The surface area is 36\pi\text{ in}^2 (about 113.1\text{ in}^2).
Check: each lid is \pi(2^2) = 4\pi; two lids = 8\pi; plus the label 28\pi gives 36\pi ✓.
Tip: A closed cylinder = one rectangle (label) + two circles. Count all three pieces.
Question 4 of 36
A concrete pillar is a cylinder with a radius of 3\text{ ft} and a height of 7\text{ ft}.
What is the volume of the pillar, in terms of \pi?
Hint
Use V = \pi r^2 h from the formula sheet. Square the radius before multiplying by the height.
Step-by-step solution
Formula: V = \pi r^2 h.
Square the radius: r^2 = 3^2 = 9.
Multiply by the height: 9 \times 7 = 63.
Keep \pi as a symbol: V = 63\pi.
The volume is 63\pi\text{ ft}^3.
Check:63\pi \approx 63 \times 3.14 \approx 198\text{ ft}^3. A box around the pillar would be 6 \times 6 \times 7 = 252\text{ ft}^3, and the cylinder should be a bit less ✓.
Tip: "In terms of \pi" means do all the arithmetic with the other numbers and leave \pi in the answer.
Question 5 of 36
Fill in
A coffee mug is shaped like the cylinder shown.
Using 3.14 for \pi, how many cubic centimeters can the mug hold?
\text{cm}^3
Type your answer in the box. Fractions like 3/4 are fine.
Hint
The label across the top is the diameter. What is the radius?
Step-by-step solution
The diameter is 8\text{ cm}, so the radius is r = 8 \div 2 = 4\text{ cm}.
r^2 = 4^2 = 16.
V = \pi r^2 h = 3.14 \times 16 \times 10.
16 \times 10 = 160, and 3.14 \times 160 = 502.4.
The mug holds 502.4\text{ cm}^3.
Check:160\pi with the π key is about 502.65, very close to 502.4 ✓.
Tip: Always look for the word or line that tells you whether a measure is a diameter or a radius. Using 8 as the radius would give 2{,}009.6 — four times too much.
Question 6 of 36
A closed cylindrical storage tank has a radius of 2\text{ m} and a height of 9\text{ m}.
What is the total surface area of the tank, in terms of \pi?
Hint
Total surface area = curved side + top + bottom: SA = 2\pi rh + 2\pi r^2.
Tip: To find a missing height, divide by the whole base area \pi r^2. On the calculator, put the base area in parentheses: 1570÷(3.14×5x²).
Question 8 of 36
Fill in
A cylindrical hot tub is 6\text{ ft} across (diameter) and is filled with water to a depth of 3\text{ ft}. One cubic foot of water is about 7.5 gallons.
To the nearest gallon, how many gallons of water are in the hot tub? Use 3.14 for \pi.
gallons
Type your answer in the box. Fractions like 3/4 are fine.
Hint
The water forms a cylinder. Find its radius from the diameter, compute V = \pi r^2 h, then convert cubic feet to gallons.
Step-by-step solution
Radius: r = 6 \div 2 = 3\text{ ft}. Height of the water: h = 3\text{ ft}.
Volume: V = \pi r^2 h = 3.14 \times 3^2 \times 3 = 3.14 \times 27 = 84.78\text{ ft}^3.
Gallons: 84.78 \times 7.5 = 635.85.
To the nearest gallon: 636 gallons.
Check: estimate \pi \approx 3: 3 \times 9 \times 3 = 81\text{ ft}^3, and 81 \times 7.5 \approx 608 gallons — close to 636 ✓. (With the π key you would get about 636.17, which also rounds to 636.)
Tip: Convert to gallons only at the end, after the volume in cubic feet is found. Multiply by 7.5 because each cubic foot holds 7.5 gallons.
Question 9 of 36
Drop-down
A company prints paper labels that wrap exactly once around the curved side of a can, with no overlap. Each can has a radius of 1.5\text{ in} and a height of 4.5\text{ in}. The labels do not cover the top or bottom.
Select the correct choice for each blank.
The size of one label is found with , so each label has an area of about (to the nearest tenth).
Choose an answer from each menu.
Hint
A label is the curved side only. Unrolled, it is a rectangle: its width is the distance around the can and its height is the height of the can.
Step-by-step solution
The label covers only the curved side, so use the lateral area 2\pi r h.
2\pi r h = 2\pi(1.5)(4.5) = 13.5\pi.
With the π key: 13.5\pi \approx 42.41, so about 42.4\text{ in}^2.
