Volume and Surface Area of Solids
Name:
_____________________________________
Important Dates and Assignments:
Resource Notebook
### Insert a resource notebook page
Common Core High School Math Reference Sheet
(Algebra I, Geometry, Algebra II)
Conversions
1 inch = 2.54 centimeters
1 kilometer = 0.62 mile
1 cup = 8 fluid ounces
1 meter = 39.37 inches
1 pound = 16 ounces
1 pint = 2 cups
1 mile = 5280 feet
1 pound = 0.454 kilograms 1 quart = 2 pints
1 mile = 1760 yards
1 kilogram = 2.2 pounds
1 gallon = 4 quarts
1 mile = 1.609 kilometers
1 ton = 2000 pounds
1 gallon = 3.785 liters
1 liter = 0.264 gallon
1 liter = 1000 cubic centimeters
Formulas
Triangle
A=
1
bh
2
Pythagorean Theorem
a 2 + b2 = c 2
x=
-b ± b2 - 4ac
2a
Parallelogram
A = bh
Quadratic Formula
Circle
A = πr2
Arithmetic Sequence
an = a1 + (n – 1)d
Circle
C = πd or C = 2πr
Geometric Sequence
an = a1rn – 1
General Prism
V = Bh
Cylinder
V = πr2h
Sphere
V=
4 3
πr
3
Cone
V=
1 2
πr h
3
Pyramid
V=
1
bh
3
a1 - a1r n
where r≠1
Geometric Series
Sn =
Radians
1 radian =
Degrees
1 degree =
Exponential Growth/Decay
A = Ao ek(t-to ) + Bo
1- r
180
p
p
180
degrees
degrees
Reference Sheet for the 2005 Standards Geometry Exam
### Insert 2005 reference page
Warmup:
Find the area of the circle and circumference with the given radius or diameter:
1. Radius = 6
2. diameter = 15
answer in terms of π
answer rounder to the nearest hundredth
Some of the solids we will see:
Volume
Volume refers to the amount of space contained
within a solid figure. We measure volume in units of
length3, such as in3, cm3, and ft3.
1in3 is the volume of a cube 1in x 1in x 1in. A cylinder
with a volume of 50in3 would be able to contain 50 of
the 1in x 1in x 1in cubes.
Solids – Basic Vocabulary
•
solid - a 3D object
we will look at
prisms
cylinders
cones
pyramids
spheres
face - one of the flat surfaces that encloses a solid
polyhedron - solid whose faces are polygons
edge - segment where 2 faces meet
vertex - point where 2 or more edges meet
•
•
•
•
Prism
1. A prism is a polyhedron with 2 parallel bases and
rectangular lateral faces
Right Prism


Oblique Prism
the lateral faces are any faces that are not bases
the height is always the perpendicular distance between the bases
For each prism:
label the bases, height, and at least one lateral face
Volume of a Prism
Volume = B h, where B is the area of one base
ex) find the volume of a prism that is
6 cm high and has a 3 cm x 2 cm
rectangular base
ex) find the volume of the triangular
prism
Cylinder - A cylinder is solid with 2 congruent parallel circular bases
Volume of a cylinder = B h
Since the base is a _________ __, V = ____________
ex) What is the volume of a cylinder
whose diameter is 6 inches and
height is 9 inches. Give your answer in terms of π.
ex) A right circular cylinder has a volume of
432cm2. If the height of the cylinder is 4.5cm,
what is the radius of the cylinder?
Practice with Volume of Prisms and Cylinders
1. A right circular cylinder has a diameter of 6in and a height of 13in. What is the volume of
the cylinder, in terms of π?
2. A cylinder has a volume of 52π in3 and a radius of 4 in. What is the height of the cylinder?
3. The volume of the prism shown in the accompanying figure is 12.5 m3. What is the length of
the dimension labeled x
2.5 m
x
2m
4. A prism has bases in the shape of a regular pentagon. If the volume of the prism is 60 in3 and
its height is 5 in, what is the area of each base?
