Problem
Volume of a solid of revolution
A 4 cm by 2 cm rectangle is spun twice, changing only the axis. In A the 4 cm side is the axis, in B the 2 cm side. Either way a cylinder is swept out. Find both volumes and say which is bigger.
Reasoning · Grade 6-2 Volume of Solid Figures
Your answer
Watch out The same rectangle suggests the same volume, but the radius is squared.
How to solve
Strategy Visualize Spatial Relationships — The whole problem is deciding which side of the rectangle becomes the radius and which becomes the height — once that is settled, the arithmetic is a one-line cylinder formula done twice. The safest way to settle it is physically: tape a paper rectangle to a pencil along one side and spin the pencil between your palms. The taped side stays put and becomes the height; the free side sweeps the circle and is the radius. Doing that for both pictures makes the two cylinders concrete before any number is written down.
1STEP 1
Spin A: name the radius and the height
A has radius 2 and height 4.
A: r = 2 cm, h = 4 cm
Whatever touches the axis cannot move, so it must be the line running up the middle of the can; the other side is the one that draws the circle.
8.G.C.9Visualize Spatial Relationships2STEP 2
Spin B: name the radius and the height
B has radius 4 and height 2.
B: r = 4 cm, h = 2 cm
It is the same rectangle both times, so the two numbers 4 and 2 simply trade jobs when the axis changes.
8.G.C.9Visualize Spatial Relationships3STEP 3
Volume of A
A holds 50.24 cm³.
2 × 2 × 3.14 × 4 = 12.56 × 4 = 50.24 cm³
Two centimeters times two centimeters gives a square measure, and one more centimeter of height makes it a cubic measure — the units confirm the setup is right.
8.G.C.9Analyze The Units4STEP 4
Volume of B
B holds 100.48 cm³.
4 × 4 × 3.14 × 2 = 50.24 × 2 = 100.48 cm³
Same formula, but the bigger number is now inside the squaring, and squaring rewards the bigger number much more than plain multiplying does.
8.G.C.9Analyze The Units5STEP 5
Compare
B is twice A.
50.24 ＜ 100.48, 100.48 ÷ 50.24 = 2
Because pi is a common factor, the comparison really only asks which of 2 x 2 x 4 and 4 x 4 x 2 is larger, and you can settle that without touching the decimals.
6.RP.A.3Draw A Diagram6STEP 6
Why the wide can wins
The squared radius makes the wide one win.
r² h: 2² × 4 = 16 vs 4² × 2 = 32
This is worth remembering as a rule: to get the biggest solid out of a spinning rectangle, put the short side on the axis.
8.G.C.9Visualize Spatial RelationshipsThe wide can wins, because widening the base counts twice over while raising the height counts only once.
Why?
The volume is the base area multiplied by the height, so the base enters through an area and the height through a single length.
🧱A prism is its base stacked upA prism is one flat face repeated all the way up, so its volume is the base area multiplied by the height.
Why?
Doubling the radius makes the base four times as large, while doubling the height only doubles the volume.
🧱Proportional quantities scale togetherWhen two quantities keep a fixed ratio, doubling one doubles the other, so a ratio you know can stand in for one you do not.
Answer
B
50.24, 100.48
Both results are in cubic centimeters, which is right for volume, and both are a few tens of cubic centimeters — about the size of a small medicine cup, which is believable for a 4 cm by 2 cm rectangle. The ratio is a strong check: the two volumes differ only in whether 4 is squared or 2 is squared, so B ought to be exactly 4 x 4 x 2 divided by 2 x 2 x 4 = 32/16 = 2 times A, and 100.48 really is exactly twice 50.24. It also fits common sense that the wide flat can is the roomier one: A is a can 4 cm long but only 4 cm across, while B is a can 8 cm across, and widening a circle counts twice.
Takeaway
Spin a rectangle about its short side and you get the bigger solid — the radius counts twice, the height only once.
- Spin A: name the radius and the height
- Spin B: name the radius and the height
- Volume of A
- Volume of B
- Compare
- Why the wide can wins