. 52
12 5 12sin ,cos , tan
13 13 5
12 5 5cos ,sin , tan
13 13 12
M M M
L L L
Use a calculator to find each ratio, round to the nearest thousands.
2.tan 87 = 19.081 3. sin 44 =.6947 4. cos 46 =.6947
Use a calculator to find each angle measure to the nearest degree.
5. tan x 0.5774 6. sin x 0.8000 7. cos x 0.7071
30 53 45
1.
16. 59
17. 28.68, 40.96
18. 44 cm
19. 10.46
20. 23.18, 6.21
21. 496.8 ft
22. 264.59 ft
23. 35.33 yds
24. 40.09 feet
1.
What are the formulas for
finding the area of: A = ½ bh
A = bh
A = ½ (b1 +b2)h Rhombus
A = ½ d1 d2
What about other
‘regular’ polygons?
Find the area of three figures –
the hexagon, the pentagon and
the octagon . . .
Think about what you do know . . .
– Use the measurements given, assume the are
regular polygons
11.6: Areas of Regular
Polygons
Class Activity
Area of a regular polygon
11.6 Area of Regular Polygons and
Circles Area of a Regular Polygon: : ______________ where a = _____________ r = ____________ and p = _________________________
*** The apothem is always perpendicular to the side of the polygon!! We are always making ISOSCELES TRIANGLES!!!!
A = ½ ap
apothem radius
perimeter of the polygon
Class Exercises: 1. Find the area of a regular octagon that has a
perimeter of 72 inches.
1. Find the central angle:
.
A B
C
D
E F
G
H
O
P
360/8 = 45
2. Redraw the isosceles triangle: Because the perimeter
is 72 in., we know that each side of our octagon is 9 in.
We also know that OP(the apothem of our polygon) is
the angle bisector, median, and altitude of our triangle
so that…
We do this to find
our apothem, a.
Class Exercises: #1 continued 1. Find the area of a regular octagon that has a
perimeter of 72 inches.
.
A = ½ ap
A = ½ (10.9)(72)
A = 392.4 square inches
3. Solve for a: We will have to use a trig function (SOH
CAH TOA) in order to solve for our apothem…which one?
tan 22.5 = 4.5/a
a = 4.5/tan22.5
a = 10.9
4. Plug in “a” and “p” values into A = ½ap to get the area:
Class Exercises: 2. Find the area of a regular pentagon
whose perimeter is 60 centimeters.
Each side will be 12 cm
The central angles = 72
Tan 36 = 6/a
a = 6/tan 36
a ≈ 8.26
A = ½ ap
A = ½(8.26)(60)
A = 247.7 square cm
36o
12
36o
36o
6
36o
a
The apothem of a regular
hexagon is cm. Find the
perimeter and area of the regular
hexagon.
9 3
Class Exercises: Find the area of the shaded region of m<BAC = m<BCA = 60.
B
C A
O
R
8
Write a Word Equation for what you are trying to do:
Area of the Circle - Area of Triangle = Shaded Area
r = 8
2)8(A
A = 201.06
201.06 -
A = 1/2 ap
Central angle =
360/3 = 120
8/2 = a/1 so a =4
AR = long leg = 4√3
Perimeter = 24√3
A = ½ (4)(24√3)
83.14 = 117.92 units
squared
1 2 √3
In pairs work on Lesson 11.6
Homework
WS 11.6 AREA of Regular Polygons