8-1. Compute the volume of the figures below.
8-2. Carefully draw the prisms below onto your paper. One way to do this is to draw
the two bases first and then to connect the corresponding vertices of the bases. Find
the total surface area of the prisms. Assume that the bases in part b are regular
hexagons with side length 4 inches and part c are equilateral triangles with side length
8 centimeters.
8-3. A butterfly house at a local zoo is a rectangular prism with dimensions
20'× 15'× 10' and contains 625 butterflies.
a. Sketch the prism on your paper.
b. What is the volume of the butterfly house? Show your work.
c. How many cubic feet of air is there for each butterfly?
d. Density is the quantity of something per unit measure, especially length, area, or
volume. For example the mass density of an element (iron has density of 7.874 grams
per cubic centimeter). Assuming the butterflies are equally distributed inside the
butterfly house, what is the density of butterflies? Explain.
8-4.A cube is a special type of rectangular prism because each face is a square.
a. What are some examples of cubes?
b. Find the volume and surface area of a cube with an edge length of 10 units.
c. Remember that a “flat side” of a prism is called a face, as shown in the diagram
above. Notice that the line segment where two faces meet is called an edge, while the
point where the edges meet is called a vertex. How many faces does a cube have?
How many edges? How many vertices? (Vertices is plural for vertex.)
8-5. A cube has an edge length of 16 units. Draw a diagram of the cube and find its
volume and surface area.
8-6. A regular hexagonal prism has a volume of 2546.13 cm3 , base area of 84.87 cm2
and base edge of 14 cm. Find the height and total surface area of the prism.
8-7. Calculate the total surface area and volume of the prism below. Assume that the
base is a regular pentagon. Show all your work.
8-8. A pyramid has a volume of 108 cubic inches and a base area of 27 square inches.
Find its height.
8-9. Find the volume of the square-based pyramid with base edge 9 units and height
48 units.
8-10. Find the volume and total surface area of a square-based right pyramid if the
base edge has length 6 units and the height of the pyramid is 4 units. Assume the
diagram below is not to scale.
8-11. Find the total surface area of the square-based pyramid. Show all work.
8-12. As Sara shopped for a tent, she came across two models that she liked best,
shown below. However, she does not know which one to pick! They are both made
by the same company and appear to have the same quality. She has come to you for
help in making her decision.
a. What are the shapes of the two tents?
b. Without doing any calculations, which tent do you think Sara should buy and why?
c. Help Sara decide which tent to buy for her backpacking trip. To make this
decision, compare the volumes, base areas, and total surface areas of both tents. Show
all your work and explain your recommendation.
8-13. The TransAmerica building in San Francisco is built of concrete and is shaped
like a square-based pyramid. The building is periodically power-washed using one
gallon of cleaning solution for every 250 square meters of surface. As the new
building manager, you need to order the cleaning supplies for this large task. The
problem is that you do not know the height of each triangular face of the building; you
only know the vertical height of the building from the base to the top vertex.
Your Task: Determine the amount of cleaning solution needed to wash the
TransAmerica building if an edge of the square base is 96 meters and the height of the
building is 220 meters. Include a sketch in your solution.
8-14. Copy the cylinder below onto your paper. Find its total surface area and volume
if the radius of the base is 5 units and the height of the cylinder is 8 units.
8-15. Hernando needs to replace the hot water tank at his house. He estimates that his
family needs a tank that can hold at least 75 gallons of water. His local water tank
supplier has a cylindrical model that has a diameter of 2 feet and a height of 3 feet. If
1 gallon of water is approximately 0.1337 cubic feet, determine if the supplier’s tank
will provide enough water.
8-16. Cindy’s cylindrical paint bucket has a diameter of 12 inches and a height of 14.5
inches. If 1 gallon ≈ 231 in3, how many gallons does her paint bucket hold?
8-17. A cork in the shape of a cylinder has a radius of 2 cm, a height of 5 cm, and
weighs 2.5 grams.
a. What is the volume of the cork?
b. What is the cork’s density in grams per cubic centimeter? That is, how many grams
of cork are there per cubic centimeter?
8-18. A rectangular prism has a cylindrical hole removed, as shown below.
a. If the length of the radius of the cylindrical hole is 0.5 cm, find the volume of the
solid.
b. What could this geometric figure represent? That is, if it were a model for
something that exists in the world, what might it be?
8-19. The Math Club has decided to sell giant waffle ice-cream cones at the Spring
Fair. Lee bought a cone, but then he got distracted. When he returned to the cone, the
ice cream had melted, filling the cone to the very top! If the diameter of the base of
the cone is 4 inches and the slant height is 6 inches, find the volume of the ice cream
in the cone.
