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.
