Volume & Surface Area

MATH 102 Contemporary Math S. Rook

Overview

• Section 10.4 in the textbook: – Volume – Surface area

Volume & Surface Area

Volume & Surface Area in General

• • • • Recall that

perimeter and area are measurements of two-dimensional figures

– e.g. rectangles, triangles, etc.

These same measurements have counterparts for three-dimensional figures

Surface Area: outside

a measurement of space on the of a three-dimensional figure

Volume:

a measurement of the space dimensional figure

inside

a three – In general

V

and

h

=

A

x

h

where

A

is the area of the base is the height (third dimension)

•

Volume & Surface Area in General (Continued)

What follows on the next few slides are the formulas for volume and surface area for common three-dimensional figures – Except for the volume of a rectangular solid and a cube, you are not required to memorize these formulas – However, you should know how to apply them to solve problems

Rectangular Solids & Cubes

• • • • Recall the formula for area of a rectangle Given a rectangular solid with length

l

, width

w

, and height

h

: –

SA

= 2

l w

+ 2

l h

+ 2

w h

–

V

=

l w h

Recall the formula for area of a square Given a cube with length

l

(length, width, and height are the same for a cube): –

SA

= 6

l

2 –

V

=

l

3

Cylinder

• • • A cylinder is a three-dimensional extension of a circle with an added height Recall the formula for area of a circle Given a cylinder with radius

r

height

h

: and –

S.A.

= 2π

r h

• + 2π

r

2 Think about “opening up” the cylinder and lying it down flat as a rectangle and then adding in the area for the two circular bottoms –

V

= π

r

2

h

Cone

• • A right circular cone has a height

h

that extends from the tip and is perpendicular to the circular base of the cone Given a right circular cone with a height

h

radius of its circular base

r

: and

SA

 

r r

2 

h

2 –

V

 1 3 

r

2

h

Sphere

• •

r

A sphere is a three-dimensional extension of a circle with radius – Think of a ball that can be cut into circles – The radius is measured from the center of the sphere – The Earth is essentially a sphere Given a sphere with radius

r

:

SA

 4 

r

2

V

 4 3 

r

3

Volume & Surface Area (Example)

Ex 1:

What is the minimum area of wrapping paper required to completely cover a box with dimensions 12.4” x 11.9“ x 7.4“?

Volume & Surface Area (Example)

Ex 2:

A packing crate in the shape of a rectangle has dimensions of 12 ft x 8 ft x 60 in. How many cubic packages with sides of length 3 ft can fit into the crate?

Volume & Surface Area (Example)

Ex 3:

An ice-cream cone in the shape of a right circular cone has a radius of 4 cm and a height of 8 cm.

a) How much ice cream can the cone hold if we completely fill it?

b) After filling the cone, a company decides to wrap it for packaging. How much wrapping is required?

Volume & Surface Area (Example)

Ex 4:

A punch bowl is in the shape of a hemisphere (half a sphere) with a radius of 9 inches. The cup part of the ladle in the bowl is also in the shape of a hemisphere with a diameter of 4 inches. If the punch bowl is filled completely, how many full ladles of punch are in the bowl?

Summary

• • • After studying these slides, you should know how to do the following: – Be familiar with the different formulas for surface area & volume of common three-dimensional figures Additional Practice: – See problems in Section 10.4

Next Lesson: – Introduction to Counting Methods (Section 13.1)