http://www.ck12.org Chapter 11. Surface Area and Volume
Surface Area
Surface areais the sum of the areas of the faces of a solid.
Surface Area of a Right Prism:The surface area of a right prism is the sum of the area of the bases and the area of
each rectangular lateral face.
Volume
Volumeis the measure of how much space a three-dimensional figure occupies. The basic unit of volume is the
cubic unit: cubic centimeter(cm^3 ), cubic inch(in^3 ), cubic meter(m^3 ), cubic foot(f t^3 ), etc. Each basic cubic unit
has a measure of one for each: length, width, and height.
Volume of a Rectangular Prism: If a rectangular prism ishunits high,wunits wide, andlunits long, then its
volume isV=l·w·h.
If we further analyze the formula for the volume of a rectangular prism, we would see thatl·wis equal to the area
of the base of the prism, a rectangle. If the bases are not rectangles, this would still be true, however we would have
to rewrite the equation a little.
Volume of a Prism:If the area of the base of a prism isBand the height ish, then the volume isV=B·h.
Recall that earlier in this Concept we talked about oblique prisms. These are prisms that lean to one side and the
height is outside the prism. What would be the area of an oblique prism? To answer this question, we need to
introduce Cavalieri’s Principle.
Cavalieri’s Principle:If two solids have the same height and the same cross-sectional area at every level, then they
will have the same volume.
Basically, if an oblique prism and a right prism have the same base area and height, then they will have the same
volume.
Example A
Find the surface area of the prism below.
Open up the prism and draw the net. Determine the measurements for each rectangle in the net.
Using the net, we have:
SAprism= 2 ( 4 )( 10 )+ 2 ( 10 )( 17 )+ 2 ( 17 )( 4 )
= 80 + 340 + 136
= 556 cm^2Because this is still area, the units are squared.
Example B
Find the surface area of the prism below.
This is a right triangular prism. To find the surface area, we need to find the length of the hypotenuse of the base
because it is the width of one of the lateral faces. Using the Pythagorean Theorem, the hypotenuse is