{
  "format": "stapler-pack",
  "version": 1,
  "kind": "quiz",
  "meta": {
    "title": "Solids: Surface Area & Volume",
    "author": "Staples Education",
    "description": "Volume and surface area for prisms, cylinders, pyramids, cones and spheres, worked in both directions, plus Cavalieri's Principle. Transcribed from the worked Surface Area & Volume section of the tutoring whiteboard archive.",
    "created": "2026-08-06T04:14:56.155Z",
    "license": "Proprietary - Staples Education"
  },
  "payload": {
    "quiz": {
      "id": "surface-area-volume",
      "title": "Solids: Surface Area & Volume",
      "blurb": "Prisms, cylinders, pyramids, cones and spheres — volume and surface area in both directions, plus Cavalieri's Principle.",
      "meta": "19 questions · easy → hard",
      "categories": [
        "three-d-figures",
        "surface-area-volume"
      ],
      "questions": [
        {
          "id": "sav-001",
          "topic": "Cylinders",
          "difficulty": "easy",
          "stem": "Find the volume of a cylinder with radius 3 and height 10, in terms of π.",
          "choices": [
            "90π",
            "30π",
            "60π",
            "180π"
          ],
          "correctIndex": 0,
          "explanations": [
            "Correct. V = πr²h = π(3²)(10) = π(9)(10) = 90π.",
            "30π uses πrh, forgetting to square the radius. The base is a circle of area πr².",
            "60π is the lateral surface area, 2πrh = 2π(3)(10). That is an area, not a volume.",
            "180π doubles the volume — perhaps from using 2πr²h. The prism/cylinder formula is base area times height, with no factor of 2."
          ],
          "hint": "Volume of any prism or cylinder is base area × height. What is the area of a circle of radius 3?",
          "diagram": "<svg viewBox=\"0 0 200 128\" width=\"100%\" style=\"max-width:200px; height:auto; display:block; margin:0 auto;\" role=\"img\"><ellipse cx=\"90\" cy=\"34\" rx=\"44\" ry=\"14\" fill=\"none\" stroke=\"#7c3aed\" stroke-width=\"2\"/><line x1=\"46\" y1=\"34\" x2=\"46\" y2=\"108\" stroke=\"#7c3aed\" stroke-width=\"2\" stroke-linecap=\"round\"/><line x1=\"134\" y1=\"34\" x2=\"134\" y2=\"108\" stroke=\"#7c3aed\" stroke-width=\"2\" stroke-linecap=\"round\"/><path d=\"M 46,108 A 44 14 0 0 0 134,108\" fill=\"none\" stroke=\"#7c3aed\" stroke-width=\"2\"/><line x1=\"90\" y1=\"34\" x2=\"134\" y2=\"34\" stroke=\"#1f2024\" stroke-width=\"1.5\" stroke-dasharray=\"5 4\" stroke-linecap=\"round\"/><text x=\"104\" y=\"30\" fill=\"#1f2024\" font-size=\"12\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">3</text><line x1=\"158\" y1=\"34\" x2=\"158\" y2=\"108\" stroke=\"#6b6b76\" stroke-width=\"1.4\" stroke-linecap=\"round\"/><text x=\"164\" y=\"76\" fill=\"#1f2024\" font-size=\"12\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">10</text></svg>",
          "diagramCaption": "Radius 3, height 10. Not drawn to scale"
        },
        {
          "id": "sav-002",
          "topic": "Cones",
          "difficulty": "medium",
          "stem": "Find the volume of a cone with radius 6 and height 8, in terms of π.",
          "choices": [
            "288π",
            "96π",
            "48π",
            "144π"
          ],
          "correctIndex": 1,
          "explanations": [
            "288π is the volume of the CYLINDER with the same radius and height, πr²h = π(36)(8). A cone holds exactly one third of that.",
            "Correct. V = ⅓πr²h = ⅓π(6²)(8) = ⅓π(36)(8) = 96π.",
            "48π halves rather than takes a third. The factor for a cone or pyramid is ⅓, not ½.",
            "144π uses ⅓π(36)(12) or a similar slip in the height. The height here is 8."
          ],
          "hint": "A cone is one third of the cylinder that contains it."
