Surface Area

The student is able to (I can):

Calculate the surface area prisms, cylinders, pyramids, and cones

The surface area is the total area of all faces and curved surfaces of a three-dimensional figure. The lateral area of a prism is the sum of the areas of the lateral faces.

Lets look at a net for a hexagonal prism:

What shape do the lateral faces make?

(a rectangle)

If each side of the hexagon is 1 in., what is the perimeter of the hexagon?

What is the length of the base of the big rectangle?

6 in.

6 in.

This relationship leads to the formula for the lateral area of a prism:

L = Ph

where P is the perimeter and h is the height of the prism.

For the total surface area, add the areas of the two bases:

S = L + 2B

We know that a net of a cylinder looks like:

The length of the lateral surface is the circumference of the circle, so the formula changes to:

L = Ch where C = pid or 2pir

and the formula for the total area is now:

S = L + 2pir2

Examples Find the lateral and total surface area of each.

1.

2. 10 cm

14 cm

4"3"

8"

5"

P = 3+4+5 = 12 in.B = (3)(4) = 6 in2

L = (12)(8) = 96 in2

S = 96 + 2(6) = 108 in2

C = 10pi cmB = 52pi = 25pi cm2

L = (10pi)(14) = 140pi cm2

S = 140pi + 2(25pi)= 190pi cm2

To find the lateral area of the pyramid, find the area of each of the faces.

Perimeter of base

slant height()

1L P

2=

For the total surface area, add the area of the base.

S = L + B

Likewise, for a cone, the lateral area is

( )1L 2 r r2

= pi = pi

and the total surface area is2S L r= + pi

Examples Find the lateral and surface area of the following:

1.

2.

8 in.

20 in.

5 m

5 m

28 3B 6

4

=

296 3 in=

1L [(6)(8)](20)

2=

2480 in=

2S 480 96 3 in= +2646.3 in

5 2 m L (5)(5 2)= pi225 2 m= pi

2S 25 2 25 m= pi + pi2189.6 m