Properties of Geometric Solids
Introduction to Engineering Design
© 2012 Project Lead The Way, Inc.
Geometric Solids
Solids are threedimensional objects.
• In sketching, twodimensional shapes are
used to create the
illusion of threedimensional solids.
Properties of Solids
Volume, mass, weight, density, and surface
area are properties that all solids possess.
These properties are used by engineers and
manufacturers to determine material type,
cost, and other factors associated with the
design of objects.
Volume
Volume (V) refers to the amount of threedimensional space occupied by an object or
enclosed within a container.
Metric
cubic
centimeter
(cc)
English System
cubic inch
(in.3)
Volume of a Cube
A cube has sides (s) of equal length.
The formula for calculating the volume (V) of
a cube is:
V=
3
s
V= s3
V= 4.0 in. x 4.0 in. x 4.0 in.
V = 64 in.3
Volume of a Rectangular Prism
A rectangular prism has at least one side
that is different in length from the other
two.
The sides are identified as width (w), depth
(d), and height (h).
Volume of Rectangular Prism
The formula for calculating the volume (V) of
a rectangular prism is:
V = wdh
V= wdh
V= 4.00 in. x 5.25 in. x 2.50 in.
V = 52.5 in.3
Volume of a Cylinder
To calculate the volume of a cylinder, its
radius (r) and height (h) must be known.
• The formula for calculating the volume (V)
of a cylinder is:
V = r2h
V= r2h
V= 3.14 x (1.50 in.)2 x 6.00 in.
V = 42.4 in.3
Volume of a Cone
• The formula for calculating the volume (V)
of a cone is:
1.50
πr2 h
V=
3
2
π(0.75 in.) (2.00 in.)
V=
3
3
V = 1.18 in.
Formula Sheet
Mass
Mass (M) refers to the quantity of matter in
an object. It is often confused with the
concept of weight in the SI system.
SI
gram
(g)
U.S. Customary
System
slug
Weight
Weight (W) is the force of gravity acting on
an object. It is often confused with the
concept of mass in the U.S. Customary
System.
SI
U.S. Customary
System
Newton
pound
(N)
(lb)
Mass vs. Weight
Contrary to popular practice, the terms mass
and weight are not interchangeable and do
not represent the same concept.
W = mg
weight = mass x acceleration due to gravity
(lb)
(slugs)
(ft/sec2)
g = 32.16 ft/sec2
Mass vs. Weight
• An object, whether on the surface of the
earth, in orbit, or on the surface of the
moon, still has the same mass.
• However, the weight of the same object
will be different in all three instances
because the magnitude of gravity is
different.
Mass vs. Weight
• Each measurement system has fallen prey
to erroneous cultural practices.
• In the SI system, a person’s weight is
typically recorded in kilograms when it
should be recorded in Newtons.
• In the U.S. Customary System, an object’s
mass is typically recorded in pounds when
it should be recorded in slugs.
Density
Mass Density (Dm) is an object’s mass per
unit volume.
SI System
grams per cubic centimeter
(g/cm3)
Weight density (Dw) is an object’s weight per
unit volume.
U.S. Customary System
pounds per cubic inch
(lb/in.3)
Examples of Density
Material
Apples
Water (Pure)
Mass Density
(g/cm3)
0.64
1.00
Weight Density
(lb/in.3)
0.023
0.036
Water (Sea)
Ice
Concrete
Aluminum
1.03
0.92
2.40
2.71
0.037
0.034
0.087
0.098
Steel (1018)
7.8
0.282
Gold
19.32
0.698
Calculating Mass
To calculate the mass (m) of any solid, its
volume (V) and mass density (Dm) must be
known.
m = VDm
Dm (aluminum) = 2.71 g/cm3
m = VDm
m = (3.81cm)(8.89 cm)(17.28 cm)(2.71 g/cm3)
m = 1586 g = 1.59 kg
Calculating Weight
To calculate the weight (W) of any solid, its
volume (V) and weight density (Dw) must
be known.
W = VDw
Dw (aluminum) = 0.098 lb/in.3
W = VDw
W = 36.75 in.3 x .098 lb/in.3
W = 3.60 lb
Area vs. Surface Area
There is a distinction between area (A) and
surface area (SA).
• Area describes the measure of the twodimensional space enclosed by a shape.
• Surface area is the sum of all the areas of
the faces of a three-dimensional solid.
Calculating Surface Area
In order to calculate the surface area (SA) of
a rectangular prism, the area (A) of each
faces must be known and added together.
Area A = 3.0 in. x 4.0 in. = 12 in.2
B
Area B = 4.0 in. x 8.0 in. = 32 in.2
C
A
D
E
Area C = 3.0 in. x 8.0 in. = 24 in.2
F
Area D = 4.0 in. x 8.0 in. = 32 in.2
Area E = 3.0 in. x 8.0 in. = 24 in.2
Area F = 3.0 in. x 4.0 in. = 12 in.2
Surface Area = 136 in.2
Calculating Surface Area
Another way to represent the formula for
surface area of a rectangular prism is given
on the formula sheet.
Calculating Surface Area
Surface Area = 2 [(8.0 in.)(4.0 in.)
+ (8.0 in.)(3.0 in.)
+ (4.0 in.)(3.0 in.)]
= 136 in.2
Surface Area Calculations
What is the surface area of this rectangular
prism?
SA = 2(wd + wh + dh)
SA = 2[(4.00 in.)(5.25 in.) + (4.00 in.)(2.50 in.) + (5.25 in.)(2.50 in.)]
SA = 2 [44.125 in.2]
SA = 88.3 in.2
Calculating Surface Area
In order to calculate the surface area (SA) of
a cube, the area (A = s2) of any one of its
faces must be known.
The formula for calculating the surface area
(SA) of a cube is:
SA = 6s2
SA = 6 (4.00 in.)2
SA = 96.0 in.2
Surface Area Calculations
In order to calculate the surface area (SA) of
a cylinder, the area of the curved face and
the combined area of the circular faces must
be known.
SA = (2r)h + 2(r2)
SA = 2()(1.50 in.)(6.00 in.) + 2()(1.50 in.)2
SA = 56.55 in.2 + 14.14 in.2
SA = 70.7 in.2