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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