        },
        {
          "id": "sav-003",
          "topic": "Cones",
          "difficulty": "hard",
          "stem": "A cone has radius 6 and height 8. Find its slant height, then its lateral area (in terms of π).",
          "choices": [
            "slant 10, lateral 60π",
            "slant 10, lateral 120π",
            "slant 14, lateral 84π",
            "slant 10, lateral 96π"
          ],
          "correctIndex": 0,
          "explanations": [
            "Correct. The slant height is the hypotenuse of the radius and the height: ℓ = √(r² + h²) = √(36 + 64) = √100 = 10. Lateral area is πrℓ = π(6)(10) = 60π.",
            "120π doubles the lateral area, using 2πrℓ. That factor of 2 belongs to a cylinder's lateral area, not a cone's.",
            "A slant of 14 adds r and h. They are perpendicular, so they combine by Pythagoras, not by addition.",
            "96π is the cone's VOLUME, not its lateral area. Volume uses ⅓πr²h; lateral area uses πrℓ."
          ],
          "hint": "The radius, the height and the slant height form a right triangle. Find the slant first, then use πrℓ."
        },
        {
          "id": "sav-004",
          "topic": "Spheres",
          "difficulty": "medium",
          "stem": "Find the volume and surface area of a sphere with radius 3, in terms of π.",
          "choices": [
            "V = 36π, SA = 36π",
            "V = 12π, SA = 36π",
            "V = 36π, SA = 12π",
            "V = 27π, SA = 9π"
          ],
          "correctIndex": 0,
          "explanations": [
            "Correct. V = (4/3)πr³ = (4/3)π(27) = 36π and SA = 4πr² = 4π(9) = 36π. They coincide numerically at r = 3, which is a coincidence of that radius rather than a general fact — the units still differ.",
            "12π for the volume drops the cube, using (4/3)πr instead. The radius is cubed for volume.",
            "12π for the surface area uses 4πr rather than 4πr². The radius is squared for area.",
            "27π is r³ times π, missing the 4/3; 9π is r² times π, missing the 4."
          ],
          "hint": "V = (4/3)πr³ and SA = 4πr². Cube for one, square for the other."
        },
        {
          "id": "sav-005",
          "topic": "Prisms",
          "difficulty": "easy",
          "stem": "A rectangular prism measures 4 × 5 × 6. Find its volume and total surface area.",
          "choices": [
            "V = 120, SA = 148",
            "V = 120, SA = 74",
            "V = 15, SA = 148",
            "V = 120, SA = 296"
          ],
          "correctIndex": 0,
          "explanations": [
            "Correct. V = 4 × 5 × 6 = 120. The three distinct faces have areas 4×5 = 20, 4×6 = 24 and 5×6 = 30, and each occurs twice: SA = 2(20 + 24 + 30) = 2(74) = 148.",
            "74 is the sum of the three distinct face areas without doubling. A box has six faces, in three matching pairs.",
            "15 adds the dimensions, 4 + 5 + 6. Volume multiplies them.",
            "296 doubles the surface area twice over. Each pair is counted once in the sum, then the whole sum is doubled — once, not twice."
          ],
          "hint": "Three different faces, each appearing twice. Add the three, then double."
        },
        {
          "id": "sav-006",
          "topic": "Prisms",
          "difficulty": "easy",
          "stem": "A cube has edge length 5. Find its volume and surface area.",
          "choices": [
            "V = 125, SA = 150",
            "V = 125, SA = 125",
            "V = 15, SA = 150",
            "V = 125, SA = 25"
          ],
          "correctIndex": 0,
          "explanations": [
            "Correct. V = s³ = 5³ = 125, and SA = 6s² = 6(25) = 150 — six square faces, each of area 25.",
            "125 for the surface area repeats the volume. Area and volume of a cube agree numerically only at s = 6.",
            "15 adds the three edge lengths. Volume is their product.",
            "25 is the area of ONE face. A cube has six."
          ],
          "hint": "A cube has six identical square faces."
        },
        {
          "id": "sav-007",
          "topic": "Cylinders",
          "difficulty": "medium",
          "stem": "A cylinder has volume 100π and height 4. Find its radius.",
          "choices": [
            "5",
            "25",
            "10",
            "√10"
          ],
          "correctIndex": 0,
          "explanations": [
            "Correct. From V = πr²h: πr²(4) = 100π, so r² = 25 and r = 5.",
            "25 is r², one step short. Take the square root.",
            "10 would give r² = 100 and a volume of 400π — four times too large.",
            "√10 would come from r² = 10. Dividing 100 by the height 4 gives 25, not 10."
          ],
          "hint": "Divide out the π and the height first, then take a square root."
