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Geometry
11.4 Areas of Regular Polygons
Definitions
• Regular polygon- a polygon that is equiangular and equilateral.
In the upper right side of your paper, please draw a regular triangle, a regular quadrilateral, a regular hexagon, and a regular octagon.
New words for the vocab list. Also add median of a trapezoid.
Definitions
• Center- the center of the circle that circumscribes the polygon.
Find the center of each polygon, you may or may not want to draw the circumscribed circle.
. center . center . center . center
Definitions
• Radius- the segment from the center to a vertex of the polygon.
Draw one radii of each regular polygon.
. . . . r r
rr
Definitions
• Central angle- the angle formed by two consecutive radii.
Draw one central angle of each regular polygon.
. . . .
Measure of a central angle = 360/n n is the number of sides
360/3
Find the measure of each central angle.
120o
360/4
90o
360/6
60o
360/8
45o
Many opportunities to use your skills of Pythagorean Theorem, 45-45-90, and 30-60-90 right triangles!
Definitions
• Apothem- The distance (perpendicular) from the center to a side
of the polygon.
Draw one apothem of each regular polygon.
. . . . a
aa
a
Area of a Regular Polygon
A = ½ a papothem
perimeter
WHY?
.apothem
x
Area of the green triangle = ½ apothem(x)
x
x
x
x
x
The regular hexagon is made up of 6 green triangles.
Area of the regular hexagon = ½ apothem(6x)
Area of the regular hexagon = ½ apothem(perimeter)
This is true for all regular polygons.
Fill in the table.
24 3
r a p A
1. 8
2.
3. 8
4. 72
1. 2. 3. 4.
.8
45o
90o
. . .4√2
4√2
8√2
P = 4(8√2)
32√2
A = ½ (4√2)(32√2)
128
6√3
6√3
6√3 6√3
3√345o
45o
3√3
3√3
3√6
3√6
A = ½ (3√3)(24√3)A = (3√3)(12√3)
108
A = (2√2)(32√2)
88√2
8√2
816
P = 4(16)
64
A = ½ (8)(64)A = (4)(64)
256
18
18
18 18
9
9
9
9√2
9√2
A = ½ (9)(72)
A = (9)(36)
324
A = ½ a p
Fill in the table.
5. 6. 7. 8.
6 3
9 3
r a p A
5. 86.
7. 88.
.8
. . .360o/3
120o60o
30o4
4
4√38√3
P = 3(8√3)
A = ½ a p
A = ½ (4)(24√3)
24√3
A = (2)(24√3)
48√3
2√3
2√3 2√3
√3
60o
30o
1
1
2
2
A = ½ (1)(6√3)
3√3
60o
30o
816
16
8√316√3
P = 3(16√3)
48√3
A = ½ (8)(48√3)A = (4)(48√3)
192√3
.3√3 3√3
3√3
60o
30o
3/23
3√32
3/23
A = ½ (3/2)(9√3)
27√34
Fill in the table. Please change some of the numbers and cross off the “Side” column.
9. 10. 11.
r a p A
1.
2.
3.5
3
5 2
. . .360o/6
60o
30o5√2
5√225√2
5√62
5√62
A = ½ a p
P = 6(5√2)
30√2
A = ½ (5√6/2)(30√2)A = (5√6/2)(15√2)
75√3
30o√3
1
2
2
2P = 6(2)
12
A = ½ (√3)(12)
6√3
30o5
52
5√32
5√32
5P = 6(5)
30
A = ½ (5√3/2)(30)
A = (5√3/2)(15)
75√32
Word Problems: Who can write these on the board? Find the area of…1) An equilateral triangle with radius 6√3.
2) A regular hexagon with perimeter of 48.
81√3 square units
96√3 square units
Word Problems: Who can write these on the board? Find the area of…3) A square with radius equal to 24.
4) A regular hexagon with apothem equal to 12√3
5) A regular dodecagon(12-sided) with side = r & apothem = s.
1152 square units
864√3 square units
6rs square units
HW
• P 443 (1-22 skip 17)