Check: the label is a rectangle 2\pi(1.5) \approx 9.42\text{ in} wide and 4.5\text{ in} tall: 9.42 \times 4.5 \approx 42.4 ✓.
Tip: Decide which parts are covered before choosing a formula: label = 2\pi r h; closed can = 2\pi r h + 2\pi r^2; contents = \pi r^2 h.
Question 10 of 36
A cylinder has a volume of 245\pi\text{ cm}^3 and a height of 5\text{ cm}. What is the radius of the cylinder?
Hint
Set \pi r^2 (5) = 245\pi. Divide both sides by 5\pi, and remember the last step to undo a square.
Step-by-step solution
V = \pi r^2 h, so 245\pi = \pi r^2 (5).
Divide both sides by 5\pi: r^2 = 245 \div 5 = 49.
Undo the square with a square root: r = \sqrt{49} = 7.
The radius is 7\text{ cm}.
Check:\pi(7^2)(5) = \pi(49)(5) = 245\pi ✓.
Tip: When the unknown is the radius, the last step is always a square root. Then reread the question: if it asks for the diameter, double the radius (here 14\text{ cm}).
Question 11 of 36
Select all
A closed cylinder has a radius of 3\text{ in} and a height of 8\text{ in}.
Select all statements that are true.
Select all that apply.
Hint
Compute \pi r^2 h, 2\pi r h, and 2\pi r h + 2\pi r^2 in terms of \pi, then judge each statement.
Step-by-step solution
Volume: V = \pi(3^2)(8) = \pi(9)(8) = 72\pi\text{ in}^3. A is true.
Curved (lateral) area: 2\pi r h = 2\pi(3)(8) = 48\pi\text{ in}^2. B is true.
Two ends: 2\pi r^2 = 2\pi(9) = 18\pi. Total: SA = 48\pi + 18\pi = 66\pi\text{ in}^2. C is true.
D (24\pi) forgets to square the radius: \pi(3)(8). False.
Check: with 3.14: 3.14 \times 75{,}000 = 235{,}500\text{ cm}^3 = 235.5 L, which also rounds to about 236 ✓.
Tip: Do the conversion as the last step and check its direction: there are 1{,}000 cubic centimeters in each liter, so the number of liters is smaller than the number of \text{cm}^3.
Question 15 of 36
Drop-down
A paint roller is a cylinder with a radius of 1\text{ in} and a length of 9\text{ in}. When it rolls one full turn, its curved surface touches the wall once.
Select the correct choice for each blank.
The area painted in one full turn is found with , and it is about .
Choose an answer from each menu.
Hint
Only the curved surface touches the wall. Unroll it in your mind: it becomes a rectangle.
Step-by-step solution
The curved surface unrolls into a rectangle. One side is the circumference 2\pi r; the other is the length h.
So the area painted in one turn is the lateral area 2\pi rh.
Tip: Dividing gives r^2, not r. The last step is always a square root. On the TI-30XS, use 2ndx² for \sqrt{\ }.
Question 17 of 36
Select all
A food company stores rice in closed cylindrical drums. Each drum is 4\text{ in} tall and measures 10\text{ in} straight across the top (its diameter). An inspector is checking the product sheet.
Select all statements about one drum that are true.
Select all that apply.
Hint
Find the radius first. Then compute V = \pi r^2 h, the curved area 2\pi rh and SA = 2\pi rh + 2\pi r^2.
Step-by-step solution
Radius: r = 10 \div 2 = 5\text{ in}. A is true.
Volume: V = \pi(5^2)(4) = 100\pi\text{ in}^3. B is true.
Total surface area: 40\pi + 50\pi = 90\pi\text{ in}^2. C is true.
D (400\pi) uses the diameter 10 as the radius. False.
E (20\pi) is \pi rh; the curved area is 2\pi rh = 40\pi. False.
The true statements are A, B and C.
Check:100\pi \approx 314\text{ in}^3, less than the box around it, 10 \times 10 \times 4 = 400\text{ in}^3 ✓.
Tip: In select-all items, find r, V, lateral area and SA once, then test each statement against your list.
Question 18 of 36
Drag and drop
A packaging designer sorts closed cylindrical containers by how much material it takes to make each one. Drag each container (measures in centimeters) into the column that shows its total surface area, in \text{cm}^2. One container will not be used.
SA = 24\pi
SA = 42\pi
SA = 56\pi
Tap a tile, then tap a box to place it.
Hint
For each cylinder compute 2\pi rh + 2\pi r^2. One cylinder has a volume that matches a column — don't be fooled.