5. A one meter long length of pipe has an inner diameter of 2.5 inch and an outer diameter of
2.8 cm. The pipe is made of a metal with a density of 7.8 grams/cm3, what is the mass of the
pipe? (density x volume = mass)
6. A book publisher is shipping books that are 6 in x 8 in x 1 in. Each book weighs 24 ounces. The
publisher doesn’t want the weight of books in a single shipping container to exceed 100 pounds.
What is the maximum volume of books the shipping container must be able to hold?
7. A hexagon - based prism has a height of 10 in. and a volume of 60 in3. Which of the following is
not necessarily true about the prism?
(1) the lateral faces are all rectangles
(2) the area of each base is 3 in2
(3) The perimeter of the hexagon equals the area of the lateral faces divided by 10 in
(4) the two hexagonal bases are congruent
8. A square-based prism and cylinder have the same volume. What is the ratio of the side length of
the square to the radius of the cylinder?
1
(1) p
(2) π
(3) π2
(4)
p
9.
An 8 m diameter round swimming pool has a uniform depth of 1.5 m. The pump for the pool’s
filter can move water at a rate of 3000 liters per hour. How long will it take to pump a volume
equal to the entire pool through the filter? Round to the nearest hour.
10. Trayvon has two beakers to use in his chemistry lab. Beaker A has a diameter of 2cm and a
height of 12cm. Beaker B has a diameter of 3cm and a height of 9cm.
a) Which beaker has the greater volume?
b) The lab instructions say to add 12 grams of a liquid into one of the beakers. If the liquid has a
density of 0.89gm/cm3, will it fit into one of his beakers? Explain your answer.
Warmup:
1. Cut either this shape or the one on the opposite side
2. Using a ruler to measure the dimensions, find the AREA of each region. Write the area on each
region!
3. Predict what solid it will make, then tape together into a solid and see if you are correct
Lateral and Surface Area of Cylinders and Prisms
Lateral Area versus Surface Area
Lateral Area
Surface Area
Lateral and Surface Area of Cones and Prisms
1. Surface area = total area of ALL faces
2. Lateral area = area of only the lateral faces
(don't count the bases!)
3. Lateral Area of a prism =
perimeter of base x height
ex) example
Find the lateral area and surface area of the rectangular prism
shown in the accompanying diagram. The bases are the top and
bottom of the prism.
3in
10in
8in
4. Lateral area of a cylinder = 2πRh
Surface area of a cylinder = ________________________
ex) Find the surface and lateral area of a cylinder with a radius of 4 in
and a height of 10 in. Give your answer in terms of π.
ex) Ella received flowers from her boyfriend today at work. She places the flowers
in a vase in the shape of a cylinder. The circumference of the base of the vase is 10
π. The volume of the vase is 75 π . Find the lateral area of the vase in terms of π .
Practice with Lateral and Surface Area of Cylinders and Prisms
1. What is the surface area of a right circular cylinder whose radius is 10 m and height
is 5 m?
2. A can of soup is 3 inches in diameter and 5 inches high. The label that goes around
the can begins ¼ inch from the bottom and comes up to ¼ inches from the top. What
is the area of the label? Round to the nearest 0.1 square inch.
3. The bases of a triangular prism are isosceles triangles with
legs 8 cm long and a base 6 cm long. The height of the prism
is 12 cm. Find the lateral and surface area of the prism.
8cm
12cm
6cm
14
3
4. Find the lateral and surface area of the trapezoidal prism.
5
6
5. Will is painting a box that is in the shape of a pentagonal prism. The lateral faces will
be blue and the bases will be green. The perimeter of base is 120 inches, and the
height of the box is 50 inches. Small cans of paint can cover 25 ft2. How many cans of
each color must be buy?
6. A right circular cylinder has a volume of 245 cm3. If its height is 12 cm, find the lateral and
surface of the cylinder. Round your answer to the nearest hundredth.
7. Find the volume, surface area, and lateral area of a right circular cylinder that has a
circumference of 18 cm and an height of 10 cm. Round to the nearest unit.