        },
        {
          "id": "sav-008",
          "topic": "Pyramids",
          "difficulty": "medium",
          "stem": "A square pyramid has base edge 6 and height 10. Find its volume.",
          "choices": [
            "360",
            "180",
            "120",
            "60"
          ],
          "correctIndex": 2,
          "explanations": [
            "360 is the volume of the PRISM with the same base and height, 36 × 10. A pyramid holds one third of it.",
            "180 halves rather than takes a third. The factor for pyramids and cones is ⅓.",
            "Correct. V = ⅓(base area)(height) = ⅓(6²)(10) = ⅓(36)(10) = 120.",
            "60 uses a base edge of 6 without squaring it, or divides by 6. The base is a 6 × 6 square of area 36."
          ],
          "hint": "Base area first — the base is a square, not an edge. Then one third of base × height."
        },
        {
          "id": "sav-009",
          "topic": "Spheres",
          "difficulty": "hard",
          "stem": "A sphere has surface area 100π. Find its radius and volume.",
          "choices": [
            "r = 5, V = 500π/3",
            "r = 25, V = 500π/3",
            "r = 5, V = 100π",
            "r = 10, V = 4000π/3"
          ],
          "correctIndex": 0,
          "explanations": [
            "Correct. From SA = 4πr² = 100π we get r² = 25 and r = 5. Then V = (4/3)πr³ = (4/3)π(125) = 500π/3.",
            "25 is r², not r. The square root is still needed before the volume can be computed.",
            "100π repeats the surface area as the volume. They are different quantities and, at r = 5, different numbers.",
            "r = 10 would give a surface area of 400π, four times too large."
          ],
          "hint": "Recover r from the surface area first, then feed it into the volume formula."
        },
        {
          "id": "sav-010",
          "topic": "Cones",
          "difficulty": "medium",
          "stem": "A cone has volume 12π and radius 3. Find its height.",
          "choices": [
            "4",
            "12",
            "3",
            "4/3"
          ],
          "correctIndex": 0,
          "explanations": [
            "Correct. From V = ⅓πr²h: ⅓π(9)h = 12π, so 3πh = 12π and h = 4.",
            "12 repeats the volume's coefficient. The height has to be extracted from the equation, not read off it.",
            "3 repeats the radius. Nothing here forces the height to equal the radius.",
            "4/3 divides by 3 one time too many. The ⅓ and the r² = 9 combine to a single factor of 3, giving 3πh = 12π."
          ],
          "hint": "Substitute what you know and simplify — the ⅓ and the 9 collapse into a single 3."
        },
        {
          "id": "sav-011",
          "topic": "Cylinders",
          "difficulty": "medium",
          "stem": "A cylinder has radius 2 and height 7. Find its total surface area, in terms of π.",
          "choices": [
            "28π",
            "36π",
            "32π",
            "44π"
          ],
          "correctIndex": 1,
          "explanations": [
            "28π is the lateral area alone, 2πrh = 2π(2)(7). The two circular ends are still missing.",
            "Correct. Total surface area is the lateral area plus the two ends: 2πrh + 2πr² = 2π(2)(7) + 2π(4) = 28π + 8π = 36π.",
            "32π adds only one end, 28π + 4π. A closed cylinder has two.",
            "44π adds the ends twice over, or uses 4πr². There are exactly two circular faces, contributing 2πr² between them."
          ],
          "hint": "Unroll it: a rectangle for the side, plus two circles for the ends."
        },
        {
          "id": "sav-012",
          "topic": "Cavalieri's Principle",
          "difficulty": "hard",
          "stem": "Two solids have the same height and the same cross-sectional area at every level. By Cavalieri's Principle, what must be true?",
          "choices": [
            "They are congruent.",
            "They have equal volumes.",
            "They have equal surface areas.",
            "They are similar."
          ],
          "correctIndex": 1,
          "explanations": [
            "Congruent means identical in size AND shape. A right cylinder and a slanted (oblique) one with the same height and radius satisfy Cavalieri's condition, yet they are clearly different shapes — so equal cross-sections cannot force congruence.",
            "Correct. Cavalieri's Principle says that if two solids have equal heights and their cross-sections at every level have equal areas, their volumes are equal. Stacking the same amount of area to the same height gives the same total, however the stack leans.",
            "Surface area is not controlled by cross-sections. Slanting a cylinder leaves its volume alone but lengthens its curved surface.",
            "Similar means the same shape at a different scale. Cavalieri's condition says nothing about shape — the two solids need not resemble each other at all."
