Three-Dimensional Geometry
Spatial Relations
Many jobs in the real-world deal with using
three-dimensional figures on two-dimensional
surfaces. A good example of this is architects
use drawings to show what the exteriors of
buildings will look like.
Three-dimensional figures have faces, edges,
and vertices. A face - is a flat surface, and
edge - is where two faces meet, and a vertex is where three or more edges meet. Volume is
measured in cubic units.
See the example below. Isometric dot paper can
be used to draw three-dimensional figures.
How many faces do most threedimensional figures have?
With your isometric dot paper, sketch the
drawing below. Make your box 3 units wide, 2
units high, and 5 units long.
Now try to sketch the box.
After you have sketched the box, try other
figures like a cube or pyramid.
Drawing threedimensional figures uses
a technique called
perspective. Here you
make a two-dimensional
figure look like it is
three-dimensional.
Let’s try to see if we can
draw some threedimensional figures of
our own.
You will need some
isometric dot paper to
sketch you drawing.
Next, we are going to make
a three-dimensional figure
using lock blocks and then
draw our figure and
determine how many blocks
are used to make the
figure.
3-dimensional objects can also be depicted as
2-dimensional drawings taken at different
views.
These representations are called orthogonal
drawings.
The 3-dimensional drawing at the left is
represented by the 2-dimensional drawings
from the top, front and right-side views.
Volume of Prisms and
Cylinders
Measured in cubic units3
Volumes of Prisms and
Cylinders
A prism is a threedimensional figure
named for the shape
of its bases.
Triangular prism has
triangles for bases.
Rectangular prism has
rectangles for bases.
If all six faces of a
prism are squares, it is
a cube.
Triangular prism
In this triangular
prism the two
bases are
triangles. The
formula for volume
of a triangular
prism is V = Bh,
where B is area of
the base and h is
height.
Here is another view of a triangular prism.
The view on the left shows you how the prism
looks in a 3-dimensional view. The view on the
right is the base of the prism.
V = Bh
B = area of the base = area
of a triangle
V = ½ bh · h
V = (.5)(16)(12) = 96 in2
V = Bh height = 12 in
V = 96 · 12
V = 1152 in3
Volume of the prism is
1152 in3. Volume is
measured in cubic units.
Find the volume of
the prism
Rectangular prism
In this rectangular
prism the two bases
are rectangles. The
volume formula is
V = Bh
V = (lw)h
length · width · height
Find the volume of the
prism
V = Bh or V = lwh
V = 12 · 8 · 3
V = 288 in3
The volume of the prism
is 288 in3. Volume is
measured in cubic
units.
CUBE
Here is a 3-dimensional view of a cube. The
view on the left is the cube. The view on the
right shows the base of the cube. The
formula for the volume of a cube:
V = Bh
V = lwh
V = Bh or V =lwh
V = 5 · 5 · 5 or 53
V = 125 units3
The volume of the cube
is 125 units3. Volume
is measured in cubic
units.
Find the volume of the
cube
A die is a cube molded from hard plastic. The edge of a
typical die measure 0.62 inches. Dice are usually
produced in a mold which holds 100 die at a time. To the
nearest cubic inch, how much plastic is needed to fill this
large mold?
When working with word problems, be sure to read
carefully to determine what the question wants you to
find. This question clearly indicates that you are to
compute the volume by stating “to the nearest cubic
inch.”
Volume of one die = lwh = (.62)(.62)(.62) = 0.238 cubic
inches
For 100 dice = 23.8 = 24 cubic inches
Cylinder: a cylinder is a three-dimensional
figure with two circular bases. The volume of
a cylinder is the area of the base B times the
height h.
V = Bh
or
V = (πr²)h
Find the volume of the
cylinder
V = Bh or V = πr2h
V = (π · 42) · 10
V = 502.4 cm3
The volume of the
cylinder is 502.4 cm3.
Volume is measured in
cubic units.
Effects of Changing
Dimensions
By changing the dimensions of a figure, it can have
an effect on the volume in different ways,
depending on which dimension you change. Lets
look at what happens when you change the
dimensions of a prism and a cylinder.
A juice box measures 3“ by 2“ by 4“. Explain
whether doubling the length, width, or height
of the box would double the amount of juice
the box holds.
Original
Double length
Double width
Double height
V = lwh
V = 3·2·4
V = 24 cu.in.
V = lwh
V = 6·2·4
V = 48 cu.in
V = lwh
V = 3·4·4
V = 48 cu.in
V = lwh
V = 3·2·8
V = 48 cu.in.
