
Solids That Aren’t Prisms
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 nonprisms. The formulas for
the volume and surface area of the nonprisms 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 =
2x, 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 πr^{2}.
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 halfcircle, 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 = πx^{2}.
The lateral surface area = πx 2x.
The total surface area therefore equals πx^{2} +
π2x^{2} = 3πx^{2}.
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.
Volume of a Pyramid
The formula for calculating the volume of a pyramid is:
where B is the area of the
base and h is the height. Try to find
the volume of the pyramid below:
The base is just a square with a side of 3,
and the height is ^{3}/_{2} . B =
3^{2} = 9, and the total volume of
the pyramid is:
Surface Area of a Pyramid
The surface area of a pyramid is rarely tested on the
Math IC test. If you come across one of those rare questions that
covers the topic, you can calculate the area of each face individually
using techniques from plane geometry, since the base of a pyramid
is a square and the sides are triangles. Practice by finding the
surface area of the same pyramid in the figure below:
To calculate the surface area, you need to add together
the area of the base and the areas of the four sides. The base is
simply a square, and we’ve seen that B =
3^{2} = 9. Each side is an equilateral triangle,
and we can use the properties of a 306090 triangle to find their
areas:
For each triangle, Area = ^{1}
/_{2} 3 ^{3}/_{2} =
^{9}/
_{4} . The sum of the areas
of the four triangles is 4 ^{9}/_{4} =
9 The total surface area of the pyramid
is 9 + 9
Spheres
A sphere is the collection of points in threedimensional
space that are equidistant from a fixed point, the center of the
sphere. Essentially, a sphere is a 3D circle. The main measurement
of a sphere is its radius, r, the
distance from the center to any point on the sphere.
If you know the radius of a sphere you can find both its
volume and surface area. The equation for the volume of a sphere
is:
The equation for the surface area of a sphere is:
