Problem
Volumes of cylinders and cones
Two solids are drawn. Solid (1) is a cone sitting on a cylinder. Solid (2) is a cone with its top sliced off. Find the volume of each.
Reasoning · Grade 6-2 Volume of Solid Figures
Your answer
Watch out The printed height is tempting, but it may be the whole height — subtract.
How to solve
Strategy Identify Subproblems — Neither solid has a formula of its own, but each one is built out of two shapes that do. Solid (1) is a sum — cylinder plus cone — and solid (2) is a difference — big cone minus the small cone that was cut off. So in both cases I solve two easy problems I already know how to do and then add or subtract. The only real trap is reading the heights off the picture correctly, so before computing anything I mark on the diagram which segment is which height.
1STEP 1
(1) Read the two heights off the picture
The cone's height in (1) is 3 cm.
cone height = 6 - 3 = 3 cm
Stacked solids share their heights the way a stack of books shares its total height: the whole minus the bottom book gives the top one.
8.G.C.9Draw A Diagram2STEP 2
(1) Volume of the cylinder
The cylinder holds 144 cm³.
4 × 4 × 3 × 3 = 48 × 3 = 144 cm³
A cylinder is just a stack of identical circles, so its volume is one circle's area repeated for the whole height.
8.G.C.9Identify Subproblems3STEP 3
(1) Volume of the cone on top
The cone on top holds 48 cm³.
1/3 × 4 × 4 × 3 × 3 = 144/3 = 48 cm³
Filling a cone-shaped cup with water and pouring it into a can of the same base and height takes exactly three cupfuls — that is the one-third rule, and it is something you can actually try.
8.G.C.9Solve An Easier Related ProblemThe cone on top holds exactly one third of what a cylinder of the same base and height would hold.
Why?
Filling a cone-shaped cup and pouring it into a matching cylinder takes exactly three cupfuls, a rule you can test with water.
🧱A cone is one third of its cylinderFill a cone-shaped cup and pour it into a cylinder of the same base and height: it takes exactly three cupfuls.
Why?
That matching cylinder is its base circle repeated all the way up, so its volume is the base area multiplied by the height.
🧱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.
4STEP 4
(1) Add the two pieces
Together that is 192 cm³.
144 + 48 = 192 cm³
The two pieces only touch along a flat face, so no space is counted twice and plain addition finishes the job.
8.G.C.9Identify Subproblems5STEP 5
(2) Rebuild the whole cone
Rebuilding (2) gives a cone 8 cm tall.
big cone height = 4 + 4 = 8 cm
Putting the cut-off tip back on is easier to picture than the odd shape that is left, and it turns an unfamiliar solid into a plain cone.
8.G.C.9Visualize Spatial Relationships6STEP 6
(2) Volume of the big cone
The big cone holds 288 cm³.
1/3 × 6 × 6 × 3 × 8 = 864/3 = 288 cm³
Same one-third rule as before, just with bigger numbers; the slant edge of 5 cm is not needed here, and noticing which measurement is irrelevant is part of the work.
8.G.C.9Identify Subproblems7STEP 7
(2) Volume of the small cone that was cut off
The removed cone holds 36 cm³.
1/3 × 3 × 3 × 3 × 4 = 108/3 = 36 cm³
The removed tip is a cone in its own right, so the very same formula applies to it.
8.G.C.9Identify Subproblems8STEP 8
(2) Subtract
Subtracting gives 252 cm³.
288 - 36 = 252 cm³
Cutting a piece off means subtracting its volume — the same idea as taking a bite out of a block.
8.G.C.9Identify SubproblemsAnswer
192, 252 cm³
144 + 48 = 192, 288 − 36 = 252
Both answers are in cubic centimeters, which is right for volumes. For (1), the cone sits on the same base as the cylinder and is the same 3 cm tall, so it should hold exactly one third as much: 48 is one third of 144, and 192 is four thirds of 144, which is sensible. For (2), the answer must be smaller than the big cone's 288 and much bigger than the removed tip's 36, and 252 sits exactly there. A sharper check: the small cone is half the big one in every direction, so it should hold one eighth as much, and 288 divided by 8 really is 36. That means the leftover piece keeps seven eighths of the big cone, 288 x 7 / 8 = 252 — the same number a second way.
Takeaway
Any solid built from a cone and a cylinder is just add-or-subtract — and a cone always holds one third of its can.
- (1) Read the two heights off the picture
- (1) Volume of the cylinder
- (1) Volume of the cone on top
- (1) Add the two pieces
- (2) Rebuild the whole cone
- (2) Volume of the big cone
- (2) Volume of the small cone that was cut off
- (2) Subtract
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