8. Jocelyn is painting the walls and ceiling of her room. The room measures 12 ft x 8 ft,
and the walls are 8 ft high. If a can of paint can cover 250 square feet, how many
cans of paint does she need to buy if she wants to apply two coats of paint?
9. Which quantity is always larger – lateral area or surface area? Explain your
reasoning.
Warmup
Use the following conversions to do these problems:
1 foot = 12 inches
and
1 inch = 2.54 cm
strategy: Make a ratio out of the conversion equation, with the unit to keep on top, and the unit to
eliminate on the bottom. Then multiply by the given dimension!
1) Fred is 64 inches tall.
What is his height in centimeters?
2. Jack is 5 1/2 feet tall.
What is his height in inches?
Volume of Cones and Pyramids
1. circular cone solid with circular base and one vertex (apex)
a) height(h) - vertical distance from vertex to base,
the height is always _______________________ to the base
b) Volume (V) =
1
Bh
3
since the base is a __________________,
apex
h
R
B = _________________
ex) Find the volume of a cone whose diameter is 6 inches and height is 4 inches.
Express your answer in terms of π.
4
6
ex) Find the radius of a cone whose volume is 420 in2 and height is 12 in. Round to the
nearest tenth.
2. Pyramid - solid with one polygon base, triangular faces, and an apex
a) Pyramid has the same volume formula as a cone: Volume (V) =
1
Bh
3
where B is the __________________________________
ex) The base of a pyramid is a right triangle with legs of length 5 and 12. The height of the pyramid
is 9. Find the volume of the pyramid.
ex) A square-based pyramid has a volume of 160 cm3. If its height is 10 cm, what is the perimeter of
its base?
Now You Try It – Cones and Pyramids
1. Find the volume of a cone with radius 6 ft
and height 8 ft. Express your answer in terms of π.
8
6
2. Find the radius of a cone whose volume is 525 in3and whose height is 5 in. Round to the nearest
tenth.
3. 3. Find the volume a pyramid whose base is a rectangle 10 ft wide and 6 feet long, and whose
height is 12 ft.
4. An Egyptian pyramid has a square base 100 ft on each side, and a height of 60 ft. It is built of
limestone blocks that are 3ft x 5ft x 2ft in dimension. How many blocks were needed to build the
pyramid (assume the pyramid is solid inside).
Practice with Cones and Pyramids
1. What is the volume of a square-based pyramid if the perimeter of the base is 60 inches
and the height is 20 inches?
2. What is the volume of a cone with a diameter of 4 ft and a height of 6 ft, in terms of π?
3. What is the radius of a cone whose volume is 455 in3 and height is 12 in? Round to the
nearest hundredth.
4. What the volume of a cone whose radius is 6 in and height 8 in, in terms of π?
5. A pyramid has a base in the shape of a trapezoid as shown in the figure. If the height of
the pyramid is 10, what is its volume? Hint – if you don’t know the formula for the area
of a trapezoid, divide it into to triangles and a rectangle.
5
6
9
6. What is the volume of a rectangular pyramid whose base is a 4 inches by 3 inches, and
is 12 inches high?
7. A cone and a cylinder have the same radius height. The volumes of the two solids are
equal. Find the ratio of the radius of the cone to the radius of the cylinder.
8. An hourglass is made of two cones, each with a radius of 5 inches and a height of 10
inches, and the sand fills one cone 75% of one cone. If it takes 5 seconds for 10 cubic
inches of sand to pass between the cones, how long will it take for all the sand to pass
through?
Warmup:
1. Cut out one of the shapes below
2. Using a ruler to measure the dimensions, find the AREA of each region. Write the area on each
region!
3. Predict what solid it will make, then tape together into a solid and see if you are correct
Lateral and Surface Area of Cones and Pyramids
1. slant height (
) - distance from vertex to edge of base
2. lateral area of a cone: L = πR
3. The slant height and radius form a ________________________________
equation: __________________________________________
4. Surface Area of cone = ________________________________________________
example
Find the lateral area and surface area of a cone whose diameter is 6 and
height is 4.
