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

1. Surface Area
The student is able to (I can):
• Calculate the surface area of prisms, cylinders, pyramids,
and cones

2. The surface areasurface areasurface areasurface area is the total area of all faces and curved
surfaces of a three-dimensional figure. The lateral arealateral arealateral arealateral area of a
prism is the sum of the areas of the lateral faces.
Let’s look at a net for a hexagonal prism:
What shape
do the
lateral faces
make?
(a rectangle)

3. 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.

4. 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

5. 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 = πd or 2πr
and the formula for the total area is now:
S = L + 2πr2

6. Examples: Find the lateral and total surface area of each.
1.
2.
10 cm
14 cm
4"3"
8"
5"

7. 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 = ½(3)(4) = 6 in2
L = (12)(8) = 96 in2
S = 96 + 2(6) = 108 in2
C = 10π cm
B = 52π = 25π cm2
L = (10π)(14) = 140π cm2
S = 140π + 2(25π)
= 190π cm2

8. To find the lateral area of the pyramid, find the area of each
of the faces.
Perimeter of base
slant
height
(ℓ)

9. To find the lateral area of the pyramid, find the area of each
of the faces.
Perimeter of base
slant
height
(ℓ)
1
2
L P= ℓ
For the total surface area, add the
area of the base.
S = L + B

10. Likewise, for a cone, the lateral area is
( )
1
2
2
L r r= π = πℓ ℓ
and the total surface area is
2
S L r= + π

11. Examples: Find the lateral and surface area of the following:
1.
2. 8 in.
20 in.
5 m
5 m
5 2 m

12. Examples: Find the lateral and surface area of the following:
1.
2. 8 in.
20 in.
5 m
5 m
2
2
8 3
6
4
96 3 in
B
 
=  
 
=
2
1
[(6)(8)](20)
2
480 in
L =
=
2
2
480 96 3 in
646.3 in
S = +
≈
5 2 m
2
(5)(5 2)
25 2 m
L = π
= π
2
2
25 2 25 m
189.6 m
S = π + π
≈