12 multiple-choice questions, progressively harder.
A great circle of a sphere has circumference 14π14\pi14π cm. What volume does the sphere enclose?
Solution
Correct answer: D
The great-circle circumference gives the sphere radius.
A plane is 9 cm from a sphere center, measured perpendicularly. Its section circle has radius 12 cm. What is the sphere surface area?
Correct answer: B
The sphere radius is the right triangle hypotenuse. Find its square before computing area.
A solid hemisphere has total boundary area 192π cm2192\pi\text{ cm}^2192π cm2, including its flat base. What is its volume?
Correct answer: C
The total area gives the positive radius 8 cm. Then use the hemisphere, rather than the whole-ball, volume.
A hemisphere encloses 2662π3 cm3\frac{2662\pi}{3}\text{ cm}^332662π cm3. What is its total boundary area, including the flat base?
Correct answer: A
Recover radius 11 cm from the half-ball volume.
A sphere has surface area 400π cm2400\pi\text{ cm}^2400π cm2. A hemisphere has the same radius. Which pair gives the hemisphere's curved area and volume, in that order?
400π cm2400\pi\text{ cm}^2400π cm2
2000π3 cm3\frac{2000\pi}{3}\text{ cm}^332000π cm3
200π cm2200\pi\text{ cm}^2200π cm2
4000π3 cm3\frac{4000\pi}{3}\text{ cm}^334000π cm3
300π cm2300\pi\text{ cm}^2300π cm2
The sphere area gives radius 10 cm. The hemisphere has half the sphere surface as its curved area, and half the ball volume.
A hemisphere has radius 5 cm. In the Cavalieri comparison, the cylinder has radius and height 5 cm and the removed cone apex is at its bottom center. At height 4 cm above the base, what are the hemisphere section area and the remaining cylinder section area?
The hemisphere section radius squared is the sphere radius squared minus the height squared. The cone section radius is 4 cm, so the remaining annulus has the same area.
A hemisphere has radius R. Height z is measured upward from its equator-level base. At height 2R/32R/32R/3, what is its filled section area?
Substitute the height into the section-area difference of squares.
In the informal radial pyramid explanation of sphere area, which step requires refinement rather than being exact for a fixed finite partition?
The curved patches are not flat. Their flat approximations have heights near the radius. Refinement is what motivates replacing their total contribution by one-third radius times sphere area. The other listed operations are exact under their stated conditions.
A solid hemisphere has total boundary area 48π cm248\pi\text{ cm}^248π cm2, including its flat base. What is the circumference of its circular rim?
The total area gives radius 4 cm. The rim is a circle of that radius.
A sphere has area 400π cm2400\pi\text{ cm}^2400π cm2. A plane cuts the ball in a disk of area 64π cm264\pi\text{ cm}^264π cm2. What is the perpendicular distance from the sphere center to the plane?
The sphere area gives its radius squared; the disk area gives the section radius squared. Subtract those squares.
A hemisphere has curved area A, where A is positive. What is its total boundary area when the flat base is included?
The curved cap has twice the area of its base disk, so the disk contributes half of A. Add that to A.
A solid hemisphere has rim circumference 18π18\pi18π cm. Its flat base rests flush against an opaque table and the curved cap is uncovered. Which pair gives its volume and exposed area, in that order?
972π cm3972\pi\text{ cm}^3972π cm3
162π cm2162\pi\text{ cm}^2162π cm2
486π cm3486\pi\text{ cm}^3486π cm3
243π cm2243\pi\text{ cm}^2243π cm2
81π cm281\pi\text{ cm}^281π cm2
The rim circumference gives radius 9 cm. Use half-ball volume and count only the curved cap, since the table hides the disk.
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