5. Lateral Area of a Pyramid
____________________________________________
Surface Area of a Pyramid
____________________________________________
ex) A pyramid has a square base with side length of 10 ft and a height of 18 ft.
Find its lateral and surface area.
Let’s Do it Together!
1. Find the surface and lateral area of a cone with radius 6 ft and height 8 ft.
Round to the nearest tenth.
2. Find the lateral area of a cone whose volume is 525 in3 and whose height is 5 in.
Round to the nearest tenth.
3. Find the lateral area and surface area of a pyramid whose base is a square 10 ft on each side,
and whose height is 12 ft.
4. Road salt storage buildings operated by the NYS highway
department are pyramids with a regular 16-sided polygon
for a base that measures 42 feet across and 35 feet high. A
painter wants to estimate the amount of paint needed to
paint the exterior of the building. If a gallon of paint will
cover 300 square feet, how many gallons of paint will be
needed.
Practice with Lateral and Surface Area of Cones and Pyramids
1. Explain in words how to find the slant height of a cone if you know the radius and
height of the cone. Include a sketch!
2. What is the lateral area and surface area of a cone whose radius is 6 in and height 8
in?
3. What is the lateral and surface area of a pyramid whose base is a square 10 inches on
each side and 12 inches high?
4. An ice cream cone has a diameter of 3 inches and height of 6 inches. A paper wrapper
starts at the tip of the cone and extends 2/3 of the way up the cone. Assuming no
overlap of the wrapper, what is the area of the wrapper? Round to the nearest tenth.
5. True Story - The external fuel tank of the space shuttle is approximately a right circular
cylinder with a diameter of 12.14 ft and a length of 45.6 ft. The tanks were painted white on
the first few missions. Then engineers examined all the space shuttle systems to look for
ways to reduce weight so they could increase payload. Someone suggested not painting the
external fuel tank. The paint, after drying, weighs 0.35 pounds per square foot. How much
weight was saved? Round to the nearest pound.
Warmup
Write a definition of a circle:
What do you think a corresponding definition of a sphere would be?
Spheres
Sphere - ______________________________________________________________________________
______________________________________________________________________________
Volume (V) =
4
πR3
3
Surface Area (SA) = 4πR2
ex) Find the volume and surface area of a sphere with
a radius of 7.5 inches. Round to the nearest tenth.
ex) The diameter of the earth is 7960 miles.
What is the surface area of the earth? Round to the nearest 1 square mile
ex) The volume of a sphere is 30π cubic inches. What is its radius?
R
ex) The surface area of a sphere is 20 in2. Find the volume of the sphere to the nearest 1 in3.
ex) You place one sphere of ice cream on top of a cone. The sphere has a radius of 7
cm. The diameter of the top of the cone is 5 cm and the height of the cone is 12 cm.
When the ice cream melts, will it overflow the cone? Explain your answer.
You Try it – Spheres
1. In terms of π, what is the volume of a sphere whose radius is 16 ft?
2. What is radius of a sphere whose volume is 130in2? Round to the nearest tenth.
3. What is the surface area of a sphere whose volume is 36π in3?
4. Find the volume and surface area of a sphere with a radius of 4 inches. (in terms of
p)
5. The volume of a sphere is 1000 cm3. What is its diameter? Round to the nearest tenth.
6. The surface area of a sphere is 16p cm2. What is its radius?
7. A craft store sells 6 inch diameter styrofoam spheres for $1. How much would you expect to
pay for a 1 foot diameter sphere? Justify your answer.
8. Jack and Jill built a snowman composed of three snowballs. The bottom snowball has a
diameter of 75 cm. Each snowball above that is a dilation of the one below by a scale factor
of 2. What is the total volume, in cubic meters of the snowman? Round to the nearest tenth.
9. Explain in words and a sketch how the definition of a sphere a circle are alike, and how they
are different. Use appropriate geometry vocabulary.