          ],
          "hint": "Picture a straight stack of coins beside the same stack pushed into a lean. What changed, and what did not?",
          "diagram": "<svg viewBox=\"0 0 240 142\" width=\"100%\" style=\"max-width:240px; height:auto; display:block; margin:0 auto;\" role=\"img\"><ellipse cx=\"60\" cy=\"30\" rx=\"30\" ry=\"10\" fill=\"none\" stroke=\"#7c3aed\" stroke-width=\"2\"/><line x1=\"30\" y1=\"30\" x2=\"30\" y2=\"110\" stroke=\"#7c3aed\" stroke-width=\"2\" stroke-linecap=\"round\"/><line x1=\"90\" y1=\"30\" x2=\"90\" y2=\"110\" stroke=\"#7c3aed\" stroke-width=\"2\" stroke-linecap=\"round\"/><path d=\"M 30,110 A 30 10 0 0 0 90,110\" fill=\"none\" stroke=\"#7c3aed\" stroke-width=\"2\"/><ellipse cx=\"185\" cy=\"30\" rx=\"30\" ry=\"10\" fill=\"none\" stroke=\"#1f2024\" stroke-width=\"2\"/><line x1=\"155\" y1=\"30\" x2=\"135\" y2=\"110\" stroke=\"#1f2024\" stroke-width=\"2\" stroke-linecap=\"round\"/><line x1=\"215\" y1=\"30\" x2=\"195\" y2=\"110\" stroke=\"#1f2024\" stroke-width=\"2\" stroke-linecap=\"round\"/><path d=\"M 135,110 A 30 10 0 0 0 195,110\" fill=\"none\" stroke=\"#1f2024\" stroke-width=\"2\"/><line x1=\"30\" y1=\"70\" x2=\"90\" y2=\"70\" stroke=\"#6b6b76\" stroke-width=\"1.5\" stroke-dasharray=\"5 4\" stroke-linecap=\"round\"/><line x1=\"145\" y1=\"70\" x2=\"205\" y2=\"70\" stroke=\"#6b6b76\" stroke-width=\"1.5\" stroke-dasharray=\"5 4\" stroke-linecap=\"round\"/><text x=\"38\" y=\"132\" fill=\"#6b6b76\" font-size=\"11\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">right</text><text x=\"152\" y=\"132\" fill=\"#6b6b76\" font-size=\"11\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">oblique</text></svg>",
          "diagramCaption": "Same height, same cross-section at every level. Not drawn to scale"
        },
        {
          "id": "sav-013",
          "topic": "Cone Slant Height",
          "difficulty": "medium",
          "stem": "A cone has radius 6 and height 8. Find its lateral surface area.",
          "choices": [
            "48π",
            "60π",
            "96π",
            "100π"
          ],
          "correctIndex": 1,
          "explanations": [
            "48π uses the HEIGHT in place of the slant height: π(6)(8). Lateral area needs the slant height along the surface, not the vertical height inside.",
            "Correct. First find the slant height: ℓ = √(r² + h²) = √(36 + 64) = √100 = 10. Then lateral area = πrℓ = π(6)(10) = 60π.",
            "96π is the cone's VOLUME, ⅓π(36)(8). Volume and lateral area are different quantities with different formulas.",
            "100π is ℓ² times π — the slant height was squared instead of being multiplied by the radius."
          ],
          "hint": "Lateral area needs the slant height, which is a hypotenuse. Compute it before applying the area formula."
        },
        {
          "id": "sav-014",
          "topic": "Sphere from Surface Area",
          "difficulty": "hard",
          "stem": "A sphere has surface area 100π. Find its volume.",
          "choices": [
            "(500/3)π",
            "(100/3)π",
            "500π",
            "(250/3)π"
          ],
          "correctIndex": 0,
          "explanations": [
            "Correct. From SA = 4πr² = 100π we get r² = 25, so r = 5. Then V = (4/3)πr³ = (4/3)π(125) = (500/3)π.",
            "(100/3)π comes from substituting the surface area into the volume formula in place of r³. The radius must be extracted first.",
            "500π drops the ⅓ from the sphere volume formula. The coefficient is 4/3, and forgetting the denominator is a listed high-risk error.",
            "(250/3)π uses r³ = 62.5, which comes from halving 125. Once r = 5 is found, r³ is 125 exactly."
          ],
          "hint": "Two formulas, one shared unknown. Use the surface area to pin down r, then move to the volume formula."
        },
        {
          "id": "sav-015",
          "topic": "Cylinder Radius",
          "difficulty": "medium",
          "stem": "A cylinder has volume 100π and height 4. Find its radius.",
          "choices": [
            "5",
            "25",
            "10",
            "2.5"
          ],
          "correctIndex": 0,
          "explanations": [
            "Correct. V = πr²h, so 100π = πr²(4). Dividing by π gives 100 = 4r², then r² = 25 and r = 5.",
            "25 is r², not r. The final square root was skipped — the same last-step slip that catches people on the tangent-secant theorem.",
            "10 comes from taking 100/(4·2.5) or otherwise mishandling the division. Divide by both π and the height before taking the root.",
            "2.5 is 10/4, which divides a length by the height. The height divides out of r², not out of r directly."
