Relating Length, Surface Area, and Volume
The Math IIC tests not only whether you’ve memorized the
formulas for the different geometric solids, but also if you understand
those formulas. The test gauges your understanding by asking you
to calculate the lengths, surface areas, and volumes of various
solids. The Math IIC will ask you about the relationship between
these three properties. The exam includes two kinds of questions
that cover these relations.
The first way the Math IIC will test your understanding
of the relationship between the basic measurements of geometric
solids is by giving you the length, surface area, or volume of various
figures and asking you to compare their dimensions. The math needed
to answer comparing dimensions questions isn’t all that hard. But
in order to get to the math, you need to have a good grasp of the
formulas for each type of solid and be able to relate those formulas
to each other algebraically. For example:
surface area of a sphere is the same as the volume of a cylinder.
What is the ratio of the radius of the sphere to the radius of the
This question tells you that the surface area of a sphere
and the volume, a cylinder are equal. A sphere’s surface area is
where rs is
the radius of the sphere.
A cylinder’s volume is π
, where rc
the radius of the cylinder and h
is its height. Therefore:
The question asks for the ratio between the radii of the
sphere and cylinder. This ratio is given by rs
Now you can solve the equation 4πrs2
= πrc2 h
for the ratio
The second way the Math IIC will test your understanding
of the relationships between length, surface area, and volume is
by changing one of these measurements by a given factor and then
asking how this change will influence the other measurements.
When the solids in the question are increased by a single
constant factor, a simple rule can help you come to an answer.
- If a solid’s length is multiplied by a given
factor, then its surface area is multiplied by the square of that
factor, and its volume is multiplied by the cube of that factor.
Remember that this rule holds true if all of
a solid’s dimensions increase in length by a given factor. So for
a cube or a sphere, just a side or the radius need change for the
rule to hold, but for a rectangular solid, cylinder, or other solid,
all of the dimensions must change. If the dimensions of the object
do not increase by a constant factor, for instance, if the height
of a cylinder doubles but the radius of the base triples, you will
have to go back to the equation for the dimension you are trying
to determine and calculate by hand.
you double the length of the side of a square, by how much do you
increase the area of that square?
If you understand the formula for the area of a square,
this question is not difficult. The formula for the area of a square
is A = s2,
where s is the length of a side. Replace s with
2s, and you see that the area of a square quadruples
when the length of its sides double: (2s)2 = 4s2.
a sphere’s radius is halved, by what factor does its volume decrease?
The radius of the sphere is multiplied by a factor of
divided by a factor of 2), and so its volume multiplies by the cube
of that factor: (1/
Therefore, the volume of the sphere is multiplied by a factor of
by 8), which is the same as decreasing by a factor of 8.
rectangular solid has dimensions xyz (these are its length,
width, and height) and has a volume of 64. What is the volume of
a rectangular solid of dimensions x
If this rectangular solid had dimensions that were all
one-half as large as the dimensions of the solid whose volume is
64, then its volume would be (1
/2)3 64 = 1
/8 64 = 8
. But dimension z
not multiplied by 1/
Hence, to answer a question like this one, you should turn to the
volume formula for rectangular solids: volume = l w h
. It is given
in the question that xyz
= 64. So,
64 = 16.