10. Challenge! The Everswift tennis ball company packages three tennis balls in a cylindrical
tube with the balls stacked one above the other. Assuming no gaps at the top, bottom and
one the sides between the balls and the tube, what fraction of the tube is occupied by the
tennis balls?
11. Challenge! A sphere is circumscribed around a right circular cylinder whose diameter is 20
cm and whose height is 24 cm. What is the volume of the cylinder? Round to the nearest 0.1
cm3.
Warmup
Convert the following units:
Remember – write the conversion factor as a ratio
with units to keep on top, and units to eliminate on the bottom!
15 cm to inches
15,800 ft to miles
2.5 tons to pounds
8 ft to centimeters
15 kg to grams
11 cm3 to in3
Modeling With Volume
We can model real world objects using our basic solids.
What types of objects can you think of that can be modeled by…
Prisms:
Cylinders:
Cones:
Pyramids:
Spheres:
You may be asked to calculate weight or cost:
weight= density x volume
cost = price/volume x volume
cost = price / mass x weight x volume
example
The density of gasoline is 0.026 lbs/in3. What is the weight of a 2ft diameter x 3.5 ft high barrel of
gasoline?
Step 1 – convert feet to inches
Step 2 – find the volume of the barrel
Step 3 - calculate the mass
A volume of material may be transformed from one shape to another shape.
The total volume is unchanged!
example
A baker is rolling out dough for a pie crust. The dough is initially round with a diameter of 15in and
a uniform height of 2in. She rolls it out so that it is still round, but now has a diameter of
1
in. How
4
many round pie crusts can she cut out of the dough if each pie crust has a diameter of 20 inches?
Design Strategies
You may be asked to find the dimensions of some figure or solid that:

makes volume or area a maximum or minimum

makes volume or area a specific value

maximize or minimize cost
Strategy - model the situation as usual and use trial and error to find the correct value!
example
A food packager is designing a cylindrical aluminum can that must hold 150in3 of tomato sauce.
What dimensions of the can will result in the least amount of aluminum being used? Round to the
nearest 0.1 inch. (the surface area of a cylinder is 2πRh + 2πR2, where R is the radius and h is the
height of the cylinder).
Design Problems can involve many steps! Make a plan:
A construction company for a construction company working on plans for
a project that call for digging a straight 0.25 mile tunnel through a
mountain. The cross section of the tunnel is shown in the accompanying
figure.
10ft
20ft
As the workers dig through the mountain, the rubble is brought to a gravel company to be
processed into gravel. The cargo beds of the dump trucks are 18ft long by 10ft wide by 6ft high, and
the trucking company charges $450 per load delivered to the gravel company. Approximately how
much money should the engineer budget for trucking costs for the project?
Plan:
What needs to be modeled, and by what shape? _______________________________________________
What are you looking for?
What quantities need to be calculated?
_______________________________________________
_______________________________________________
Practice with Modeling
1. An orange-juice processing plant fills 375 mL bottles with orange juice. The average orange
has a diameter of 2.5 in and is 10% juice by volume. How many bottles are needed to
process 15,000 oranges?
2. Jack is making a scaled drawing of the floor plan of his home. The scale factor is 1in:4ft. The
drawing of his living room is a rectangle measuring 5 inches by 3 inches. He is planning to
purchase new carpet for the living room that costs $4 per square foot. How much will the
carpet cost?
3. A construction worker is building a brick wall using bricks that are 8in x 4in by 2in. Each
brick weighs 4 lbs. How many pound of bricks are needed to build a wall 1 ft wide, 4ft high
and 8ft long?
4. Carrie is designing a new patio for her backyard. The patio is to be 24 ft by 18ft, which
includes a 4 in border made using 8 in by 4 in red bricks. How many red bricks will she need
for the border if no bricks are to be cut?