Step-by-step solution
Use SA = 2\pi rh + 2\pi r^2:
r = 2, h = 4: 2\pi(2)(4) + 2\pi(4) = 16\pi + 8\pi = 24\pi → 24\pi.
r = 3, h = 4: 2\pi(3)(4) + 2\pi(9) = 24\pi + 18\pi = 42\pi → 42\pi.
r = 4, h = 3: 2\pi(4)(3) + 2\pi(16) = 24\pi + 32\pi = 56\pi → 56\pi.
r = 2, h = 6: 24\pi + 8\pi = 32\pi → not used (its volume, \pi \cdot 4 \cdot 6 = 24\pi, is the trap).
Check: cylinders 2 and 3 have the same curved area (24\pi) but different bases: 18\pi vs 32\pi ✓.
Tip: Swapping r and h changes the answer, because r is squared in the base areas.
Question 19 of 36
Fill in
A cylindrical water tank has a radius of 2\text{ ft} and a height of 5\text{ ft}. It is empty, and a pump fills it at 15 gallons per minute. One cubic foot holds about 7.5 gallons.
Using 3.14 for \pi, how many minutes will it take to fill the tank? Round to the nearest tenth.
minutes
Type your answer in the box. Fractions like 3/4 are fine.
Hint
Three steps: volume in cubic feet → gallons → minutes.
Check: in 31.4 minutes the pump delivers 31.4 \times 15 = 471 gallons, and 471 \div 7.5 = 62.8\text{ ft}^3 ✓.
Tip: Write the units next to every number. Cubic feet × (gallons per cubic foot) = gallons, and gallons ÷ (gallons per minute) = minutes. If the units don't cancel, the operation is wrong.
Question 20 of 36
A closed cylindrical tin has a radius of 4\text{ in} and a total surface area of 96\pi\text{ in}^2.
What is the height of the tin?
Hint
Start from SA = 2\pi rh + 2\pi r^2. Find the two base areas and subtract them first.
Tip: When everything is "in terms of \pi", the \pis cancel. Just work with the numbers in front of \pi: 96 - 32 = 64, then 64 \div 8 = 8.
Question 21 of 36
A bakery uses a round cake pan with a radius of 4\text{ in}. It buys a new pan with the same depth but a radius of 8\text{ in}.
The new pan holds how many times as much batter as the old pan?
Hint
The height stays the same, so compare r^2 for the two pans.
Step-by-step solution
Volume of each pan: V = \pi r^2 h, with the same h.
Old pan: r^2 = 4^2 = 16. New pan: r^2 = 8^2 = 64.
Compare: 64 \div 16 = 4.
The new pan holds 4 times as much batter.
Check: with a depth of 2\text{ in}: old = \pi(16)(2) = 32\pi, new = \pi(64)(2) = 128\pi; 128 \div 32 = 4 ✓.
Tip: Doubling the radius multiplies the volume by 2^2 = 4, because the radius is squared. Doubling the height only doubles the volume.
Question 22 of 36
Fill in
A section of drain pipe is a cylinder with an inside radius of 1.5\text{ ft} and a length of 8\text{ ft}.
What volume of water does the section hold when it is full? Give your answer in terms of \pi.
\pi\ \text{ft}^3
Type your answer in the box. Fractions like 3/4 are fine.
Hint
Use V = \pi r^2 h, where the length of the pipe plays the role of h. Square the decimal radius carefully.
Step-by-step solution
A pipe lying on its side is still a cylinder; its length is the "height" h = 8.
r^2 = 1.5^2 = 1.5 \times 1.5 = 2.25.
V = \pi \times 2.25 \times 8 = 18\pi.
The pipe holds 18\pi\text{ ft}^3 (about 56.5\text{ ft}^3).
Check:18 \div 8 = 2.25 = 1.5^2 ✓.
Tip:1.5^2 is 2.25, not 3 (that would be 1.5 \times 2). Squaring a decimal is a common place to slip — use the x² key if unsure.
Question 23 of 36
A round hay bale stands on one end, as shown.
To the nearest tenth, what is the volume of the bale? (Use the π key or \pi \approx 3.14.)
Hint
The 5\text{ ft} is measured all the way across. Halve it before using V = \pi r^2 h.
Step-by-step solution
Radius: r = 5 \div 2 = 2.5\text{ ft}.
r^2 = 2.5^2 = 6.25.
V = \pi \times 6.25 \times 4 = 25\pi.
25\pi \approx 78.54, so V \approx 78.5\text{ ft}^3 (with 3.14: 78.5 exactly).
The volume is about 78.5\text{ ft}^3.
