Answer Key for Surface Area problems
Math 302B-006
Fall 2005
1b. Answer: 35628 ft2
Explanation/Notes: The height that the problem gives you is the height of the pyramid.
If you are going to use the formula that is given in the book (to find the surface area of a
pyramid, i.e., SA  B 
1
pl , where B is the area of the base of the pyramid, p is the
2
perimeter of the base of the pyramid, and l is the slant height of the pyramid), you’ll need
to do some calculations.
So,
since the dimensions of the base are given, all that is needed to find is the slant height
of they pyramid to be able to perform the calculations needed to use the formula. To find
the slant height, you can use the Pythagorean theorem since you have the height and base
of the right triangle, formed by the height of the pyramid, half of the length of the base,
and the slant height of the pyramid. The calculations are
60  65
2

2
 c 2  3600  4225  c 2  c  88.45
Now that we know the slant height of the pyramid or l, all that’s left to calculate is the
area of the Base [which is 120120  14400 ], and the perimeter of the base of the
pyramid [which is 4120  480 ].
Therefore, the final calculation of the surface area is

1
480
88.45
 

 14400  21228 14400  35628. [Do not forgot to include units in
2
your final answer.]
 1c. Answer: 242 ft2 or 34816.32 in2

Explanation/Notes: To calculate the surface area of the cylinder, you can use the
2
formula 2r  2rh . Another way to think about finding the surface area of a cylinder
is to find the area of the two bases (which are circles and whose area is represented by
2r 2 ) and the surface area of the lateral face (which when ‘flattened out’ is a rectangle
and whose area is represented by the height of the rectangle –which is the height of the
cylinder, h, and the width of the rectangle – which is the circumference of the base of the
cylinder, 2r )
Before you can calculate any of the areas mentioned above, you need to first convert the
given dimensions into either inches or feet (your choice). The given diameter can be
converted to 66 inches and the given height can be converted to 135 inches. Using these
dimensions will result in a total surface area in inches squared.
Therefore, the final calculations of the surface area is
66 135  233    4838.92  27977.4  34816.32 in2
2

To convert this value in to feet squared, you must remember to divide by 144 not 12,
since 1ft   12in   144in 2 . Dividing 34816.32 in2 results in 242 ft2.
2
2
2b. Answer: 695.2 ft2 or 100108.8 in2

Explanation/Notes: Since this shape is a prism, you can use the formula given to
find the total surface area of a prism or SA  2B  ph , where B is the area of the
base, p is the perimeter of the base, and h is the height of the prism. In order to
use this formula, we will need to find the area of the base, the perimeter of the
base, and the height of the prism. Once again, you will also need to convert the
given units to either inches orto feet. I will convert them to inches.
Converting the given units to inches, you will get 108 inches for the height of the
prism, 105 inches for one of the lengths of the base of the prism, 148 inches for
the other length of the prism, and finally 156 for the slant height of the base of
the prism.
To find the area of the base of the prism, you will need to find the area of a
trapezoid, or
1
h(a  b), where h is the height of the trapezoid and a and b are
2
the lengths of the base of the prism. We have a and we have b; what we do not
have is h. To find h, we need to use the Pythagorean theorem, using the slant
height of the trapezoid, ¼ of the length of the trapezoid, and the height of the
trapezoid
to form a right triangle. We have the slant height; to find the length of

the base of the right triangle, you need to first divide the longer base of the
trapezoid (148) by 4. This gives you 21.5 as the length of the base of the right
triangle. Using the Pythagorean theorem, results in
21.5  b
2


2
 156  24336  462.25  b 2  b  15.45
2
Now that we have all the required pieces of information, we can now calculate
total surface area of the prism:
1
(15.45)(105 148)  (105 148 156 156)(108) 100108.8 in 2
2
Again, if you would like to convert this answer to feet squared, you will need to
divide this answer by 144 (see the explanation for this given for problem #1c).
Doing this division, you would get 695.2 ft2.
3a. Answer: 1085 ft2
Explanation/Notes: The easiest way to calculate the total surface area of this
object is to calculate the area of each of the individual pieces that make up the
object.
The two longer sides of the object can each be thought of as a rectangle adjoined
with a trapezoid. Thus, you can find the area of each of the rectangles, find the
area of each of the trapezoids, and add them together. The area of each of the
rectangles is 20(8) = 160, but since there are 2 of them, you would double 160 to
get 320. The area of each of the trapezoids is
1
10(8  4)  60, but since there
2
are 2 of them, you would double 60 to get 120.
To get the area of the base, you multiply the length and the width and get
20(15)=300.

To get the area of one of the sides, you multiply its length and width and get
15(8)=120.
To get the area of the other side, you multiply its length and width and get
15(4)=60.
The last calculation to make, before adding up the individual areas to get the total
surface area, is the area of the angular base rectangle (the part that slants
upward). When you do this you get 15(11)=165.
Now adding up all the individual areas, 320+120+300+120+60+165, gives you
the total surface area of 1085 ft2.
5. Answer: 137.16 cm2, but since the question asks for square millimeters, your final
answer should be 13716 mm2.
Explanation/Notes: What is given is the diameter of the base of the inner
cylinder, the distance from the edge of the inner base to the edge of the outer
base, and the height of both cylinders.
To calculate this total surface area, you will need to calculate three areas: the area
of the lateral face of the outer cylinder; the area of the lateral face of the inner
cylinder; and the area of the ‘rim portion of the base” (i.e., the difference
between the areas of the larger base and smaller base).
To calculate the area of the lateral face of the outer cylinder, you need the height
of the cylinder and the circumference of the larger base (see the explanation for
problem 1c to see why these dimensions are necessary). These dimensions are
5.4   and 4 [the diameter of the larger base is 5.4 because you need to add 0.4
to 5 (the diameter of the inner base)]. Multiplying these values together gives
67.86.


To calculate the area of the lateral face of the inner cylinder, you need the height
of the cylinder and the circumference of the smaller base (see the explanation for
problem 1c to see why these dimensions are necessary). These dimensions are
5  and 4 . Multiplying these values together gives 62.8.
To find the area of the ‘rim portion of the base’, you need to find the difference
of the area of the larger base and the area of the smaller base. Doing so results in
 2.7   2.5  22.89 19.62  3.27 . Since we have 2 of these ‘rims’, we
2
2
need to multiply 3.27 by 2 and get 6.54.

Adding up the quantities, 6.54+62.8+67.86, gives the answer of 137.16 cm2, but
to get the final answer in mm2, you need to multiply this answer by 100 (using
the same logic used in 1c). This gives the final answer of 13716 mm2.