5. Patrick uses heating oil to heat his house in the winter. He has an oil storage tank in his
basement that is 2 m long and 1 m in diameter. The energy content of heating oil is 0.034
million BTU/liter, and he is charged $1300 to completely fill the tank. He is considering
switching to natural gas, which costs $10 per million BTU. Which fuel would be less
expensive to use? Justify your answer.
6. The Metro Zoo is designing a new enclosure for their 6 polar bears. It has a central island
surrounded by a moat 5 ft deep and 5 ft wide, as shown in the accompanying figure. The
population density of polar bears on the island is specified to be 0.004 polar bears per
square foot. What will be the volume of water in the moat? Round to the nearest 1 ft3.
Island
5 ft
7. A cardboard box is to have a height of 5 cm and a square base. What is the maximum
volume of the box that can be constructed using 400 cm3 of cardboard?
8. Kevin wants to install solar panels capable of providing 3000 Watts of electrical power to
his home. The supplier he selected sells panels that are rectangular with a length of 60 in
and a width of 40 in, and cost $275 each. Kevin has determined that these solar panels
installed on his roof could produce electrical power with an energy density of 14 Watts/ft2.
How much money will Kevin have to spend on solar panels?
9. A winery ages its wine in barrels that have a diameter of 25 in and a height of 35 in. There
are currently 125 full barrels of wine in the cellar of the winery. If the winery can sell a 750
ml bottle of wine for $14, what is the value of wine in the cellar? Round to the nearest dollar.
10. The towns of Marionville and Springfield are shown on
the map below. Marionville has a population of 8100
people. If Springfield has the same population density
as Marionville, estimate the the total population of
Springfield, rounding to the nearest thousand.
Marionville
1 grid unit =
1 mile
Springfield
11. A cylindrical piece of aluminum has a diameter of 30cm inches and a length of 75cm. The
cylinder will be rolled into aluminum foil, and the foil will be packaged in 1-pound boxes to
be sold in a grocery store. The density of aluminum is 2.7 grams/cm3. How many boxes of
aluminum foil can be produced from the piece of aluminum?
12. The town of Bakersfield has hired the APEX construction company to install a new water
tower in their town. The Bakersfield tow board specified a water tower with a capacity of
100,000 gallons and a total height of no more than 75 ft. The designers at APEX have a
water tower design that consists of a cylinder sitting on top of a cone, where the height of
the cone is ½ its radius and the height of the cylinder is 3 times the radius of the cone. If the
apex of the cone must be at least 30 ft above the ground to provide proper water pressure,
can the designers at APEX use their design? Justify your answer.
13. Challenge! An engineer is designing a hinge to be used on
the landing gear of a new airplane. She sketches the cross
section of the hinge on a computer, which is shown in the
figure below. The circle represents a hole for a hinge-pin.
y
(5, 9)
(9, 9)
(2, 6)
(7, 7)
The engineer next uses a function on the computer that
creates a solid by translating the cross-section in a
(2, 4)
direction perpendicular to the sketch. The engineer enters
(5,2)
(9,2)
a translation distance of 10 units. In the final step the
engineer enters a scale factor of 1 unit = 2 centimeters. The
x
resulting computer image of the solid is then sent to a
factory to be manufactured out of a high strength metal. If the density of the metal is 8.44
grams/cm3 and the metal costs $32 per kilogram, what is the expected cost for the metal to
make one hinge? Round to the nearest dollar.
Warmup
What is the area of a regular octagon with side length 12?
Hint - Complete a central angle, then sketch an altitude of that triangle. You
will then need to apply trigonometry!
Cross Sections
A Cross Section is the 2-dimensional figure created when a plane intercepts a solid.
cross
section
Q
base
P
Cross sections taken parallel to the base of common solids:
cross
sections
cross
section
Q
base
P
Properties of Cross Sections Taken Parallel to the Base
Solid
Cross Section Shape
Prism
Cylinder
Cone
Pyramid
Sphere
Are Cross Sections Congruent
or Similar?
example
A right circular cylinder has a radius of 4 and a height of 10. What is the area of a cross section
taken parallel to the bases? Write your answer in terms of π.
example
The base of a pyramid is a regular octagon with side length 24. The height of the pyramid is 12.
