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Prisms
Most of the solids you’ll see on the Math IC test are
prisms, or variations on prisms. A prism is defined
as a geometric solid with two congruent bases that lie in parallel
planes. You can create a prism by dragging any two-dimensional triangle,
circle, or polygon through space without rotating or tilting it.
The three-dimensional space defined by the moving triangle or polygon
is the body of the prism. The prism’s two bases are the planes where
the two-dimensional shape begins and ends. The perpendicular distance
between the bases is the height of the prism.
The figures below are all prisms. The bases of these prisms
are shaded, and the altitude (the height) of each prism is marked
by a dashed line:
There are two main aspects of geometric solids that are
relevant for the Math IC: volume and surface area.
Volume of a Prism
The volume of a prism is the amount of space
taken up by that prism. The general formula for calculating the
volume of a prism is very simple:
where B is the area of the
base, and h is the prism’s height.
Certain geometric solids have slightly different formulas for calculating
volume that we will cover on a case-by-case basis.
Surface Area
The surface area of a prism is the sum of
the areas of all the prism’s sides. The formula for the surface
area of a prism therefore depends on the type of prism with which
you are dealing. As with volume, we cover the specifics of calculating
surface area as we cover each type of geometric solid.
Rectangular Solids
A rectangular solid is a prism with a rectangular
base and lateral edges that are perpendicular to its base. In short,
a rectangular solid is shaped like a box.
A rectangular solid has three important dimensions: length
(l), width (w),
and height (h). If you know these
measurements, you can find the solid’s surface area, volume, and
diagonal length.
Volume of a Rectangular Solid
The volume of a rectangular solid is given by the following
formula:
where l is the length, w is
the width, and h is the height. Notice how this
formula corresponds with the general formula for the volume of a
prism: the product lw is the area
of the base. Now try to find the volume of the prism in the following
example:
In this solid, l =
3x, w = x,
and h = 2x.
Simply plug the values into the formula given for volume, and you
would find Volume = (3x)(2x)(x)
= 6x3.
Surface Area
The surface area of a rectangular solid is given by the
following formula:
where l is the length, w is
the width, and h is the height.
The six faces of a rectangular solid consist of three
congruent pairs. The surface area formula is derived by simply adding
the areas of the faces—two faces have areas of l 
w, two faces have
areas of l 
h, and two faces
have areas of w 
h.
w, two faces have
areas of l h, and two faces
have areas of w h.To practice, try to find the surface area of the rectangular
solid we used as an example for volume. Here’s the figure again:
All you have to do is enter the given values into the
formula for surface area:
Diagonal Length of a Rectangular Solid
The diagonal of a rectangular solid, d,
is the line segment whose endpoints are opposite corners of the
solid. Every rectangular solid has four diagonals, each with the
same length, that connect each pair of opposite vertices. Here’s
one diagonal drawn in:
The formula for the length of a diagonal is:
where l is the length, w is
the width, and h is the height.
You can look at this formula as the Pythagorean theorem
in three dimensions. In fact, you can derive this formula using
the Pythagorean theorem. First, find the length of the diagonal
along the base. This is 
. Then use the
Pythagorean theorem again, incorporating height to find the length
of the diagonal from one corner to the other: 
. Thus, 
and 
. A Math IC question might ask you:
. Then use the
Pythagorean theorem again, incorporating height to find the length
of the diagonal from one corner to the other: . Thus, and . A Math IC question might ask you:
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The question gives the length, width, and height of the
rectangular solid, so you can just plug those numbers into the formula:
The length of the diagonal AH (as well
as BG, CF, and DE)
is
.
.Cubes
Just as a square is a special kind of rectangle, a cube
is a special kind of rectangular solid. A cube is a
rectangular solid whose edges are each the same length. In other
words, the length, width, and height are equal, and each of its
six faces is a square.
Volume of a Cube
The formula for finding the volume of a cube is essentially
the same as the formula for the volume of a rectangular volume.
However, since a cube’s length, width, and height are all equal,
the cube volume formula is:
where s is the length of one edge of
the cube.
Surface Area of a Cube
Since a cube is just a rectangular solid whose
sides are all equal, the formula for finding the surface area of
a cube is the same as that for a rectangular solid, except with s
= l = w = h:
where s is the length of one edge of
the cube.
Diagonal Length of a Cube
The same is true for measuring the diagonal length of
a cube. The formula for the diagonal of a cube is simply adapted
from the formula for the diagonal length of a rectangular solid, with s
= l = w = h:
where s is the length of one edge of
the cube.
Cylinders
A cylinder is a prism with circular bases.
The height of a cylinder, h, is the length
of the line segment whose endpoints are the centers of the bases.
The radius of a cylinder, r, is the radius of its
base. If you know the height and radius of a cylinder, you can easily
calculate its volume and surface area.
Volume of a Cylinder
The volume of a cylinder is the product of the area of
its base and its height. Because a cylinder has a circular base,
the volume of a cylinder is:
where r is the radius of the circular
base and h is the height. Try to find the volume
of the cylinder below:
This cylinder has a radius of 4 and a height of 6. Using
the volume formula:
2(6).gif)
Surface Area of a Cylinder
The surface area of a cylinder is the sum of the areas
of the two bases and the lateral face of the cylinder. The bases
are congruent circles, so their areas can be found easily. The lateral face
is the tubing that connects the two bases. When “unrolled,” the
lateral base is simply a rectangle whose length is the circumference
of the base and whose width is the height of the cylinder. Therefore,
the surface area of a cylinder is given by this formula:
where r is the radius and h is
the height. As with finding the volume of a cylinder, finding the
surface area involves plugging the height and radius of the base
into the formula. To find the surface area of the cylinder in the
practice example on volume,
just plug the values into the formula:
2.gif)
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