A rectangular prism and an oblique prism share the same 6 × 4 base and the same perpendicular height of 10 cm, but the oblique prism's sides are slanted. Which has greater volume?
AThe rectangular prism, because it is upright and its sides are not wasted on slant
BThe oblique prism, because its slant height is longer than 10 cm, adding more material
CThey have equal volumes, because they have equal cross-sectional areas at every height
DCannot be determined without knowing the exact slant angle
Cavalieri's principle: if two solids have the same height and identical cross-sectional area at every level, they have equal volumes — regardless of how the solid is tilted. Both prisms stack the same 24 cm² base area over the same 10 cm perpendicular height, so both have volume 240 cm³. The slant height is longer, but it only tells you how far the side travels, not how many layers of base area are stacked.
Question 2 Multiple Choice
A student calculates the volume of a cylinder with radius 3 and height 5 as π × 3 × 5 = 15π. What is wrong with this calculation?
AThe student should have used the diameter (6) instead of the radius
BThe student forgot to square the radius — πr²h requires r², not r, so the result has the wrong value and wrong units
CThe student should have multiplied by 2π to account for the full circumference
DThe formula is correct; the student just needs to add the correct units (cm³)
The base of a cylinder is a circle with area πr², not πr. Forgetting to square the radius gives πrh, which has units of length² (not length³) — dimensionally wrong for volume. With r = 3 and h = 5, the correct volume is π(3²)(5) = 45π, not 15π. This is three times larger — a significant error. Always check: volume must have cubic units, confirming you multiplied an area by a length.
Question 3 True / False
The formula V = Bh works for oblique (tilted) prisms as long as you use the slant height rather than the perpendicular height.
TTrue
FFalse
Answer: False
This is backwards: V = Bh requires the *perpendicular* height — the straight-up distance between the two bases. The slant height is longer and would give an inflated, incorrect volume. Cavalieri's principle explains why: volume is the sum of identical cross-sectional layers stacked vertically; 'height' in V = Bh counts how many vertical layers there are, not how long the slanted side runs.
Question 4 True / False
A pyramid has the same base and perpendicular height as a prism. The pyramid's volume is less than the prism's volume.
TTrue
FFalse
Answer: True
Yes — a pyramid's volume is exactly (1/3)Bh, compared to Bh for the corresponding prism. They share the same base area and height, but the pyramid tapers to a point, so its cross-sectional area shrinks as you move upward; the prism's cross-section stays constant. This one-third relationship holds for all pyramids relative to their matching prisms (and cones relative to cylinders), which Cavalieri's principle also helps justify.
Question 5 Short Answer
Why must you use perpendicular height — not slant height — when calculating the volume of an oblique prism or cylinder?
Think about your answer, then reveal below.
Model answer: Because V = Bh counts how many layers of base area are stacked to reach the solid's full height. 'Height' means the perpendicular distance between the two bases — how far up the layers are stacked. The slant height measures the distance along the tilted edge, which is longer, but it doesn't change the number of horizontal layers or their size. Using slant height would overcount the layers and give an incorrect (inflated) volume.
Cavalieri's principle formalizes this: volume depends on cross-sectional area at each horizontal level. When a prism is tilted, those cross-sections shift sideways but stay the same size — the tilting doesn't add or remove material. Only the perpendicular height captures the true number of identical layers being stacked.