Volume & Surface Area

MATH 102Contemporary 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: a measurement of space on the outside of a three-dimensional figure

• Volume: a measurement of the space inside a three-dimensional figure– In general V = A x h where A is the area of the base and h 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 = 2lw + 2lh + 2wh– V = lwh

• 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 = 6l2 – V = l3

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 and height h:– S.A. = 2πrh + 2πr2

• 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 = πr2h

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 and radius of its circular base r:

–

22 hrrSA

hrV 2

3

1

Sphere

• A sphere is a three-dimensional extension of a circle with radius r– 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:24 rSA

3

3

4rV

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)