Problem
Volumes of assorted solids
A lying cylinder is sliced off at one end by a slanted plane. The circular end has diameter 2 cm; the other end is a slanted oval. The side edges measure 3 cm on top and 5 cm below. Find the volume of what is left.
Reasoning · Grade 6-2 Volume of Solid Figures
Your answer
Watch out Using one edge as the height is wrong: take the average of the two.
How to solve
Strategy Create a Physical Representation — There is no formula for a cylinder with a slanted top, so instead of attacking this solid directly I make a second copy of it, turn that copy upside down, and fit the two slanted faces together. Two identical pieces snap into one ordinary cylinder whose height is 3 + 5, and a plain cylinder is something I can measure. Then the solid I want is exactly half of it. This is the same trick as cutting a sandwich diagonally: the two pieces are the same, so each is half the whole.
1STEP 1
Get the radius from the figure
The radius reads as 1 cm.
r = 2 ÷ 2 = 1 cm
A radius runs from the center dot to the edge, but this brace starts at the top edge and ends at the bottom edge, passing the center on the way — that is a diameter.
7.G.B.4Draw A Diagram2STEP 2
Make a second copy and flip it over
Make a copy and flip it over.
Turning a shape over never changes its size, so the flipped copy still matches its partner face for face — this is the same reason two slices of a diagonally cut sandwich fit back together.
8.G.A.1Create A Physical Representation3STEP 3
See that the two copies make a whole cylinder
Together they make a cylinder 8 cm long.
3 + 5 = 8 cm
Wherever one copy is short the other is long by exactly the same amount, so every side of the joined solid comes out the same length, which is what makes it a straight cylinder.
8.G.C.9Visualize Spatial RelationshipsA second copy turned upside down fits against the first to make one complete cylinder.
Why?
Turning the copy over keeps its volume, so the two pieces together hold exactly twice what one of them holds.
🧱Rigid motion preserves length and angleSliding, turning, or flipping a shape lays it exactly onto a copy, so every length and angle stays the same (folding is the flip case).
Why?
The slanted top gives away as much above the middle as it takes below, so the solid holds what a cylinder of the average height would.
🧱The mean is the total shared out equallyThe average is the total divided by how many there are, so the total is the average times the count — and everything above that level gives away exactly what everything below it needs.
4STEP 4
Volume of that whole cylinder
That cylinder holds 24 cm³.
1 × 1 × 3 × 8 = 3 × 8 = 24 cm³
The awkward solid has been traded for a plain can, and a can's volume is just the circle at its end repeated all the way along.
8.G.C.9Change Focus Count The Complement5STEP 5
Take half
Halving gives 12 cm³.
24 ÷ 2 = 12 cm³
Two equal pieces make a whole, so one piece is half — no other information is needed.
8.G.C.9Create A Physical Representation6STEP 6
The same thing as an average height
The average height also gives 12 cm³.
1 × 1 × 3 × (3+5)/2 = 3 × 4 = 12 cm³
The slanted top takes away as much on the short side as it leaves extra on the long side, so the height at the middle is what really counts.
8.G.C.9Change Focus Count The ComplementAnswer
12 cm³
24 ÷ 2 = 12
The answer is in cubic centimeters, which is right for a volume. It also has to sit between the two cylinders that bracket the solid: a cylinder of radius 1 cm and height 3 cm holds 3 x 1 x 1 x 3 = 9 cubic centimeters and one of height 5 cm holds 15 cubic centimeters, and the slanted solid fits snugly between those two, so its volume must be between 9 and 15. The answer 12 lies right in the middle, exactly where a straight slanted cut should put it. It is also worth re-checking the radius: with the 2 cm misread as a radius the answer would come out 48, four times too big, and 48 would fall far outside the 9-to-15 bracket, which is a quick way to catch that mistake.
Takeaway
Two copies of a slant-cut can stack into one straight can — so the slanted piece is just half of it.
- Get the radius from the figure
- Make a second copy and flip it over
- See that the two copies make a whole cylinder
- Volume of that whole cylinder
- Take half
- The same thing as an average height
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