Inscribed Solids
An inscribed solid is a solid fit inside another solid,
with the edges of the two solids touching. The figures below are,
from left to right, a cylinder inscribed in a sphere, a sphere inscribed
in a cube, and a rectangular solid inscribed in a sphere.
Math IIC questions that involve inscribed solids don’t
require any additional techniques than those you’ve already learned.
What these questions do require is an ability to visualize inscribed
solids and the awareness of how certain line segments relate to
both solids in a given figure.
Most often, an inscribed solid question will present a
figure of an inscribed solid and give you information about one
of the solids. For example, you may be given the radius of a cylinder
and then be asked to find the volume of the other solid using the
figure as your guide. Use what you know about the radius of the
cylinder to find the dimensions of the other solid. Take a look
at this example:
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In
the figure below, a cube is inscribed in a cylinder. If the length
of the diagonal of the cube is 4 and the height of the cylinder is 5, what
is the volume of the cylinder? |
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The formula for the volume of a cylinder is πr2(h).
It is given in the question that h = 5, but there
is no value given for r. So in order to solve for
the volume of the cylinder, we need to first find the value of r.
The key step in this problem is to recognize that the
diagonal of a face of the cube is also the diameter, or twice the
radius, of the cylinder. To see this, draw a diagonal, d,
in either the top or bottom face of the cube.
In order to find this diagonal, which is the hypotenuse
in a 45-45-90 triangle, we need the length of an edge of the cube,
or
s. We can find
s from the diagonal
of the cube (not to be mistaken with the diagonal of a face of the
cube), since the formula for the diagonal of a cube is
s
with
s being the
length of an edge of the cube. The question gives us the diagonal
of the cube as 4
so it follows
that
s = 4. This means that the diagonal along
a single face of the cube is 4
(using the special properties of
a 45-45-90 triangle). Therefore, the radius of the cylinder is
4
/
2 = 2
Plug that back into the formula
for the volume of the cylinder, and you get
π 
(2
)
2 
5 = 40
π.
The Rules of Inscribed Solids
Math IIC questions involving inscribed solids are much
easier to solve when you know how the lines of different solids
relate to one another. For instance, the previous example showed
that when a cube is inscribed in a cylinder, the diagonal of a face
of the cube is equal to the diameter of the cylinder. The better
you know the rules of inscribed solids, the better you’ll do on
these questions. Here are the rules of inscribed solids that most
commonly appear on the Math IIC.
Cylinder Inscribed in a Sphere
The diameter of the sphere is equal to the diagonal of
the cylinder’s height and diameter.
Sphere Inscribed in a Cube
The diameter of the sphere is equal to the length of cube’s
edge.
Sphere Inscribed in a Cylinder
The cylinder and the sphere have the same radius.
