Geometry Quiz: Informal Argument For Volume Of Sphere
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Informal Argument For Volume Of SphereQuestion 1 of 20

A class is discussing why the volume of a sphere of radius rr can be justified without calculus by comparing it to a cylinder of radius rr and height 2r2r with two congruent cones removed (each cone has base radius rr and height rr). Which statement justifies the sphere's volume formula?

If the cross-sectional areas are equal for every height, then the volumes are equal by Cavalieri's principle.
If the cross-sectional circumferences are equal for every height, then the volumes are equal by Cavalieri's principle.
If the solids share the same top and bottom points, then the volumes are equal by Cavalieri's principle.
If the formula for the sphere is memorized, then the comparison is automatically justified.
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Geometry Quiz: Informal Argument For Volume Of Sphere

Practice Informal Argument For Volume Of Sphere in Geometry with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

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This quiz focuses on Informal Argument For Volume Of Sphere, giving you a quick way to practice the rules, question types, and explanations that matter most for Geometry.

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Question 1

A class is discussing why the volume of a sphere of radius rr can be justified without calculus by comparing it to a cylinder of radius rr and height 2r2r with two congruent cones removed (each cone has base radius rr and height rr). Which statement justifies the sphere's volume formula?

  1. If the cross-sectional areas are equal for every height, then the volumes are equal by Cavalieri's principle. (correct answer)
  2. If the cross-sectional circumferences are equal for every height, then the volumes are equal by Cavalieri's principle.
  3. If the solids share the same top and bottom points, then the volumes are equal by Cavalieri's principle.
  4. If the formula for the sphere is memorized, then the comparison is automatically justified.
Explanation: 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.

Question 2

In using Cavalieri's principle to find the volume of a sphere, a student sets up the comparison incorrectly and uses a cylinder of radius rr and height rr (instead of height 2r2r) with one cone removed. At what height hh from the bottom would the cross-sectional areas first fail to match?

  1. The areas fail to match immediately at h=0h = 0 because the sphere's base area differs from the cylinder's base area (correct answer)
  2. The areas fail to match at h=r2h = \frac{r}{2} because this is where the geometric relationships break down completely
  3. The areas never match at any height because the fundamental setup violates the conditions needed for Cavalieri's principle
  4. The areas fail to match at h=rh = r because the cylinder ends while the sphere continues beyond this height
Explanation: 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.

Question 3

A student applies Cavalieri's principle to find the volume of a sphere by comparing it to a cylinder with two cones removed. The student claims that since the cross-sections match at every height, the volume formula V=43πr3V = \frac{4}{3}\pi r^3 follows immediately. What additional step is actually required?

  1. The student must verify that the principle applies by checking that all cross-sections are perpendicular to the same axis
  2. The student must calculate the volume of the cylinder-minus-cones combination to determine the sphere's volume explicitly (correct answer)
  3. The student must prove that the sphere and comparison solid have the same height before applying the principle
  4. The student must demonstrate that the cross-sectional areas are continuous functions of height throughout the solids
Explanation: 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.

Question 4

When using Cavalieri's principle to argue for the sphere volume formula, a student makes an error in setting up the comparison solid. Instead of using a cylinder with two cones removed, the student uses a cylinder with two hemispheres removed. What is the fundamental problem with this approach?

  1. The comparison solid has variable cross-sections that depend on orientation, violating the requirements for Cavalieri's principle
  2. The resulting comparison solid has negative volume, which makes Cavalieri's principle inapplicable to this configuration
  3. The cross-sectional areas match perfectly, but the comparison solid is identical to the original sphere, providing no new information
  4. The cross-sectional areas will never match because hemispheres and spheres have different geometric properties than cones (correct answer)
Explanation: 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 hh from the center, you get a circle with area π(r2−h2)\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.

Question 5

Consider the informal argument for sphere volume using Cavalieri's principle with a hemisphere, cylinder, and cone. If the hemisphere has radius RR, and we slice at distance dd from the center (where 0≤d≤R0 \leq d \leq R), which expression correctly represents the cross-sectional area of the hemisphere?

  1. π(R2−d2)\pi(R^2 - d^2) because the cross-section is a circle with radius determined by the Pythagorean theorem (correct answer)
  2. π(R−d)2\pi(R - d)^2 because the cross-section radius decreases linearly from the center to the edge
  3. πR2−πd2\pi R^2 - \pi d^2 because we subtract the area lost due to the distance from center
  4. πR2−d2\pi \sqrt{R^2 - d^2} because the cross-section radius involves the square root from Pythagorean theorem
Explanation: 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.