          ],
          "hint": "Strip away π and the height first to isolate r², then remember one more step remains."
        },
        {
          "id": "sav-016",
          "topic": "Cavalieri's Principle",
          "difficulty": "hard",
          "stem": "Two solids have the same height, and every horizontal cross-section at the same level has equal area. What does Cavalieri's principle conclude?",
          "choices": [
            "They are congruent.",
            "They have equal surface area.",
            "They have equal volume.",
            "They are similar with scale factor 1."
          ],
          "correctIndex": 2,
          "explanations": [
            "Congruence is far stronger than the hypothesis supports. A vertical stack of coins and the same stack pushed into a slant have matching cross-sections at every level but are not congruent.",
            "Surface areas can differ substantially. Shearing a prism leaves every cross-section unchanged while lengthening the slanted lateral faces, so the surface area grows even though the cross-sections do not.",
            "Correct. Cavalieri's principle states that if two solids have equal-area cross-sections at every level, their volumes are equal. It is what justifies the oblique-prism volume formula and underlies the derivation of the sphere's volume.",
            "Similarity with scale factor 1 would mean congruence, which the hypothesis does not give. The two solids may have entirely different shapes."
          ],
          "hint": "The principle concerns one measurement only — the one built by stacking cross-sections."
        },
        {
          "id": "sav-017",
          "topic": "Cone Height",
          "difficulty": "medium",
          "stem": "A cone has volume 12π and radius 3. Find its height.",
          "choices": [
            "4",
            "12",
            "1.33",
            "36"
          ],
          "correctIndex": 0,
          "explanations": [
            "Correct. V = ⅓πr²h, so 12π = ⅓π(9)h = 3πh. Dividing by 3π gives h = 4.",
            "12 comes from omitting the ⅓ and solving 12π = π(9)h incorrectly, or from reading the volume as the height. The ⅓ factor for cones is a listed high-risk item.",
            "1.33 is ⅓ applied to the answer rather than to the formula. The ⅓ belongs on the volume side of the equation, which makes h larger, not smaller.",
            "36 comes from multiplying by 3 instead of dividing. Solve step by step: 12π = 3πh means h = 12π/3π."
          ],
          "hint": "Write out the cone volume formula including its fractional coefficient before substituting anything."
        },
        {
          "id": "sav-018",
          "topic": "Reasoning: Scaling",
          "difficulty": "hard",
          "stem": "Two cubes are similar, with edges in the ratio 2:3. What is the ratio of their volumes, and why?",
          "choices": [
            "2:3, because volume scales with edge length.",
            "4:9, matching the ratio of their surface areas.",
            "8:27, because volume involves three dimensions, each scaled by the same factor.",
            "6:9, because a cube has six faces."
          ],
          "correctIndex": 2,
          "explanations": [
            "Only LENGTHS scale by the linear ratio. Volume is built from three of them.",
            "4:9 is the ratio of the SURFACE AREAS, which scale by the square of the linear ratio.",
            "Correct. Volume scales by the cube of the linear ratio: 2³ : 3³ = 8 : 27.",
            "The number of faces does not enter into it, and it is the same for both cubes anyway."
          ],
          "hint": "Length scales by k, area by k², volume by k³. Which one is being asked for?",
          "kind": "reasoning"
        },
        {
          "id": "sav-019",
          "topic": "Reasoning: Error Analysis",
          "difficulty": "medium",
          "stem": "A student reports the volume of a cylinder as 96π cm². What is wrong before any arithmetic is checked?",
          "choices": [
            "Volume cannot contain π.",
            "The units are square centimetres, but volume must be in CUBIC units — so this is an area, or the units are wrong.",
            "Volume must be a whole number.",
            "Nothing is wrong."
          ],
          "correctIndex": 1,
          "explanations": [
            "π appears in the volume of any cylinder or cone; there is nothing wrong with it.",
            "Correct. Volume is a three-dimensional measure and must be reported in cubic units — cm³. Square centimetres signal that an area formula was used, or that the units were copied carelessly.",
            "Volumes are frequently irrational, especially with π involved.",
            "The units contradict the quantity, which is exactly the kind of error a unit check is for."
          ],
          "kind": "reasoning"
        }
      ]
    }
  }
}