A juice can has a radius of 1.5 in. and a height
of 5 in.. Explain whether doubling the height
of the can would have the same effect on the
volume as doubling the radius
Original
Double
radius
Double
height
V=
V=
V=
V=
V=
V=
V=
V=
V=
πr²h
π·1.5²·5
11.25π cu.in.
πr²h
π·3²·5
45π cu.in.
πr²h
π·1.5²·10
22.5π cu.in.
Volumes of Pyramids
and Cones
1/3 of prisms and cylinders
A pyramid is named for the shape of its base.
The base is a polygon, and all the other faces
are triangles.
A cone has a circular base.
The height of a pyramid or cone is a
perpendicular line measured from the highest
point to the base.
A cone has a circular base. The height of a
pyramid or cone is perpendicular line measured
from the highest point to the base.
In the cone to the left the height is h and the
radius of the circular base is r.
The s is the slant height which is used to
measure surface area of a cone or pyramid.
The volume formula for a cone is
V = 1/3Bh or
V = 1/3πr²h
A pyramid is named for its base. The base is a
polygon, and all the other faces are triangles
that meet at a common vertex. The height is a
perpendicular line from the base to the
highest point.
The volume formula for a pyramid is
V = 1/3Bh
V = 1/3(lw)h
The volumes of cones and pyramids are related
to the volumes of cylinders and prisms.
V = πr²h
V = Bh
V = 1/3πr²h
V = 1/3Bh
A cone is 1/3 the size of a cylinder with the
same base and height. Also, a pyramid is 1/3
the size of a prism with the same height and
base.
Finding Volumes
A practical application
Find the volume of the
cylinder to the
nearest tenth.
V = Bh
V = πr2 · h
V = 3.14 · 32 · 8.6
V = 243.036 cm3
V = 243 cm3
Find the volume of the
prism to the nearest
tenth
V = Bh
V=6·8·2
V = 96 cm3
Find the volume of the
triangular prism
V = Bh
V = ½bh · h
V = ½(12 · 16) · 12
V = ½(192) · 12
V = ½(2304)
V = 1152 in3
Surface Area of Prisms
and Cylinders
Back to areas2 again
Surface area of objects are
used to advertise, inform,
create art, and many other
things. On the left is an
anamorphic image, which is
a distorted picture that
becomes recognizable
when reflected onto a
cylindrical mirror.
One of the most
recognizable forms of
advertising that uses
surface area of an
object is the cereal
box.
If you find the volume,
you will find the amount
of cereal the box will
hold.
If you find the surface
area of the box you
determine how much
cardboard is needed to
make the box.
When you flatten-out a three-dimensional
object the diagram is called a net. Which of
the following answers is the correct net for
the cube. Choose a, b, c, or d.
Finding surface area of figures, for example
the box below, can be relatively simple. All is
needed is to visualize the faces and then use
the appropriate area formulas for rectangles
and circles.
Surface area is the
sum of areas of all
surfaces of a figure.
The figure to the
left is a rectangular
prism. Notice how
many surfaces there
are. Lateral surfaces
of a prism are
rectangles that
connect the bases.
Top and bottom
Left and right
Front and back
Surface area - is the sum of the areas of all
surfaces of a figure. Lateral surfaces - of a
cylinder is the curved surface.
Surface Area:
is the number of square
units needed to cover all surfaces of a threedimensional figure.
Surface area is the
sum of the areas of
all surfaces of a
figure. The lateral
surfaces of a
triangular prism are
triangles and
rectangles.
Two triangular bases
and three
rectangles.
Finding Surface Areas
Unfolding the figure
Find the surface area of the figure
SA = (top & bottom)
+ ( front & back)
+ (left & right)
= 2(8 · 6) + 2(8 · 2) + 2(6 · 2)
= 96 + 32 + 24
SA = 152 cm2
Find the surface area of the figure
SA = 2(πr2) + lw
= 2(area of circle) + (circumference · height)
= 2(3.14 · 3.12) + (π6.2) · 12
= 60.3508 + 233.616
= 293.9668 in2
Find the surface area of the figure
SA = 2(area of triangle) + (lw) + (lw) + (lw)
= 2(½ · 12 · 16) + (20 · 12) + (16 · 12) + (12 · 12)
= 192 + 240 + 192 + 144
= 768 in2
New Year’s Eve ball dropped in New York city
each year. The ball is made of 2,668
Waterford crystals with 32,256 LED’s that
produce about 16 million different colors.
So the next time you see
an unusual shape, just
remember geometry is all
around us.
US Pavilion at the 1967 World
Expo in Montreal, Canada.