Question 6

A student wants to apply Cavalieri's principle to compare a sphere of radius rr with a cylinder of radius rr and height 2r2r minus two cones (each cone radius rr, height rr). Which step is essential for the argument?

  1. Show that the solids have the same total surface area.
  2. Show that cross-sectional areas match for every height measured from the same reference level. (correct answer)
  3. Show that the cross-sectional perimeters match for one carefully chosen height.
  4. State the sphere volume formula and conclude the comparison must work.
Explanation: 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.

Question 7

A sphere of radius RR is compared to a solid formed by taking a cylinder of radius RR and height 2R2R and removing two congruent cones (each cone has base radius RR and height RR). Slices are taken by horizontal planes at equal heights yy from the center. Which claim is NOT supported by the slicing argument?

  1. If the cross-sectional areas match for every yy between −R-R and RR, then the volumes are equal.
  2. The sphere's volume equals the cylinder's volume minus the volumes of the two cones if the slice areas match at each height.
  3. If the solids have the same surface area, then their volumes must be equal by Cavalieri's principle. (correct answer)
  4. Matching slice areas must be done at the same height yy measured from the same reference plane.
Explanation: 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.

Question 8

A student uses Cavalieri's principle to find the volume of a sphere by comparing cross-sections of the sphere to cross-sections of a cylinder with two cones removed. If the sphere has radius rr, and the cylinder has radius rr and height 2r2r, what must be true about the areas of corresponding cross-sections at height hh above the center for this method to work?

  1. The sphere's cross-sectional area equals πr2−πh2\pi r^2 - \pi h^2, which matches the cylinder minus cones at the same height
  2. The sphere's cross-sectional area equals π(r2−h2)\pi(r^2 - h^2), which matches the cylinder minus cones at the same height (correct answer)
  3. The sphere's cross-sectional area equals πr2+πh2\pi r^2 + \pi h^2, which matches the cylinder minus cones at the same height
  4. The sphere's cross-sectional area equals π(r+h)2\pi(r + h)^2, which matches the cylinder minus cones at the same height
Explanation: 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.

Question 9

A slicing argument compares a sphere of radius RR with a cylinder of radius RR and height 2R2R minus two congruent cones inside the cylinder. Slices are taken by horizontal planes at height hh from the center of the sphere, and the same height is used on the cylinder-and-cones solid.

Which reasoning matches the cross-sections described?

  1. It is enough that the cross-sectional areas match at h=0h=0 to conclude the volumes are equal.
  2. If the cross-sectional areas match for every hh between −R-R and RR, then the volumes are equal by Cavalieri's principle. (correct answer)
  3. If the perimeters of the cross-sections match for every hh, then the volumes are equal by Cavalieri's principle.
  4. If the cross-sections match but the heights are measured from different starting points, Cavalieri's principle still guarantees equal volumes.
Explanation: 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.

Question 10

In a Cavalieri comparison, a sphere of radius rr is matched to a cylinder of radius rr and height 2r2r with two congruent cones removed (each cone has base radius rr and height rr). Which comparison uses the correct height reference for the slices?

  1. Measure slice height hh from the center for both solids, and compare cross-sectional areas at that same hh. (correct answer)
  2. Measure slice height from the top of the sphere but from the center of the cylinder-minus-cones, and compare areas.
  3. Measure slice height from the bottom of the sphere but from the top of the cylinder-minus-cones, and compare areas.
  4. Measure slice height from any convenient point; the reference level does not matter for Cavalieri's principle.
Explanation: 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.

Question 11

In the cylinder-minus-two-cones model, the remaining cross-section at height hh is described as "a disk from the cylinder with two smaller disks removed (one from each cone)." Which reasoning correctly connects that description to the sphere using Cavalieri's principle?

  1. At each height hh, subtract the two cone slice areas from the cylinder slice area and match the result to the sphere slice area at the same hh. (correct answer)
  2. At each height hh, add the two cone slice areas to the cylinder slice area and match the result to the sphere slice area at the same hh.
  3. At one height hh, subtract the two cone slice areas from the cylinder slice area; that single match proves equal volumes.
  4. Use the sphere volume formula first, then choose cones so the subtraction works out numerically.
Explanation: This question tests understanding of how cross-sectional areas combine when applying Cavalieri's principle to the sphere volume derivation. At height h, the cylinder contributes a disk of area πr², while each cone contributes a disk of area π(h²) at that height. The remaining cross-section after removing both cone contributions is πr² - 2·π(h²/2) = πr² - πh² = π(r²-h²), which matches the sphere's cross-section at height h. This matching must occur at every height h, not just one as option C suggests. Option B incorrectly adds instead of subtracts, while D reverses the logical process by starting with the formula. The common error is thinking one matching cross-section proves volume equality. The transfer strategy emphasizes subtracting the cone areas from the cylinder area at each height to match the sphere's cross-sectional area.

