Some of the solids that appear on the Math IC do not have two congruent bases that lie in parallel planes, so they cannot be considered prisms. As with prisms, you need to know how to calculate the volume and surface area of these non-prisms. The formulas for the volume and surface area of the non-prisms are a little more complex than those for the prisms, but not too difficult.
Cones
A cone is not a prism, but it is similar to a cylinder. A cone is essentially a cylinder in which one of the bases is collapsed into a single point at the center of the base.
The radius of a cone is the radius of its one circular base. The height of a cone is the distance from the center of the base to the apex (the point on top). The lateral height, or slant height, of a cone is the distance from a point on the edge of the base to the apex. In the figure above, these three measurements are denoted by r, h, and l, respectively.
Notice that the height, radius, and lateral height of a cone form a right triangle. This means that if you know the value for any two of these measurements, you will always be able to find the third by using the Pythagorean theorem.
Volume of a Cone
Since a cone is similar to a cylinder except that it is collapsed to a single point at one end, the formula for the volume of a cone is a fraction of the formula for the volume of a cylinder:
where r is the radius and h is the height.
For practice, find the volume of the cone pictured below:
To answer this question, just use the formula for the volume of a cone with the following values plugged in:
r =
x,
l = 2
x, and
h =
x
. The volume is:
Surface Area of a Cone
The surface area of a cone consists of the lateral surface area and the area of the base. Because the base is a circle, it has an area of πr2. The lateral surface is the cone “unrolled,” which, depending on the shape of the cone, can be the shape of a triangle with a curved base, a half-circle, or a “Pacman” shape. The area of the lateral surface is related to the circumference of the circle times the lateral height, l. This is the formula:
where r is the radius and l is the lateral height.
The total surface area is the sum of the base area and lateral surface area:
When you are finding the surface area of a cone, be careful not to find only the lateral surface area and then stop. Students often forget the step of adding on the area of the circular base. Practice by finding the total surface area of the cone pictured below:
The total surface area is equal to the area of the base plus the area of the lateral surface. The area of the base = π
x2. The lateral surface area = π
x 
2x. The total surface area therefore equals π
x2 + π2
x2 = 3π
x2.
Pyramids
A pyramid is like a cone, except that it has a polygon for a base. Though pyramids are not tested very often on the Math IC test, you should be able to recognize them and calculate their volume.
The shaded area in the figure above is the base, and the height is the perpendicular distance from the apex of the pyramid to its base.
