unit 6 area of regular polygons
TRANSCRIPT
The Apothem
The apothem (a) is the segment drawn from the center of the polygon to the midpoint of the side (and perpendicular to the side)
aaa
Deriving the Formula - Squares
sa
The diagonals of the square divide it into four triangles with base s and height a. The area of each triangle is sa.
2
1
Since there are 4 triangles, the total area is 4( )sa or (4s)a. Since the perimeter (p) = 4s the formula becomes A = ap
2
1
2
1
2
1
Deriving the Formula - Triangles
sa
Connecting the center of the equilateral triangle to each vertex creates three congruent triangles with area A = sa. Since there are 2
1
three triangles, the total area is 3( )sa,
or (3s)a. Since the perimeter = 3s, the
formula may be written A = ap
2
1
2
1
2
1
Deriving the Formula - Regular Hexagons
sa
Connecting the center of the regular hexagon to each vertex creates six congruent
triangles with area A = sa.
2
1
Since there are six triangles, the total
area is 6( )sa, or (6s)a. Since the
perimeter = 6s, the formula may be
written A = ap
2
1
2
1
2
1
Finding the apothem - Square
The apothem of a square is one-half the length of the side.
sa
If s = 15, a = ?
a = 7.5If a = 14, s = ?
s = 28
Find the apothem - Triangles
The apothem of an equilateral triangle is the short leg of a 30-60-90 triangle where s/2 is the long leg.
s/ 2
a
sa
30
60
90
Then a = (s/2)/3
or
3
32s
Find the apothem - Triangles
sa
If s = 18, a = ?
a = 3 3
If s = 24, a = ?
a = 4 3
If s = 10, a = ?
a = 5
33
Find the side - Triangles
sa
If the apothem is 6 cm, the side = ?
s = 12 3
If the apothem is
2.5 cm, the side = ?
s = 5 3
Finding the apothem - Hexagons
sa
The apothem of a regular hexagon is the long leg of a 30-60-90 triangle.
60 90
30
Therefore, the apothem is (s/2)
s/ 2a
3
Finding the apothem - hexagons
sa
If the side = 12
the apothem = ?
a = 6 3
If the side = 5
the apothem = ?
a = 5
23
Finding the side - hexagons
sa
If the apothem = 12, the side = ?
s = 8 3
If the apothem = 16, the side = ?
s = 32
33
Finding the area - Squares
sa
a = 6 cm
Find the area
A = 144 cm2
A = 288 cm2
A = 50 cm2
s = 5 2 cmFind the area
a = 6 2 cmFind the area
Finding the Area - Triangles
as
If a = 3 cm, find the area of the triangle
A = 27 3 sq cm
Finding the Area - Triangles
as
If a = 5 cm, find the area of the triangle
A = 75 3 sq cm
Finding the Area - Triangles
as
I f a = 3 2 cm findthe area of the triangle
A = 54 3 sq cm
Finding the Area - Triangles
as
If the side of the triangle = 10 cm, find the area of the triangle
A = 25 3 sq cm
Finding the Area - Triangles
as
I f s = 8 3 cm findthe area of the triangle
A = 48 3 sq cm
Finding the area - hexagons
s a
If the a = 6 cm, find the area of the hexagon.
A = 72 3 sq cm
Finding the area - hexagons
s a
I f a = 8 3 cm findthe area of the hexagon
A = 384 3 sq cm
Finding the area - hexagons
s a
I f s = 8 cm findthe area of the hexagon
A = 96 3 sq cm
Finding the area - hexagons
s a
I f s = 8 2 cm findthe area of the hexagon
A = 192 3 sq cm