Question 12

A student claims: "A sphere of radius rr has the same volume as a cylinder of radius rr and height 2r2r with two cones removed, because the remaining solid looks about the same as the sphere." The intended justification is to use Cavalieri's principle by comparing cross-sections at equal heights.

Which argument explains why the sphere has this volume?

  1. Match each horizontal slice of the sphere to the cylinder slice minus the two cone slices at the same height, so equal slice areas imply equal volumes. (correct answer)
  2. Compare the curved surface areas of the solids; equal surface areas imply equal volumes.
  3. Check only the middle slice; if the areas match there, then the volumes must match.
  4. Use the sphere volume formula directly; since it is known, that is the justification.
Explanation: The skill involves deriving the volume of a sphere using Cavalieri's principle through an informal argument. We compare the sphere of radius r to the solid formed by a cylinder of radius r and height 2r with two congruent cones removed, each with base radius r and height r. At any height h from the center, the cross-sectional area of the sphere equals the cylinder's cross-sectional area minus the cross-sectional areas of the two cones' slices. Cavalieri's principle states that solids of equal height with matching cross-sectional areas at every corresponding height have equal volumes. Therefore, the sphere's volume equals the volume of the cylinder minus the volumes of the two cones. A misconception is equating volumes based on similar curved surface areas, but Cavalieri focuses on cross-sectional areas, not surfaces. To apply this strategy more broadly, always compare slices at the same height from a common reference point.

Question 13

In the pictured comparison, a sphere of radius RR is aligned with a cylinder of radius RR and height 2R2R, and two congruent cones are positioned so their bases lie in the cylinder's midplane. Cross-sections are taken by horizontal planes at the same height yy in all solids. Which statement justifies the sphere's volume formula?

  1. At each height yy, the area of the sphere's cross-section equals the area of the cylinder's cross-section minus the combined areas of the two cone cross-sections. (correct answer)
  2. At height y=0y=0 only, the areas match, so the volumes must match.
  3. Because the sphere is symmetric, its volume must be half the cylinder's volume.
  4. Because the cylinder has constant cross-sectional area, it must have the same volume as the sphere.
Explanation: This problem tests understanding of how cross-sectional area comparison justifies the sphere volume formula via Cavalieri's principle. The setup aligns a sphere of radius R with a cylinder (radius R, height 2R) containing two cones whose bases meet at the midplane. At height y, the sphere's cross-section is a circle with area π(R² - y²), the cylinder's cross-section has constant area πR², and each cone's cross-section has area πy². The crucial observation is that π(R² - y²) = πR² - 2πy², meaning the sphere's area equals the cylinder's area minus the two cones' areas at every height. By Cavalieri's principle, this equality of cross-sectional areas implies equality of volumes. Option B incorrectly claims matching areas at just one height suffices, while option C wrongly assumes the sphere has half the cylinder's volume.

Question 14

A class is proving the sphere volume formula without calculus by comparing cross-sections. They use a sphere of radius rr and a cylinder of radius rr and height 2r2r with two cones removed. They agree to compare horizontal slices at equal heights.

Which argument explains why the sphere has this volume?

  1. If two solids have equal cross-sectional areas at every height, then Cavalieri's principle implies their volumes are equal. (correct answer)
  2. If two solids have equal surface areas, then Cavalieri's principle implies their volumes are equal.
  3. If two solids share the same radius, then Cavalieri's principle implies their volumes are equal.
  4. If two solids look similar in a diagram, then Cavalieri's principle implies their volumes are equal.
Explanation: The skill involves deriving the volume of a sphere using Cavalieri's principle through an informal argument. We compare the sphere of radius r to the solid formed by a cylinder of radius r and height 2r with two congruent cones removed, each with base radius r and height r. At any height h from the center, the cross-sectional area of the sphere equals the cylinder's cross-sectional area minus the cross-sectional areas of the two cones' slices. Cavalieri's principle states that solids of equal height with matching cross-sectional areas at every corresponding height have equal volumes. Therefore, the sphere's volume equals the volume of the cylinder minus the volumes of the two cones. A misconception is assuming equal surface areas imply equal volumes via Cavalieri, but the principle uses cross-sectional areas, not surfaces. To apply this strategy more broadly, always compare slices at the same height from a common reference point.

