Lesson 9
Three-Dimensional Geometry
1
Planes
• A plane is a flat surface (think tabletop) that
extends forever in all directions.
• It is a two-dimensional figure.
• Three non-collinear points determine a plane.
• So far, all of the geometry we’ve done in these
lessons took place in a plane.
• But objects in the real world are threedimensional, so we will have to leave the plane
and talk about objects like spheres, boxes,
cones, and cylinders.
2
Boxes
• A box (also called a right
parallelepiped) is just what the name
box suggests. One is shown to the
right.
• A box has six rectangular faces, twelve
edges, and eight vertices.
• A box has a length, width, and height
(or base, height, and depth).
• These three dimensions are marked in
the figure.
H
L
W
3
Volume and Surface Area
• The volume of a three-dimensional object
measures the amount of “space” the object
takes up.
• Volume can be thought of as a capacity and
units for volume include cubic centimeters
(cm3 ), cubic yards, and gallons.
• The surface area of a three-dimensional
object is, as the name suggests, the area of
its surface.
4
Volume and Surface Area of a Box
• The volume of a box is found by
multiplying its three dimensions
together:
V  L W  H
• The surface area of a box is found by
adding the areas of its six rectangular
faces. Since we already know how to
find the area of a rectangle, no formula
is necessary.
H
L
W
5
Example
• Find the volume and surface area of
the box shown.
• The volume is
8  5  4  40  4  160
• The surface area is
85  85  5 4  5 4  8 4  8 4
 40  40  20  20  32  32
 184
4
8
5
6
Cubes
• A cube is a box with three equal
dimensions (length = width = height).
• Since a cube is a box, the same formulas
for volume and surface area hold.
• If s denotes the length of an edge of a
3
cube, then its volume is s and its surface
area is 6s 2 .
7
Prisms
• A prism is a three-dimensional solid with
two congruent bases that lie in parallel
planes, one directly above the other, and
with edges connecting the corresponding
vertices of the bases.
• The bases can be any shape and the name
of the prism is based on the name of the
bases.
• For example, the prism shown at right is a
triangular prism.
• The volume of a prism is found by
multiplying the area of its base by its height.
• The surface area of a prism is found by
adding the areas of all of its polygonal faces
including its bases.
8
Cylinders
• A cylinder is a prism in which the bases
are circles.
• The volume of a cylinder is the area of
its base times its height:
V   r 2h
h
• The surface area of a cylinder is:
A  2 r 2  2 rh
r
9
Pyramids
• A pyramid is a three-dimensional solid
with one polygonal base and with line
segments connecting the vertices of
the base to a single point somewhere
above the base.
• There are different kinds of pyramids
depending on what shape the base is.
To the right is a rectangular pyramid.
• To find the volume of a pyramid,
multiply one-third the area of its base
by its height.
• To find the surface area of a pyramid,
add the areas of all of its faces.
10
Cones
• A cone is like a pyramid but with a
circular base instead of a polygonal
base.
• The volume of a cone is one-third the
area of its base times its height:
1 2
V  r h
3
• The surface area of a cone is:
A   r 2   r r 2  h2
h
r
11
Spheres
• Sphere is the mathematical word for
“ball.” It is the set of all points in space a
fixed distance from a given point called
the center of the sphere.
• A sphere has a radius and diameter, just
like a circle does.
• The volume of a sphere is:
r
4 3
V  r
3
• The surface area of a sphere is:
A  4 r 2
12