Inscribed Solids
An inscribed solid is a solid placed 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 IC questions that involve inscribed solids don’t
require any techniques other than those you’ve already learned.
These questions do require an ability to visualize inscribed solids
and an awareness of how certain line segments relate to both of
the 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, say a rectangular
solid. Using the figure as your guide, you need to use the radius
of the cylinder to find the dimensions of the other solid so that
you can answer the question. Here’s an 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).
The question states 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 confused with the diagonal of
a face of the cube), since the formula for the diagonal of a cube
is
s
where
s is
the length of an edge of the cube. The question states that the
diagonal of the cube is 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 into the formula for the
volume of the cylinder, and you get π
(2
)
2 
5 = 40π.
Helpful Tips
Math IC 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. So without further ado, here are the rules of inscribed
solids that most commonly appear on the Math IC.
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 the
cube’s edge.
Sphere Inscribed in a Cylinder
Both the cylinder and the sphere have the same diameter
and radius.
