Competition · AMC preparation · step 4 of 4
AMC 8 · 2008 · #21
Grade 8 geometry-3d
Pick an answer.
AMC 8 2008 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
The dashed oval looks fancy, but it is just the slanted view of a flat cut that goes straight through the cylinder's middle line (its axis). Tool #15 (Visualize / Symmetry) lets us see that this single slice splits the cylinder into two mirror-image halves of equal size. Once we believe that, the hard 3-D shape question shrinks to Tool #9 (Easier Related Problem): find the whole cylinder's volume and take half. No calculus, no fancy slicing.
Read the dimensions off the figure
The 8 cm across is the diameter, so r = 4 cm, and the label along the cylinder gives h = 6 cm.
Grade 7 geometry: pull the radius and the height straight off a labeled solid before plugging into a formula.
7.G.B.6Organize Information In More WaysSee the half-cylinder symmetry
The cut is a flat plane through the central axis, so reflection makes the two wedges congruent — each is exactly half the cylinder.
Grade 8 rigid-motion thinking: a reflection across the cutting plane sends one wedge onto the other, so their volumes match.
One wedge is exactly half of the whole cylinder's volume.
▸ Why?
The flat cut passes through the cylinder's center line, splitting it into just two wedges, and those two wedges have equal volume.
▸ Why?
Folding the cylinder over the cutting plane lays one wedge exactly onto the other, and a fold is a flip that never stretches or shrinks a shape, so the two wedges are the same size.
▸ Why?
That fold really does land the cylinder back on itself, because the round face looks the same on both sides of any line through its center — every point of it sits the same distance from that center.
▸ Why?
The two wedges together fill the whole cylinder with no gap and no overlap, so their volumes add back to the whole; two equal parts that build a whole are each half of it.
Compute the cylinder's volume
Cylinder volume is V = π r² h = π · 16 · 6 = 96π cm³.
Grade 8 "volume of a cylinder" is just area of the circle times the length.
8.G.C.9Solve An Easier Related ProblemHalve it and approximate
Halving gives the wedge: 48π ≈ 150.8 cm³, closest to choice (C).
Grade 7 "know the formulas for area and circumference of a circle" includes using π ≈ 3.14 for numerical estimates.
7.G.B.4Solve An Easier Related ProblemWhen a cut goes straight through the center of a round shape, both pieces are equal — so this AMC 8 problem is really just "half of π r² h".
- Read the dimensions off the figure
- See the half-cylinder symmetry
- Compute the cylinder's volume
- Halve it and approximate
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