Unit 3: Geometry

Lesson #5: Volume & Surface Area

• In the design of containers and packages, two of the most important measurements to consider are volume and surface area. The volume of a three-dimensional object is a measure of how much space it occupies.

• The mathematical name for a box, with right angles at every corner, is rectangular prism. The volume of a rectangular prism can be calculated using the formula

V= l x w x h

• Volume is measured in CUBIC UNITS, because volume is a three-dimensional measurement. Typical cubic units for volume are cubic centimetres (cm3), cubic metres (m3), and cubic millimetres (mm3)

• Another unit that is commonly used is the litre (L), which is defined as the volume of a cube with sides of length 10 cm.

• Litres are generally used when measuring the volume of liquid. They are often used to describe the capacity of a container. Capacity is the greatest volume that a container can hold. For example, your family may have a milk jug with a capacity of 1 L.

A PrismCylinder

Rectangular Prism

Triangular Prism

base

Area Formulas

Area Circle = π x r2

r

Area Rectangle = Length x Width

l

w

b

h

Area Triangle = ½ x Base x height

Notice the first part of the formula for the volume of a rectangular prism.

V= l x w x h

This part, in red type, is the formula for the area of a rectangular base of the prism.

Therefore, the volume is the area of the base times the height:

Volume=base area x height l

w h

Volume Rectangular

Prism

• The volume relationship always applies to other types of prisms. Think of a prism as a flat shape, such as a rectangle or triangle, stretched through space to form a three-dimensional solid.

• The volume of any prism is the base area times the height (or depth).

Volume =base area x height

V=A x h

IMPORTANT NOTE ABOUTVOLUME OF PRISMS

TURN TO PAGE 82• First, calculate the area of

the triangular end of the prism (the front of the shape).

• Then, multiple the base area by the depth of the prism (the length of the shape from front to back)

VolumeTriangular Prism

2.8 cm2.4 cm

2.0 cmV=Aendhprism

Aend=bhend

TURN TO PAGE 82-83

• A cylinder is like a prism with a circular base. To determine the area of a circular base, you need to know its radius or diameter.

• For a cylinder, calculate area of the base, then calculate the volume by dividing it by the height.

Volume of a Cylinder

V=AendhAbase =Πr2

TURN TO PAGE 86-87• A cylinder holds three times the

volume of a cone with the same radius and height. The formula for the volume of a cone is:

Volume of a Cone

Abase =Πr2

Vcone=Aendh4.2 m

1.75 m

TURN TO PAGE 88

• A sphere is a ball-shaped object. The volume of a sphere can be calculate if you know its radius, using the formula:

Volume of a Sphere

V=Aendh

You have seen how the volume of a container or package can tell you its storage capacity. Another important measure is surface area, or how much material is needed to build or paint an object, such as an airplane wing.

When you carefully take a box part, and lay it flat, you can see way it is constructed. The flat pattern is called a net.

l

wh

Surface Area of a

Rectangular Prism

TURN TO PAGE 92

l

wh

Surface Area of a

Rectangular Prism

TURN TO PAGE 94

Atop, bottom =lwAfront, back=lwAsides=lwTo determine the TOTAL surface area, add up the areas of all six rectangles. Since there are two of each side, double each of the areas calculated.

Surface Area of a

Rectangular Prism

TURN TO PAGE 95

S= 2lw + 2wh + 2lh

or S= 2(lw + wh + lh)

Notice that, even though you are working with a 3-D object, surface area is a sum of 2-D area measurements. The units for areas are square units.

Surface Area of a

CylinderTURN TO PAGE 102

S= 2Πr2 + 2Πrh

The wing of a large passenger airplane has two functions. It holds the fuel for the planes, and provides lift to keep it in the air. So, the plane’s design should consider both the volume and surface area of the wing

.

Surface Area of a

Triangular PrismTURN TO PAGE 103

S=2Aend + 2Aside + Abottom

Each end is a right triangle. Recall the formula for the area of a triangle. Aends=bh

Apply the formula for the area of a rectangle for one side faces and bottom.

Asides=

Abottom=