Topic 5: Visualization and Volumes
page 19
CYLINDER
LA = ph (as above, since a cylinder is a prism)
= (2  r)h
B =  r2
TA = LA + 2B = 2  rh + 2  r2
PYRAMIDS
1
1
1
1
la + lb + la + lb
2
2
2
2
1
= l (2a + 2b)
2
1
= pl
2
LA =
l = slant height
CONE
LA =
1
1
pl which becomes LA = (2 r )l
2
2
B =  r2
TA = LA + B =  rl   r 2
We will use a model to visualize how we can calculate the surface area of a sphere. (There is
no need to distinguish between lateral area and total area for a sphere, as a sphere really does not
have a base. What we calculate is called just the surface area.)
Picture this scenario. Fill a sphere with golf tees, so that he pointy end of each tee meets at the
center of the sphere. We want the tees to be as close together as possible. If there were enough
Topic 5: Visualization and Volumes
page 20
tees, and they were exceptionally long and narrow, we can imagine that the part of the tee where
the gold ball goes would together completely become the surface area of the sphere. Okay?
If the number of tees is n, then the volume is the volume of one tee (a cone, almost)
times n.
1
V n tees = n ( Bh)
3
which will approach the volume of the sphere
Make sure that you understand each step. The best way to do this is to be explain
it to someone else until they understand it, too.
1
4 3
r
n ( Bh) =
3
3
1
4
(nB)h =  r 3
3
3
1
4
(nB)r =  r 3
3
3
nB
= n times the area of the base of one tee
= one coverage of the sphere
= SA of the sphere
1
4
(SA)r =  r 3
3
3
(SA)r = 4  r 3
SA = 4  r 2
Exercises.
12.
Calculate the LA and TA of the solid (a measurement skill).
Topic 5: Visualization and Volumes
13.
14.
15.
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Topic 5: Visualization and Volumes
16.
17.
18. (Find SA only)
19.
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Topic 5: Visualization and Volumes
20.
The volume of a sphere is 6  cubic cm. What is the surface area?
21.
The volume of a square pyramid is 50 in3. The height is 6 in. What is the LA?
22.
Suppose that all of the dimensions of a 3 x 4 x 8 prims are doubled?
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a) What happens to the volume?
b) What happens to the lateral area and to the total area?
23.
In a square pyramid whose base is 4 cm x 4 cm and whose height is 6 cm, suppose that a
plane which is parallel to the base slices the top of the pyramid off. What remains is
called the frustum of the pyramid. Suppose that the parallel plane is located halfway
from the base and the apex. What is the volume of the frustum? (Provide a strategy to
find this volume.