Check: a box around the bale is 5 \times 5 \times 4 = 100\text{ ft}^3; a cylinder fills about 79\% of its box, and 78.5 is about 79\% of 100 ✓.
Tip: The "box check" is a fast way to catch a diameter mistake: if your cylinder volume is bigger than the box around it, you used the diameter as the radius.
Question 24 of 36
A cylindrical drink cooler has a radius of 6\text{ in} and a height of 15\text{ in}. It is completely full of lemonade. The lemonade is poured into cylindrical cups that each have a radius of 1.5\text{ in} and a height of 4\text{ in}.
How many cups can be filled completely?
Hint
Find the volume of the cooler and the volume of one cup, then divide. You can leave both in terms of \pi; it cancels.
Step-by-step solution
Cooler: V = \pi(6^2)(15) = \pi(36)(15) = 540\pi\text{ in}^3.
One cup: V = \pi(1.5^2)(4) = \pi(2.25)(4) = 9\pi\text{ in}^3.
Number of cups: 540\pi \div 9\pi = 540 \div 9 = 60.
The cooler fills 60 cups.
Check:60 \times 9\pi = 540\pi, exactly the cooler's volume ✓.
Tip: In "how many fit" problems, divide volume by volume. Comparing only the radii and heights (lengths) gives the wrong ratio, because the radius is squared.
Question 25 of 36
Fill in
A closed cylinder has a radius of 5\text{ cm} and a total surface area of 170\pi\text{ cm}^2.
What is the height of the cylinder, in centimeters?
h = cm
Type your answer in the box. Fractions like 3/4 are fine.
Hint
Write SA = 2\pi r h + 2\pi r^2 with r = 5. Find the area of the two circles first, subtract, then solve for h.
Step-by-step solution
SA = 2\pi r h + 2\pi r^2, with r = 5: 170\pi = 2\pi(5)h + 2\pi(25).
Tip: When every term contains \pi, you can divide it out at the start: 170 = 10h + 50. Remove the two lids first, then divide by 2\pi r.
Question 26 of 36
Fill in
A closed cylindrical oatmeal container has a radius of 6\text{ cm} and a height of 18\text{ cm}. A company wants to know how much cardboard is used for the whole container, including the top and bottom.
Using 3.14 for \pi, find the total surface area to the nearest square centimeter.
\text{cm}^2
Type your answer in the box. Fractions like 3/4 are fine.
Hint
SA = 2\pi rh + 2\pi r^2. Work in terms of \pi first, then multiply by 3.14 at the end.
Step-by-step solution
Curved side: 2\pi rh = 2\pi(6)(18) = 216\pi.
Two bases: 2\pi r^2 = 2\pi(36) = 72\pi.
Total: 216\pi + 72\pi = 288\pi.
288 \times 3.14 = 904.32 \approx 904.
The container uses about 904\text{ cm}^2 of cardboard.
Tip: Gallons are not cubic feet. Convert the capacity into the same units as the dimensions before using V = \pi r^2 h.
Question 28 of 36
Fill in
A heating-oil tank is a cylinder lying on its side. It has a diameter of 4\text{ ft} and a length of 6\text{ ft}. One cubic foot holds 7.5 gallons, and heating oil costs \$3.50 per gallon.
Using 3.14 for \pi, how much does it cost to fill the empty tank? Round to the nearest cent.
\$
Type your answer in the box. Fractions like 3/4 are fine.
Hint
A cylinder on its side still has V = \pi r^2 h; the length is h. Then go from cubic feet to gallons to dollars.
Tip: Think of the label as a rectangle: its width is the can's circumference. Dividing the area by the height gives the circumference (18.84); dividing that by 2\pi gives the radius.
Question 30 of 36
Fill in
A round backyard pond is a cylinder 2\text{ ft} deep. It holds 2{,}512\text{ ft}^3 of water when full.
Using 3.14 for \pi, what is the diameter of the pond, in feet?
ft
Type your answer in the box. Fractions like 3/4 are fine.
Hint
Find r^2 by dividing the volume by \pi h, take the square root to get r, then double it.
Step-by-step solution
V = \pi r^2 h \Rightarrow 2{,}512 = 3.14 \times r^2 \times 2.
So they hold the same amount, but B needs more material.
Check:32\pi = 32\pi ✓ and 48\pi > 40\pi ✓.
Tip: Equal volumes do not mean equal surface areas. Wide, short cylinders use more material for the same capacity because their two big circles add a lot of area.
