Surface Area of Prisms, Cylinders, and
Spheres
A prism is a solid figure which has two congruent polygonal bases
joined by lateral faces. In the figure below, the trapezoidal prism is
sitting on a lateral face.
The hei ght of the pri sm i s the length of thi s edge.
b1
h
b2
The lateral faces are all rectangles that connect the two bases of the
prism. The distance from one base to the other is the height of the
prism. Recall that the base, which is a trapezoid in this case, also has
a height (the dotted line segment that is perpendicular to the bases
of the trapezoid), but it is a segment within the base of the prism. Do
not confuse the base segment of the polygonal region with the
polygon that forms the base of the prism. One is a segment length
while the other is a region that has area.
Now, to find the surface area of the prism, we need the area of the
polygonal base (times 2) and the areas of the lateral faces. The area
of the lateral faces can be found separately as a number of
rectangles/parallelograms (in some prisms). Another way to think
about the lateral area is folding out the rectangles that form the
lateral faces. The figure would look similar to the one below.
So, the lateral area is just the perimeter of the polygonal base
multiplied by the height of the prism. The formula for the surface
area becomes
SA = 2 B + p h
where B is the base area (area of the polygon that forms the base of
the prism), p is the perimeter of that base, and h is the height of the
prism.
How does the surface area of a cylinder compare to the surface area
of a prism? Grab a can of soup or an oatmeal canister. This figure is
a cylinder. Notice that the base of the cylinder is a circle. Two other
solids have the circle as its base: the cone (comparable to the
pyramid) and the sphere (beach ball).
Notice that the cylinder has two bases just as the prism does. The
lateral area cannot be called a face because it has no edges. Imagine
slitting the cylinder down its side perpendicular to the bases. If we
open this out, we get the figure below:
So, the surface area of the cylinder is the two circular bases and the
lateral area in rectangle form. This is very similar to the surface area
of the prism. So, the formula becomes
SA = 2Πr2 + 2Πrh
Where r is the radius of the circle of the cylinder and h is the height
of the cylinder.
The sphere is the third solid determined by a circle. If we slice the
sphere through its center point, we get a circle that has the same
center and radius as the sphere. This circle is called the great circle.
So, if we know the area of the great circle is 81Π, this tells us that
the radius of the sphere is 9 and its diameter is 18. Considering the
horizontal great circle and the vertical great circle as dividing the
surface of the sphere into 4 equal areas, it can be shown that these
areas equal the area of the great circle. So, the formula for the
surface area of a sphere is
SA = 4Π r2