Transcript ppt
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)