Question 15

A sphere of radius rr is compared to a cylinder of radius rr and height 2r2r with two cones removed (each cone has radius rr and height rr). Consider slicing all solids with planes perpendicular to the cylinder's axis. Which comparison uses Cavalieri's principle correctly to conclude the sphere's volume equals the cylinder-minus-cones volume?​

  1. Match the cross-sectional areas at every height; equal areas for all slices imply equal volumes. (correct answer)
  2. Match one cross-section at the equator; one equal slice implies the volumes are equal.
  3. Compare slices taken from the sphere's center but from the cylinder's top; the heights do not need to align.
  4. Use the fact that the solids look balanced in a picture to conclude the volumes are equal.
Explanation: This question tests understanding of how to correctly apply Cavalieri's principle when comparing a sphere to a cylinder with cones removed. The comparison involves a sphere of radius r and a cylinder (radius r, height 2r) with two cones removed, where all slices are taken perpendicular to the cylinder's axis. Cavalieri's principle states that if two solids have equal cross-sectional areas at every height, then they have equal volumes. Option A correctly states this requirement: we must match cross-sectional areas at every height, not just at one height. Option B incorrectly suggests that matching one cross-section (at the equator) is sufficient, which violates Cavalieri's principle. Option C incorrectly proposes comparing slices at different heights in the two solids, which makes no sense for area comparison. When applying Cavalieri's principle, always ensure you're comparing cross-sections at corresponding heights in both solids.

Question 16

A student tries to justify the sphere volume formula by comparing a sphere of radius rr to a cylinder of radius rr and height 2r2r with two congruent cones removed (each cone has base radius rr and height rr). Which argument explains why the sphere has this volume?

  1. Because the solids have equal heights and equal radii, they must have equal volumes.
  2. Because the cross-sectional areas match at every height, the sphere and the cylinder-minus-cones have equal volumes. (correct answer)
  3. Because the sphere has the same diameter as the cylinder, the volumes must be equal after removing cones.
  4. Because 43πr3\frac{4}{3}\pi r^3 is the sphere's formula, the comparison solid must have that volume too.
Explanation: This problem tests understanding of the logical justification for the sphere volume formula using Cavalieri's principle. The student compares a sphere to a cylinder with two cones removed. At each height h from the center, the sphere's cross-sectional area π(r²-h²) equals the cylinder's area πr² minus the two cones' combined area 2πh², giving π(r²-h²). Since these areas match at every height, Cavalieri's principle guarantees equal volumes. Option B correctly states this reasoning, while A incorrectly assumes equal dimensions imply equal volumes, C wrongly focuses on diameter equality, and D circularly assumes the formula. The misconception to avoid is thinking geometric similarity alone determines volume - the key is verifying equal cross-sectional areas throughout.

Question 17

When applying Cavalieri's principle to derive the sphere volume formula, a key step involves showing that cross-sections at the same height have equal areas. If this principle is applied to a hemisphere of radius 6 and a cylinder of radius 6 and height 6 with a cone removed, what is the ratio of their volumes?

  1. The ratio is 23\frac{2}{3} because the hemisphere volume is 23\frac{2}{3} of the cylinder-minus-cone volume
  2. The ratio is 34\frac{3}{4} because the hemisphere volume is 34\frac{3}{4} of the cylinder-minus-cone volume
  3. The ratio is 11 because Cavalieri's principle guarantees that solids with equal cross-sections have equal volumes (correct answer)
  4. The ratio is 43\frac{4}{3} because the hemisphere volume is 43\frac{4}{3} of the cylinder-minus-cone volume
Explanation: Cavalieri's principle states that if two solids have the same height and their cross-sections at every level have equal areas, then the solids have equal volumes. Since the construction is specifically designed so that corresponding cross-sections of the hemisphere and the cylinder-minus-cone have equal areas at every height, their volumes must be equal, giving a ratio of 1. The other choices give incorrect ratios that would contradict Cavalieri's principle.

Question 18

A student tries to use Cavalieri's principle to compare a sphere (radius rr) with a cylinder (radius rr, height 2r2r) minus two cones. The student takes a slice in the sphere at height hh above the center, but takes slices in the cylinder and cones at height hh above the bottom of the cylinder.

Which statement justifies the sphere's volume formula correctly?

