Volume & Surface Area
MATH 102
Contemporary Math
S. Rook
Overview
• Section 10.4 in the textbook:
– Volume
– Surface area
Volume & Surface Area
Volume & Surface Area in General
• Recall that perimeter and area are measurements of
two-dimensional figures
– e.g. rectangles, triangles, etc.
• These same measurements have counterparts for
three-dimensional figures
• Surface Area: a measurement of space on the
outside of a three-dimensional figure
• Volume: a measurement of the space inside a threedimensional figure
– In general V = A x h where A is the area of the base
and h is the height (third dimension)
Volume & Surface Area in General
(Continued)
• What follows on the next few slides are the
formulas for volume and surface area for
common three-dimensional figures
– Except for the volume of a rectangular solid and a
cube, you are not required to memorize these
formulas
– However, you should know how to apply them to
solve problems
Rectangular Solids & Cubes
• Recall the formula for area of a rectangle
• Given a rectangular solid with length l, width w,
and height h:
– SA = 2lw + 2lh + 2wh
– V = lwh
• Recall the formula for area of a square
• Given a cube with length l (length, width, and
height are the same for a cube):
– SA = 6l2
– V = l3
Cylinder
• A cylinder is a three-dimensional
extension of a circle with an
added height
• Recall the formula for area of a circle
• Given a cylinder with radius r and
height h:
– S.A. = 2πrh + 2πr2
• Think about “opening up” the cylinder and lying it
down flat as a rectangle and then adding in the area for
the two circular bottoms
– V = πr2h
Cone
• A right circular cone has a
height h that extends from
the tip and is perpendicular
to the circular base of the
cone
• Given a right circular cone with a height h and
radius of its circular base r:
SA  r r 2  h2
–
1 2
V  r h
3
Sphere
• A sphere is a three-dimensional
extension of a circle with radius
r
– Think of a ball that can be cut
into circles
– The radius is measured from the center of the sphere
– The Earth is essentially a sphere
• Given a sphere with radius r:
SA  4r 2
4 3
V  r
3
Volume & Surface Area (Example)
Ex 1: What is the minimum area of wrapping
paper required to completely cover a box with
dimensions 12.4” x 11.9“ x 7.4“?
Volume & Surface Area (Example)
Ex 2: A packing crate in the shape of a rectangle
has dimensions of 12 ft x 8 ft x 60 in. How
many cubic packages with sides of length 3 ft
can fit into the crate?
Volume & Surface Area (Example)
Ex 3: An ice-cream cone in the shape of a rightcircular cone has a radius of 4 cm and a height
of 8 cm.
a) How much ice cream can the cone hold if we
completely fill it?
b) After filling the cone, a company decides to wrap
it for packaging. How much wrapping is required?
Volume & Surface Area (Example)
Ex 4: A punch bowl is in the shape of a
hemisphere (half a sphere) with a radius of 9
inches. The cup part of the ladle in the bowl is
also in the shape of a hemisphere with a
diameter of 4 inches. If the punch bowl is
filled completely, how many full ladles of
punch are in the bowl?
Summary
• After studying these slides, you should know
how to do the following:
– Be familiar with the different formulas for surface
area & volume of common three-dimensional
figures
• Additional Practice:
– See problems in Section 10.4
• Next Lesson:
– Introduction to Counting Methods (Section 13.1)