This final question reinforces the core principle behind the sphere volume derivation. Cavalieri's principle states that if two solids have equal cross-sectional areas when sliced by parallel planes at every height, then they have equal volumes. For the sphere and cylinder-minus-cones comparison, at height h the areas are both π(r²-h²), confirming equal volumes. Option A correctly states this principle. Option B incorrectly focuses on circumferences rather than areas, C wrongly emphasizes endpoint alignment, and D avoids the geometric reasoning entirely. The key misconception to address is confusing other measurements (like perimeter) with area - Cavalieri's principle specifically requires equal cross-sectional areas at all heights for the volume conclusion.

At h = 0 (the bottom), the sphere's cross-section has area 0 (since we're at the bottom point), but the cylinder has area πr² and the cone contributes 0, so cylinder-minus-cone has area πr². Since 0 ≠ πr², the areas don't match from the very beginning. Choice B suggests a specific intermediate point but the failure occurs immediately. Choice C is incorrect because some setups could work with modifications. Choice D focuses on the wrong issue - the mismatch occurs much earlier.

Cavalieri's principle tells us that if cross-sections match, then volumes are equal, but it doesn't give us the actual volume value. The student must compute the volume of the cylinder (πr² × 2r = 2πr³) minus the volume of two cones (2 × ⅓πr² × r = ⅔πr³), which gives 2πr³ - ⅔πr³ = ⁴⁄₃πr³. Choice A describes a condition that's already assumed. Choice C is incorrect since the heights already match by construction. Choice D describes a mathematical technicality not essential to the basic application.

Cavalieri's principle states that if two three-dimensional solids have equal cross-sectional areas at every height, then they have equal volumes. To derive the sphere volume formula, you need a comparison solid whose cross-sections you can easily calculate and that match those of a sphere at every level.

The correct approach uses a cylinder with two cones removed because this creates cross-sections that are annuli (rings) whose areas exactly match the circular cross-sections of a sphere at corresponding heights. When you slice a sphere horizontally at height $h$ from the center, you get a circle with area $\pi(r^2 - h^2)$. The cylinder-minus-cones solid produces the same cross-sectional area at that height.

However, if you use a cylinder with two hemispheres removed, the cross-sectional areas will never match those of the sphere. A hemisphere has curved surfaces that create circular cross-sections of varying sizes, but these don't align with the sphere's cross-sections in the way needed for Cavalieri's principle. The geometric properties of hemispheres simply don't produce the required annular areas that match the sphere's circular cross-sections.

Choice A is wrong because orientation isn't the issue here—both solids would be sliced the same way. Choice B is incorrect because negative volume doesn't invalidate Cavalieri's principle; the method depends on cross-sectional area equality, not volume signs. Choice C misses the point entirely—the solids aren't identical, and the cross-sections don't match.

Study tip: For Cavalieri problems, always verify that your comparison solid produces cross-sections with calculable areas that genuinely match the original solid at every level.

At distance d from the center, the cross-section of the hemisphere is a circle whose radius is √(R² - d²) by the Pythagorean theorem. The area is therefore π(√(R² - d²))² = π(R² - d²). Choice B incorrectly assumes linear decrease. Choice C has the right numbers but wrong grouping of the π terms. Choice D gives the radius (without squaring) rather than the area.

The skill is deriving the volume of a sphere using an informal argument based on Cavalieri’s principle. The comparison involves a sphere of radius r and a cylinder of radius r and height 2r from which two congruent cones, each of radius r and height r, have been removed, with the cones' tips meeting at the cylinder's center. For any height h from the center, the cross-sectional area of the sphere equals the cross-sectional area of the cylinder minus the cross-sectional areas of the cones at that height, where typically only one cone contributes a non-zero area. By Cavalieri’s principle, solids with the same height and identical cross-sectional areas at every corresponding level have the same volume. This equality of cross-sections at every h establishes that the volume of the sphere equals the volume of the cylinder minus the volumes of the two cones. A distractor misconception is believing that matching cross-sectional perimeters at one height is essential, but the principle requires area matching at every height. When applying this strategy to other volume derivations, always compare cross-sections at the same height from a consistent reference point, such as the center for symmetric solids.

This question asks which claim is NOT supported by the Cavalieri slicing argument for sphere volume. The valid approach shows that at each height y, the sphere's cross-sectional area π(R² - y²) equals the cylinder's area πR² minus the two cones' areas 2πy². This equality at every height allows Cavalieri's principle to conclude equal volumes. Option A correctly states this principle, option B properly describes the volume relationship, and option D correctly emphasizes matching slices at the same height. However, option C incorrectly claims that equal surface area implies equal volume by Cavalieri's principle - this is false, as Cavalieri's principle concerns cross-sectional areas, not surface areas. The sphere and cylinder-minus-cones have equal volumes but different surface areas.

Using Cavalieri's principle, at height h above the center, a sphere of radius r has a circular cross-section with radius √(r² - h²), giving area π(r² - h²). The cylinder has cross-sectional area πr², and each cone contributes πh², so the cylinder minus two cones has area πr² - 2πh². For this to equal π(r² - h²), we need πr² - 2πh² = π(r² - h²), which simplifies correctly. Choice A has incorrect parentheses placement. Choice C uses addition instead of subtraction. Choice D uses an incorrect formula (r + h)².

The skill is deriving the volume of a sphere using Cavalieri’s principle. We compare the sphere of radius R to the solid formed by a cylinder of radius R and height 2R minus two congruent cones, each with apex at the center and base at the end of the cylinder. At every height h from the center, the cross-sectional area of the sphere is equal to that of the cylinder minus the relevant cone's cross-sectional area in that half. Since the cross-sectional areas are equal at every corresponding height, Cavalieri’s principle states that the volumes are equal. Therefore, the sphere's volume equals the volume of the cylinder minus the volumes of the two cones. A common misconception is that matching cross-sections at just one height suffices for equal volumes, but Cavalieri’s principle requires matching at every height. To apply this strategy to other solids, always compare cross-sectional areas at the same corresponding heights.

This question addresses the critical detail of consistent height measurement in Cavalieri's principle. For the sphere-to-cylinder-minus-cones comparison to work, slices must be taken at the same height h measured from the same reference point - the center of both solids. At height h from center, the sphere has cross-sectional area π(r²-h²), and the cylinder minus cones has area πr² - 2πh² = π(r²-h²). Option A correctly identifies measuring from the center for both solids. Options B and C incorrectly mix reference points between solids, which would yield different cross-sectional areas. Option D wrongly claims the reference doesn't matter, but consistent measurement is essential for Cavalieri's principle. The key insight is that corresponding slices must be at the same signed distance from the same reference level.