  1. The comparison is valid because both solids have total height 2r2r, so the choice of reference point does not matter.
  2. The comparison is valid if the slice heights are measured from the same reference plane in all solids, so corresponding slices are at equal heights. (correct answer)
  3. The comparison is valid because the sphere's surface area equals the cylinder's lateral area minus the cones' lateral areas.
  4. The comparison is valid because the formulas for cylinder and cone volumes can be subtracted to get the sphere's formula.
Explanation: The skill involves deriving the volume of a sphere using Cavalieri's principle through an informal argument. We compare the sphere of radius r to the solid formed by a cylinder of radius r and height 2r with two congruent cones removed, each with base radius r and height r. At any height h from the center, the cross-sectional area of the sphere equals the cylinder's cross-sectional area minus the cross-sectional areas of the two cones' slices. Cavalieri's principle states that solids of equal height with matching cross-sectional areas at every corresponding height have equal volumes. Therefore, the sphere's volume equals the volume of the cylinder minus the volumes of the two cones. A misconception is equating volumes based on matching surface areas of the solids, but Cavalieri relies on cross-sections, not surfaces. To apply this strategy more broadly, always compare slices at the same height from a common reference point.

Question 19

A student attempting to use Cavalieri's principle for sphere volume makes the following argument: 'Since a sphere can be approximated by many thin cylindrical slices, and each slice has volume πr2Δh\pi r^2 \Delta h, the total volume is πr2⋅2r=2πr3\pi r^2 \cdot 2r = 2\pi r^3.' What is the main flaw in this reasoning?

  1. The student used the wrong height (2r2r instead of rr) in the final calculation, leading to an incorrect volume formula
  2. The student applied Cavalieri's principle incorrectly by not comparing the sphere to another solid with known volume
  3. The student assumed all cylindrical slices have the same radius rr, but sphere cross-sections have variable radii depending on height (correct answer)
  4. The student used cylindrical slices instead of conical slices, which are the appropriate shapes for spherical volume calculations
Explanation: When applying Cavalieri's principle or any slicing method to find volume, you must carefully consider how the cross-sectional area changes as you move through the solid. The correct answer is C because the student's fundamental error is assuming all horizontal slices of the sphere have the same radius rr. In reality, when you slice a sphere horizontally at different heights, each circular cross-section has a different radius. Only the slice through the very center (the equator) has radius rr. Slices closer to the top or bottom of the sphere have progressively smaller radii, eventually approaching zero at the poles. This means the area πr2\pi r^2 is not constant for all slices—it varies with position. Answer A is wrong because using height 2r2r (the sphere's diameter) is actually correct for the total height being integrated over. The error isn't in the height calculation. Answer B misunderstands Cavalieri's principle. While the principle does involve comparing solids, the student's approach of summing slice volumes is a valid application of integration concepts, not necessarily requiring comparison to another solid. Answer D is incorrect because cylindrical slices (thin disks) are perfectly appropriate for finding volumes of revolution and spherical volumes. Conical slices aren't necessary or standard for this type of calculation. Study tip: When using slicing methods for volume, always ask yourself: "Does my cross-sectional area change as I move through the solid?" For spheres, pyramids, and cones, the answer is always yes, so you need calculus or more sophisticated geometric relationships to account for these changes.

Question 20

Consider the classic Cavalieri comparison: a sphere of radius RR and a cylinder of radius RR and height 2R2R with two congruent cones removed (each cone has height RR). Horizontal slices are taken at the same height yy. Which argument explains why the sphere has this volume?

  1. Because the sphere is curved while the cylinder is not, subtracting cone volumes must produce the sphere's volume.
  2. Because the cross-sectional areas match for every height yy, Cavalieri's principle gives equal volumes for the sphere and the cylinder minus two cones. (correct answer)
  3. Because the cross-sectional areas match at y=0y=0 and y=Ry=R, the volumes must match.
  4. Because the formula V=43πR3V=\tfrac{4}{3}\pi R^3 is known, the comparison must be true.
Explanation: This question tests understanding of the classic Cavalieri argument for sphere volume using cross-sectional area comparison. The setup compares a sphere of radius R with a cylinder (radius R, height 2R) from which two cones of height R are removed. At each height y, the sphere's cross-section is a circle with area π(R² - y²), while the cylinder has constant area πR² and each cone contributes area πy². The crucial calculation shows πR² - 2πy² = π(R² - y²), proving the cross-sectional areas match at every height. By Cavalieri's principle, equal cross-sectional areas at all heights implies equal volumes. Option C incorrectly claims that matching areas at just two heights suffices, while option A relies on vague geometric intuition rather than precise area calculations.