What is the area of a cross section taken parallel to the base and 2/3 of the way from the base to
the apex? (hint – use the result from the warmup!)
example
A length of pipe has a wall thickness of 0.25 inches and an inner diameter of 2 inches. Sketch the
cross section of the pipe, including relevant dimensions.
An I-beam is a solid piece of steel with a cross section in the shape of an “I”. What is the volume of a
12 foot long I-beam with the cross section shown in the accompanying figure. Round to the nearest
0.1 ft3.
6 in
2 in
2 in
8 in
Cross sections do not have to be parallel to the base:
Can you sketch the following? Describe the shape of the cross section.
The cross section of a square-based pyramid,
perpendicular to the base.
The cross section of a cube that makes a 45 angle
with base.
The cross section of a cone perpendicular to
the base.
The cross section of a cone that makes a 45 angle
with the base.
Solids of Revolution
Certain solids can be generated by rotating a planar figure around line. We call these solids of
revolution. Some examples of solids of revolution and the figures that generate them are given in
the following table.
Sketch the figure and the solid of revolution of revolution formed:
Figure Rotated
Solid
right triangle,
rotate 360 about leg AB
rectangle ABCD,
rotate 360 around side CD
circle P,
rotate 180 about diameter AB
example
A right triangle JGH has a right angle at H and legs with lengths GH = 7cm and GJ = 6cm. What solid
figure is generate when ∆GHJ is rotated 360 about JG ? What is the volume in terms of π?
example
A quadrilateral with coordinates A(0, 3) B(5, 3) C(5, 0) and D(0, 0) is rotated 360 about the x-axis.
Describe the solid generated and find the volume.
Practice with Cross Sections and Solids of Revolution
1. The bases of a prism are right triangles. Which of the following describes the cross sections
of the prism taken parallel to the base?
(1) right triangles
(2) isosceles triangles
(3) rectangles
(4) trapezoids
2. A cross section of a cylinder is in the shape of a square. What must be true about the
cylinder?
(1) The cylinder is oblique
(2) The height of the cylinder is twice the length of its radius
(3) The cross section was formed by the intersection of the cylinder and a plane parallel to
one of its bases
(4) The area of the square is equal to the area of one of the bases of the cylinder
3. Which of the following statements is not true about the pentagonal
prism shown in the accompanying figure?
(1) All cross sections taken parallel to the base are congruent
(2) All cross sections taken perpendicular to the base are congruent
(3) All cross sections taken parallel to the base are pentagons
(4) All cross sections perpendicular to the base are rectangles
4. A solid has a square cross section when it is intercepted by a plane parallel to the base. The
cross section is triangle when the plane is perpendicular to the base. Which of the following
could be the solid?
(1) rectangular prism
(2) circular cone
(3) triangular prism
(4) rectangular pyramid
5. Sarah is designing plumbing system for a new building using a computer-aided design
program. She wants to create a hollow pipe using the program by sketching a cross section
and then rotating it 360 around a line. Which of the following cross sections and lines
would result in a hollow cylindrical pipe?
(1)
(2)
(3)
(4)
6. Which of the following is not true about the cross sections of a sphere?
(1) All the cross sections have the same radius
(2) All the cross sections are circles
(3) All the cross sections are similar
(4) The cross sections with the largest diameter pass through the center of the sphere
7. ∆JKLis a right triangle with a right angle at K, JK = 6 and KL = 10. If it is rotated 360 about
KL, describe the solid generated and find the volume.
8. In which of the following solids are all cross-sections similar?
(1) cube
(2) sphere
(2) cylinder
(4) triangular-based pyramid
9. The triangle with vertices A(0, 0) B(4, 0) C(0, 6) is rotated about the y-axis. Describe the
solid formed and find its volume in terms of π.
10. The bases of triangular prism ABCD are equilateral triangles with side
length 8. The height of the prism is 6. What is the area of a cross
section passing though C, F, and the midpoint of D? Express your
answer in simplest radical form.
