MA120S
Unit: Surface Area & Volume
Work for Music Tour students
At the beginning of April you will be learning about surface area and volume of
different shapes and objects. Those students that are going on Music Tour
should complete the homework list below. If you have questions, refer to the
notes pages included here as well as the examples in the textbook. Feel free to
work together with friends who are also working on this. Also come talk to me
before Spring Break if you want.
There will be a quiz or test in late April. You will be responsible to know this
material for then as well as on the exam at the end of the year.
Good luck!
Page
p. 34
Topic
1.4 - Suri:ace Areas of Right
Assigned Questions Due On
#4-8, 12, 16,20
Pyramids and Right Cones
p. 42
1.5-Volumes of Right
Pyramids and Right Cones
#4,5,8, 10, 18(ab)
p. 51
1.6 - Surface Area and Volume
#3-5,8, 11, 16
p. 59
of a Sphere
1.7-Solving Problems
Involving Objects
#3, 5, 6, 9
p. 64
Review
#9-27
There will be a
Quiz or Test in
late April
Useful Formulae for this Unit
Cylinder
SA = 2nr2 + 27rrh
V=7Tr2h
Sphere
SA = 47rr2
y=l^r!
Right Pyramid with Regular Base
1
SA = -^s(perimeter of base) + (base area)
Right Pyramid
,:Y
Right Cone
SA = Trrs + nr2
nr2h
v =
3
MA120S - Mr. S. Koslowsky
1.4 - Surface Areas of Right Pyramids and Right Cones
Pyramids
is a 3-dimensional
object that has an
directly above the center of the
base. It also has triangular faces and a base that is a
height
Slant height
_. The shape of the _ tells you
the name of the pyramid. The
of the
pyramid is the distance from the center of the base to the
apex. When the base is a _ polygon, the
triangular faces are congruent (same shape and size).
The _ _ is the height of a triangular face.
Finding Surface Areas of Right Pyramids
Example #1 - Regular Tetrahedron
Calculate the surface area of a regular tetrahedron, with a side length of 5 0 m, to the
nearest square meter
MA120S - Mr. S. Koslowsky
Example #2 - Right Rectangular Pyramid
A right rectangular pyramid has base dimensions 4 m by6 m, and a height of 8 m.
Calculate the surface area of the pyramid to the nearest square meter.
Now let's figure out the surface area of a regular right pyramid in general (no numbers)
We'll do a right pyramid with a _ base.
The base has length _ and the slant height is
Here's the general formula for the surface area of a right pyramid with a regular polygon
base and slant height s:
MA120S - Mr. S. Koslowsky
Cones
is a 3-
dimensional object that has a
lateral area. The
base and a
is directly
above the center of the base. Similar to a pyramid, the height
and slant height are as shown in this diagram
The general formula for the surface area of a right cone with slant height s and base
radius r is:
Example #3 - Surface Area of a Right Cone
A right cone has a base radius of4m and a height of 10 m. Calculate the surface area
of this cone to the nearest square meter.
MA120S - Mr. S. Koslowsky
Example #4 - Finding an Unknown Measurement
The lateral area of a cone is 100 cm2. The diameter of the cone is 6 cm. Determine the
height of the cone to the nearest tenth of a centimeter.
Example #5 - Finding an Unknown Measurement (part 2)
A model of the Great Pyramid of Giza is constructed for a museum display. The surface
area of the triangular faces is 3000 in2. The side length of the square base is 50 in.
Determine the height of the model to a tenth of an inch.
MA120S - Mr. S. Koslowsky
1.5 -Volumes of Right Pyramids and Right Cones
Pyramids
When thinking about the volume of a right pyramid, it's helpful to first consider a
.. Right prisms can have a base that is any shape.
The Volume of a right prism is:
The volume of a
with the same base and the same
height is 1/3 that of the prism. So the formula is:
Example #1 - Calculating the Volume of a Right Square Pyramid
Calculate the volume of a right square pyramid with a base length of 2 ft. and a slant
height of 7 ft.
MA120S - Mr. S. Koslowsky
We could write a formula for the volume of a right rectangular pyramid:
Example #2 - Calculating the Volume of a Right Rectangular Pyramid
Determine the volume of a right rectangular pyramid with base dimensions 3.6 m by
4.7 m and height 6.9 m. Answer to the nearest tenth of a cubic meter
Cones
Right cylinders and
their
cylinder with the same
Volume of right cylinder-
Volume of right cone:
have the same relationship between
The volume of a right cone is 1/3 the volume of a right
and
MA120S - Mr. S. Koslowsky
Example #3 - Calculating the Volume of a Cone
Determine the volume of a right cone with a diameter of 8 mm and a height of 13 mm.
Answer to the nearest cubic millimeter.
Example #4 - Finding an Unknown Measurement
A right cone has a height of 8 m and a volume of 300 m3. Determine the radius of the
base of the cone to the nearest meter
MA120S - Mr. S. Koslowsky
1.6 - Surface Area and Volume of a Sphere
A sphere is an infinite set of
in space that are the same distance away
from a fixed point, which is the
.. A line segment that joins the center
to any of these points is called a
Surface Area of a Sphere
To think about a sphere's surface area, relate it to the curved surface of a
(not including the top and bottom circles). Draw a
cylinder and a sphere that have the same diameter. The height of the cylinder should
be the same as the diameter. These 2 surfaces have the same surface area!!!
The surface area of the curved part of a cylinder is:
Since the height = diameter, we can call the height 2r (2 times radius).
So this is the formula for the surface area of a sphere with radius r-
MA120S - Mr. S. Koslowsky
Example #1 - Determine the Surface Area of a Sphere
The diameter of a softball is approximately 4 in. Determine the surface area of a
softball to the nearest square inch.
Example #2 - Determine the Diameter of a Sphere
The surface area of a soccer ball is approximately 250 square inches. What is the
diameter of a soccer ball to the nearest tenth of an inch?
MA120S - Mr. S. Koslowsky
Volume of a Sphere
Imagine a sphere covered with very small
Each square has lines from it's corners to the center, which
forms a _ _ . The volume of
the sphere would be the
of all these pyramids.
Let's derive the formula!
Example #3 - Determine the Volume of a Sphere
The moon approximates a sphere with diameter 2160 mi. What is the approximate
volume of the moon? Express your answer in scientific notation with 2 decimals.
MA120S - Mr. S. Koslowsky
Example #4 - Determine the Surface Area and Volume of a Hemisphere
A hemisphere has radius 5.0 cm. Calculate the surface area and volume of the
hemisphere to the nearest tenth each.
MA120S - Mr. S. Koslowsky
^LZ-SoLvinci Problems Involving Composite Objects
A composite object consists of
or more distinct objects.
To find the volume of a composite object, find the volume of each part, then
the volumes.
Example #1 - Volume
Consider a right cylinder with a hemisphere placed on top. Both objects have a radius
of 18.0 cm. The cylinder has a height of 32.0 cm. Determine the volume of this object
to the nearest tenth of a cubic centimeter
To calculate the surface area of a composite object, only include the surfaces that are
on the _ of the object. Find these areas, then them.
Example #2 - Surface Area
Consider a cube with a right square pyramid on top of it. The cube has side length 5 m.
The right pyramid has a slant height of 4 m. Find the surface area of this composite
object.
MA120S - Mr. S. Koslowsky
Example #3
Consider a right rectangular prism with a right cylinder sitting on top. The prism has
dimensions 5 cm by 6 cm by 2 cm. The cylinder has radius 2 cm and height 10 cm.
Find the volume of this composite object.
Find the surface area of this composite object.