Surface Area
Volume
MATH 113
Section 10.3: Surface Area and Volume
Prof. Jonathan Duncan
Walla Walla College
Winter Quarter, 2007
Conclusion
Surface Area
Outline
1
Surface Area
2
Volume
3
Conclusion
Volume
Conclusion
Surface Area
Volume
Conclusion
Measuring Three Dimensional Shapes
In the last section we looked at two measurements for two
dimensional shapes.
Two Dimensional Shapes
Two dimensional shapes can be measured in two different ways:
1
Perimeter (one dimensional)
2
Area (two dimensional)
In the same way, there are two important types of measurements for
three dimensional shapes.
Three Dimensional Shapes
Three dimensional shapes have two important measurements:
1
Surface Area (two dimensional)
2
Volume (three dimensional)
Surface Area
Volume
Conclusion
What is Surface Area?
Surface area is a term which is mainly used with three dimensional
objects. It is like the perimeter of a two dimensional object.
Surface Area
The surface area of a three dimensional object is the amount of
exposed area on the figures. For a two dimensional object, the
surface area is simply the area of the object.
With many three dimensional objects we can think of a net for the
object, find the area of the net, and this will be the surface area of
the object.
Surface Area
Volume
Conclusion
Surface Area of Right Prisms
We begin by examining the surface area of right prisms.
Example
Find a general formula for the surface area of a right rectangular
prism with a base of length l, width h, and a height of h.
Is this formula unique to rectangular prisms?
Example
Find a general formula for the surface area of a right equilateral
triangular prism with base side length s and height h.
General Formula
In general the surface area of a right prism is 2B + ph where B is
the area of the base, h is the height, and p is the perimeter of the
base.
Surface Area
Volume
Conclusion
Surface Area of Cylinders
In many ways cylinders and prisms are alike. In fact, we can use a
net to analyze the surface area.
Example
Find the surface area of a right circular cylinder with radius r and
height h.
Surface Area Formula
Using the formula for prisms, 2B + ph, we can set B = πr 2 and
p = 2πr to get 2πr 2 + 2πrh = 2πr (r + h).
Surface Area
Volume
Conclusion
Surface Area of Oblique Prisms
In each of the previous examples, we dealt with right objects. That
is, the sides made right angles with the base. How does surface
area change in oblique prisms?
Example
Can we develop a surface area formula for an oblique prisms which
is independent of the slant angle?
Slant and Height
Note that the length of the lateral faces in an oblique prism will
vary depending on the height and the slant angle.
Surface Area
Volume
Conclusion
Surface Area of Pyramids
By using nets and the formula for the area of a triangle we can
develop a formula for the surface area of a pyramid.
Example
Develop a formula for the surface area of a right square pyramid
with a base side length s and a slant height l.
General Formula
In general, a pyramid with an area of base B and perimeter p with
slant height l has a surface area B + 21 pl.
Surface Area
Volume
Conclusion
Surface Area of Cones
Just as we extended the formula for the surface area of a prism to
one for a cylinder, we can extend the surface area for a pyramid to
one for a cone.
Example
Find the surface area for a right circular cone with base radius r
and slant height h.
Surface Area Formula
Using the formula for a pyramid with a base area of B = πr 2 and
perimeter, or in this case circumference, of 2πr , the surface area is
πr 2 + 12 (2πr )l = πr (r + l).
Surface Area
Volume
Conclusion
Surface Area of a Sphere
The final object which we will examine is the sphere.
Surface Area of a Sphere
The surface area of a sphere of radius r is 4πr 2 .
Radius of a Sphere?
What is the radius of a sphere? It is the radius of a “great circle”
or largest possible circle taken as a cross-section of the sphere.
Unfortunately, to show that this formula works we would need to
use calculus. This is beyond the scope of this class.
Surface Area
Volume
Conclusion
What is Volume?
Just as area is measured in “square” units, so volume is measured
in a different sort of unit called a “cubic” unit.
Volume
The volume of an object is the amount of space contained in the
object. It can be thought of as the number of cubes of unit length
which can be fit into the object leaving no empty space and with
no overlap.
Just as area can be found by multiplying the side length of a polygon
by the height of the polygon, we can use multiplication to find the
volume of certain objects.
Surface Area
Volume
Conclusion
Volume of Right Prisms
The first, and simplest, object we will examine is a right
rectangular prism.
Example
Find a formula for the volume of a right rectangular prism of base
length l, width w and with height h.
Example
How many little one by one by one inch cubes could be packed
into a box which is 3 × 5 × 2 inches?
Volume Formula
In general the volume of a right prism is B × h where B is the area
of the base and h is the height.
Surface Area
Volume
Volume of Oblique Prisms
What happens to the volume formula in an oblique prism?
Example
Find a formula for the volume of an oblique rectangular prism of
base length l, width w , and with height h.
Example
Find a formula for the volume of a right triangular prisms with a
triangular base with base length 3 and height 2 having prism
height 5.
Conclusion
Surface Area
Volume
Conclusion
Volume of Cylinders and Pyramids
How do these formulas change for cylinders and pyramids?
Example
Show that the volume of a circular cylinder is πr 2 h where r is the
radius of the base and h is the height of the prism.
Example
In can be shown that three congruent pyramids each with square
base B can be fit into a cube with sides B. Using this fact, find a
formula for the volume of a square pyramid with base area B.
Surface Area
Volume
Conclusion
Volume of a Sphere
As with surface area, the volume of a sphere is difficult to derive.
Volume of a Sphere
The volume of a sphere is 43 πr 3 .
Spheres and Cylinders
From previous work, we know that:
The volume of a right circular cylinder of radius r and height
2r is 2πr 3
The volume of a sphere of radius r is 43 πr 3 .
A sphere of radius r can be inscribed in a right circular
cylinder of radius r and height 2r .
Therefore, the volume of that sphere is
cylinder.
2
3
the volume of the
Surface Area
Volume
Conclusion
Volume of Irregular Objects
While the formulas above are necessary, their applicability in real live can
be limited as we seldom have perfect prisms, cylinders, pyramids or
spheres to work with.
Example
Archimedes was an ancient Greek mathematician. In one story
Archimedes was asked to determine if a crown presented to his king was
solid gold as the giver claimed or had sliver mixed into the crown.
Archimedes knew that the mass of an object is given by
mass = volume × density
and he knew that gold is about twice as dense as silver. Finding the
volume and weight of the crown would let him solve the problem. But
how could he determine the volume of the irregularly shaped crown?
The solution came to Archimedes as he sat is his bath one day. He is
said to have leapt out of the bath and run out into the street shouting
“Eureka!”
Surface Area
Volume
Important Concepts
Things to Remember from Section 10.3
1
Surface Area and Volume for:
1
2
3
4
2
Right Prisms
Right Cylinders
Right Pyramids
Spheres
Strategies for finding the volume of irregular objects
Conclusion