8-20. Find the total surface area of each solid below. Show all work.
8-21. Which has greater volume: a cylinder with radius 38 units and height 71 units
or a rectangular prism with dimensions 34, 84, and 99 units? Show all work and
support your reasoning.
8-22. Talila is planning on giving her geometry teacher a gift. She has two containers
to choose from. A cylinder tube with diameter 6 inches and height 10 inches. A
rectangular box with dimensions 5 inches by 6 inches by 9 inches. Assuming that her
gift can fit in either box, which will require the least amount of wrapping paper?
8-23. A silo (a structure designed to store grain) is designed as a cylinder with a cone
on top, as shown in the diagram above.
a. If a farmer wants to paint the silo, how much surface area must be painted?
b. What is the volume of the silo?
8-24. As Shannon peeled her orange for lunch, she realized that it was very close to
being a sphere. If her orange has a diameter of 8 centimeters, what is its approximate
surface area (the area of the orange peel)? What is the approximate volume of the
orange? Show all work.
8-25. Perhaps you think the Earth is big? Consider the sun! Assume that the radius
of the Earth 4000 miles. The sun is approximately 109 times as wide.
a. Find the sun’s radius.
b. What is the volume of the sun?
c. What is the volume of the earth?
d. If the sun were hollow, how many Earths would fill the inside of it?
8-26. An ice-cream cone is filled with ice cream. It also has ice-cream on top that is
in the shape of a cylinder. It turns out that the volume of ice cream inside the cone
equals the volume of the scoop on top. If the height of the cone is 6 inches and the
radius of the scoop of ice cream is 1.5 inches, find the height of the extra scoop on
top. Ignore the thickness of the cone.
8-27. When considering new plans for a covered football stadium, Santa Maria
looked into a design that used a cylinder with a dome in the shape of a hemisphere.
The radius of the proposed cylinder is 200 feet and the height is 150 feet. See a
diagram of this below.
a. One of the concerns for the citizens of Santa Maria is the cost of heating the space
inside the stadium for the fans. What is the volume of this stadium? Show all work.
b. The citizens of Santa Maria are also interested in having the outside of the new
stadium painted teal. What is the surface area of the stadium? Do not include the
base of the cylinder.
8-28. The Germany Historical Society has just acquired a castle on the Rhine River
and wishes to turn it into a museum. But for it to be inviting to guests, they must heat
the castle. If one commercial heater can heat about 16,000 cubic feet, how many
heaters will the Society need to purchase? The Society has modeled the castle with
the following diagram:
8-29. Describe the solid formed by the net below. What are its dimensions (length,
width, and height)?
On graph paper, carefully draw the net above. Then cut out your net and build the
solid (so that the gridlines end up on the outside the solid) using scissors and tape.
Next, draw the net of a similar solid using the enlargement factor times 2 and times 3.
Then cut out these two nets and build the solids .
a. Find the volumes of your solids and compare it to the volume of the original solid.
What is the ratio of these volumes?
b. How does the volume change when a three-dimensional solid is enlarged or
reduced to create a similar solid? For example, if a solid’s length, width, and depth
are enlarged by a linear scale factor of 10, then how many times bigger does the
volume get? What if the solid is enlarged by a linear scale factor of r? Explain.
8-30. Examine the 1 × 1 × 3 solid below.
a. If this shape is enlarged by a linear scale factor of 2, how wide will the new shape
be? How tall? How deep?
b. What if the 1 × 1 × 3 solid is enlarged with a linear scale factor of 3? How many
times larger would the volume of the new solid be? Explain how you found your
answer.
8-31. The Mona Lisa, by Leonardo da Vinci, is arguably the most famous painting in
existence. The rectangular artwork, which hangs in the Musée du Louvre, measures
77 cm by 53 cm. When the museum created a billboard with an enlarged version of
the portrait for advertisement, they used a linear scale factor of 20. What was the area
of the billboard?
8-32. A big warehouse carrying tents has a miniature model that is similar to the fullsized tent. The tent is a triangular-based prism and the miniature model has
dimensions shown in the diagram below.
a. How much fabric does the small tent use? That is, what is its surface area?
b. What is the volume of the small model?
c. If the volume of the full-sized model is 72 ft3, how tall is the full-sized tent?
d. How much fabric does the full-sized tent use?
8-33. Draw a cylinder on your paper. Assume the radius of the cylinder is 6 inches
and the height is 9 inches.
a. What is the surface area of the cylinder? What is the volume?
b. If the cylinder is enlarged with a linear scale factor of 3, what is the volume of the
enlarged cylinder? How do you know?