**SA**

**V**

Properties of

Geometric Solids

Solids are three-dimensional objects.

In sketching, two-dimensional shapes are used to create the illusion of three-dimensional solids.

Geometric 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

V= s

V= 4 in x 4 in x 4 in

V = 64 in

Volume of a Cube

A cube has sides (s) of equal length.

The formula for calculating the volume (V) of a cube is:

V = s

3

3

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).

V= 3.14r h

V= 3.14 x (1.5 in)2 x 6 in

V = 42.39 in

Volume of a

Rectangular Prism

The formula for calculating the

volume

(

V

) of a rectangular prism is:

V= wdh

V= 4 in x 5.25 in x 2.5 in

V = 52.5 in

V = wdh

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:

2

3

**M**

Mass

vs.

**W**

Weight

Mass

(

M

) refers to the quantity of matter (stuff) in an object. It is often confused with the concept of weight.

Mass

Metric English System

gram slug

(g)

Weight

(

W

) is the force of gravity acting on an object. It is often confused with the concept of mass.

Weight

Metric English System

Newton Pound

(N)

Mass vs. Weight

Contrary to popular practice, the terms

mass

and

weight

are not interchangeable, and do not represent the same concept.

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

.

The

weight

of that same object will be different in all three instances, because the force of gravity is different.

Calculating Mass

Weight

density

(

Wd

)

is an object’s weight (force) per unit volume.

Weight Density

English System

pounds per cubic inch

(lbs/in3)

(lbs/in. )

Calculating Weight

Surface Area

Area vs. Surface Area

Surface Area Calculations

There is a distinction between area (A) and

surface area

(

SA

).

Area describes the measure of the two-dimensional space enclosed by a shape.

Surface area is the sum of all the areas of the faces of a three-dimensional solid.

In order to calculate the

surface area

(

SA

) of a

cube

, the area (A) of any one of its faces must be known.

The formula for calculating the surface area (SA) of a cube is:

SA = 6A

SA = 6 x (4 in x 4 in)

SA = 96 in

2

SA = 2(wd + wh + dh)

SA = 2 x 44.125 in

SA = 88.25 in

SA = (2 r)h + 2( r )

SA = 56.52 in + 14.13 in

SA = 88.25 in

SA = 6A

Surface Area Calculations

In order to calculate the

surface area

(

SA

) of a

rectangular prism

, the area (A) of the three different faces must be known.

2

SA = 2(wd + wh + dh)

2

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.

Surface Area Calculations

2

2

SA = (2 r)h + 2( r )

2

2

2

2

SA = (2 r)h + 2( r )

SA = 6A

SA = 2(wd + wh + dh)

SA = r h

2

W = VWd

Volume

Volume

(

V

) refers to the amount of three-dimensional space occupied by an object or enclosed within a container.

Metric English System

cubic centimeter cubic inch

(cc)

(in. )

3

(lbs)

When an object is placed on a digital scale, the scale is measuring the force of the object and calculates mass.

W = VD

weight = volume x weight density

When an object is placed on a triple beam balance scale, the scale is measuring the mass because the force on both sides of the beam is equal.

2

W

(in. )

(lbs)

3

V = r h

3