C
B
A
F
D
11. Rectangle MNOP has length MN = 10 and NO = 6. If MNOP is rotated 360 about OP . What
solid figure is generated, and what is its volume?
12. A semicircle with a radius of 12cm is rotated 180 about the line that contains the radius.
What volume is generated and what is its volume? Express your answer in terms of π.
13. Challenge! A sphere has a radius of R. In terms of R, what is the area of a cross section
taken at a distance of ½R from the center of the sphere?
E
Warmup
Sketch a cylinder and show two cross sections that are both parallel to base. What solid is formed
by the region bounded by the two cross sections?
Cavalieri’s Principle
There are two versions of Cavalieri’s principle you should know:
If

two solids are bounded between two parallel planes,
AND

every parallel plane between these two planes intercepts regions of equal area
Then
then the solids have equal volume.
T
A solid is bounded by two planes if
the solid intersects each plane in at
least one point,
S
R
AND
No region of the solid extends beyond
the planes
In other words - If every square cross section is equal in area to every corresponding triangular
cross section, then the two prisms have equal volume.
Cavalieri’s Principle – version 2
If

two solids are bounded between two parallel planes,
AND

every parallel plane between these two planes intercepts regions of equal area
Then
Any pair of planes parallel to the bounding planes intercepts solids of equal volume
If every round cross section is equal in area to the corresponding rectangular cross section, then the
truncated cone and truncated pyramid formed by planes S and R are equal in volume.
The converse of Cavalieri’s principle is useful!
Form the converse by switching the second part of the “if” statement and the and “then” statement:
example
Explain why a “crooked” stack of 7 quarters has the same volume as a “straight” stack of 7 quarters,
in terms of Cavalieri’s principle.
example
The right circular cylinder and the oblique circular cylinder shown below have the same equal
heights and congruent bases. Are their volumes equal? Justify your answer.
h
h
example
If two solids are bounded by the same two parallel planes, and their volumes are equal, can you
conclude that any plane parallel to the bounding planes will intercept cross sections of equal area?
If yes, justify your reasoning. If no, sketch a counterexample.
Practice with Cavalieri’s Principle
1. Solids A and B, shown in the figure below, each have uniform cross sections and equal heights
perpendicular to the shaded bases.
A
B
If the heights of the two solids are equal, which of the following statements represents an
application of Cavalieri’s principal?
(1) The surface areas of solid A and solid B may be different even though their volumes are
equal
(2) Solid A can be generated by rotating a rectangle, while solid B is not a solid of revolution
(3) If the area of base A equals the area of base B then the volumes of the solids are equal
(4) If the volume and height of solid A and B are equal, then the their surface areas must be
equal
2. Two right circular cones each have a height of 4 in and a volume of 30 in3. Which of the
following is not an application of Cavalieri’s principle?
(1) The two cones have the same volume
(2) The two cones have the same surface area
(3) A cross section parallel to and ¾ in above the base of one cone must be equal in area to a
cross section taken ¾ in above and parallel to the base of the other cone
(4) Plane A is parallel to and 1 in above the base of each cone. Plane B is parallel to and ¾ in
above plane A. The two disks intercepted by planes A and B are equal in volume.
3. A cylinder and prism have the same height and the same volume. Which of the following is
not necessarily true?
(1) the area of their bases are equal
(2) a plane parallel to the bases of both the cylinder and prism intercepts two
corresponding figures with equal areas
(3) two planes parallel to the bases of each solid, intercept corresponding solids with equal
volumes
(4) the cylinder and prism are congruent
4. A stack of identical 10 playing cards is arranged as shown in the figure on the left. Anne
picks up the cards and straightens the stack as shown in the figure on the right. Explain why
the two stacks have the same volume.
5. A right rectangular prism and an oblique rectangular prism each have bases that are 5 cm
by 2 cm and heights of 10 cm. Explain, in terms of Cavalieri’s principle, why the two solids
have the same volume.