Question 32 of 36
Drag and drop
Drag each cylinder into the column that shows its volume, rounded to the nearest cubic inch. One cylinder will not be used.
about 113\text{ in}^3
about 157\text{ in}^3
about 226\text{ in}^3
Tap a tile, then tap a box to place it.
Hint
Find each volume in terms of \pi first, then multiply by \pi. Check whether each cylinder gives a radius or a diameter.
Step-by-step solution
Use V = \pi r^2 h:
r = 3, h = 4: \pi(9)(4) = 36\pi \approx 113.1 → about 113.
r = 5, h = 2: \pi(25)(2) = 50\pi \approx 157.1 → about 157.
r = 3, h = 8: \pi(9)(8) = 72\pi \approx 226.2 → about 226.
d = 6, h = 2: r = 3, so \pi(9)(2) = 18\pi \approx 56.5 → not used.
Check: doubling the height of cylinder 1 gives cylinder 3, and 113 \times 2 \approx 226 ✓.
Tip: Read every label: "diameter 6" and "radius 6" are very different cylinders. Halve a diameter before squaring.
Question 33 of 36
The two coolers shown are both cylinders.
Which cooler holds more, and by about how many cubic inches? (Use the π key or \pi \approx 3.14.)
Hint
Both measurements across the tops are diameters. Find each radius, compute both volumes, then subtract.
Step-by-step solution
Cooler A: r = 6 \div 2 = 3, h = 10. V_A = \pi(3^2)(10) = 90\pi \approx 282.7\text{ in}^3.
Cooler B: r = 10 \div 2 = 5, h = 4. V_B = \pi(5^2)(4) = 100\pi \approx 314.2\text{ in}^3.
Tip: A wide, short cylinder can hold more than a tall, thin one because the radius is squared while the height is not. Never judge volume by height alone.
Question 34 of 36
A cylindrical pitcher with a radius of 4\text{ in} is filled with lemonade to a height of 9\text{ in}. All of the lemonade is poured into an empty cylindrical punch bowl with a radius of 6\text{ in}.
How deep is the lemonade in the punch bowl?
Hint
The volume of lemonade stays the same. Find it in terms of \pi, then solve \pi(6^2)h = that volume.
Step-by-step solution
Volume of lemonade: V = \pi(4^2)(9) = 144\pi\text{ in}^3.
In the bowl: \pi(6^2)h = 36\pi h.
Set equal: 36\pi h = 144\pi.
h = 144 \div 36 = 4.
The lemonade is 4\text{ in} deep.
Check:\pi(36)(4) = 144\pi ✓. The bowl is wider, so the liquid should be shallower than 9\text{ in} — and it is.
Tip: The depth changes by the ratio of the squares of the radii: 9 \times \left(\tfrac{4}{6}\right)^2 = 9 \times \tfrac{16}{36} = 4. Using the plain ratio \tfrac{4}{6} gives the wrong answer 6.
Question 35 of 36
Fill in
A soup company makes each closed can from 207.24\text{ cm}^2 of metal (the curved side plus the top and bottom). The can's radius is 3\text{ cm}.
Using 3.14 for \pi, what is the height of the can, in centimeters?
h = cm
Type your answer in the box. Fractions like 3/4 are fine.
Hint
Subtract the area of the two circles from the total; what remains is 2\pi rh.
Tip: Working backwards from a total surface area always has two stages: remove the bases, then divide by the circumference. Skipping the first stage gives 207.24 \div 18.84 = 11, which fails the check.
Question 36 of 36
A store sells oatmeal in two cylindrical cans.
Can A: diameter 4\text{ in}, height 6\text{ in}, price \$2.40.
Can B: diameter 6\text{ in}, height 4\text{ in}, price \$3.30.
Which can is the better buy (lower cost per cubic inch)?
Hint
Find each volume (remember to halve the diameters), then divide price by volume.
Step-by-step solution
Can A: r = 2, V = \pi(2^2)(6) = 24\pi \approx 75.40\text{ in}^3.
Can B: r = 3, V = \pi(3^2)(4) = 36\pi \approx 113.10\text{ in}^3.
Cost per cubic inch, A: \$2.40 \div 75.40 \approx \$0.0318.
Cost per cubic inch, B: \$3.30 \div 113.10 \approx \$0.0292.
Can B is the better buy — it costs less per cubic inch.
Check: B holds 36\pi \div 24\pi = 1.5 times as much as A. At A's rate, 1.5 times as much would cost 1.5 \times \$2.40 = \$3.60; B costs only \$3.30 ✓.
Tip: Same two numbers (4 and 6) do not give the same volume — the one that is the radius gets squared. Compare unit prices, not sticker prices.
