Unit 6 area of regular polygons

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