{
  "format": "stapler-pack",
  "version": 1,
  "kind": "quiz",
  "meta": {
    "title": "Area, Perimeter & Measurement",
    "author": "Staples Education",
    "description": "Heron's formula, regular-polygon area via the apothem, rhombus and kite diagonals, sectors and arc length, and the geometric-mean segments of a right triangle. Transcribed from worked problems in the tutoring whiteboard archive, including its measurement and trigonometry board.",
    "created": "2026-08-06T04:14:56.057Z",
    "license": "Proprietary - Staples Education"
  },
  "payload": {
    "quiz": {
      "id": "area-perimeter-measurement",
      "title": "Area, Perimeter & Measurement",
      "blurb": "Heron's formula, the apothem and regular-polygon area, rhombus and kite diagonals, sectors and arc length, geometric-mean segments, and composite figures.",
      "meta": "19 questions · easy → hard",
      "categories": [
        "area-perimeter",
        "measurement"
      ],
      "questions": [
        {
          "id": "apm-001",
          "topic": "Heron's Formula",
          "difficulty": "hard",
          "stem": "A triangle has side lengths 5, 6 and 7. What is its area, using Heron's formula?",
          "choices": [
            "6√6",
            "15",
            "216",
            "3√21"
          ],
          "correctIndex": 0,
          "explanations": [
            "Correct. The semi-perimeter is s = (5 + 6 + 7)/2 = 9. Then A = √(s(s−a)(s−b)(s−c)) = √(9 · 4 · 3 · 2) = √216 = 6√6 ≈ 14.7.",
            "15 is close to the true value but is not it. Heron's product here is 216, and √216 is irrational — a whole-number area would be a coincidence.",
            "216 is the value under the radical, s(s−a)(s−b)(s−c). The square root is the final step.",
            "3√21 = √189, which would come from a slip in one of the four factors. Check s − a, s − b and s − c: they are 4, 3 and 2."
          ],
          "hint": "Half the perimeter first, then subtract each side from it in turn. Four numbers multiply under the root."
        },
        {
          "id": "apm-002",
          "topic": "Regular Polygons",
          "difficulty": "medium",
          "stem": "A regular hexagon has a side length of 10 units and an apothem of 5√3 units. What is its area?",
          "choices": [
            "150√3 sq units",
            "50√3 sq units",
            "75√3 sq units",
            "300√3 sq units"
          ],
          "correctIndex": 0,
          "explanations": [
            "Correct. A regular polygon's area is ½ × apothem × perimeter. The perimeter is 6 × 10 = 60, so A = ½(5√3)(60) = 150√3 ≈ 259.8 sq units.",
            "50√3 is the area of just ONE of the six triangular segments the hexagon splits into: ½(10)(5√3) = 25√3 — and even that is half of 50√3. Either way it is one piece, not the whole.",
            "75√3 halves the correct answer, which happens if the ½ is applied twice or the perimeter is taken as 30.",
            "300√3 is apothem × perimeter without the ½. That counts each triangular segment as a parallelogram."
          ],
          "hint": "Cut the hexagon into six triangles from the centre. The apothem is each one's height, and their bases add up to the perimeter.",
          "diagram": "<svg viewBox=\"0 0 240 164\" width=\"100%\" style=\"max-width:240px; height:auto; display:block; margin:0 auto;\" role=\"img\"><polygon points=\"120,18 178,51 178,117 120,150 62,117 62,51\" fill=\"none\" stroke=\"#7c3aed\" stroke-width=\"2\" stroke-linejoin=\"round\"/><line x1=\"120\" y1=\"84\" x2=\"178\" y2=\"51\" stroke=\"#6b6b76\" stroke-width=\"1.2\" stroke-linecap=\"round\"/><line x1=\"120\" y1=\"84\" x2=\"178\" y2=\"117\" stroke=\"#6b6b76\" stroke-width=\"1.2\" stroke-linecap=\"round\"/><line x1=\"120\" y1=\"84\" x2=\"120\" y2=\"150\" stroke=\"#6b6b76\" stroke-width=\"1.2\" stroke-linecap=\"round\"/><line x1=\"120\" y1=\"84\" x2=\"62\" y2=\"117\" stroke=\"#6b6b76\" stroke-width=\"1.2\" stroke-linecap=\"round\"/><line x1=\"120\" y1=\"84\" x2=\"62\" y2=\"51\" stroke=\"#6b6b76\" stroke-width=\"1.2\" stroke-linecap=\"round\"/><line x1=\"120\" y1=\"84\" x2=\"120\" y2=\"18\" stroke=\"#6b6b76\" stroke-width=\"1.2\" stroke-linecap=\"round\"/><line x1=\"120\" y1=\"84\" x2=\"149\" y2=\"100\" stroke=\"#1f2024\" stroke-width=\"1.8\" stroke-linecap=\"round\"/><path d=\"M 144,92 L 150,102 L 140,108\" fill=\"none\" stroke=\"#6b6b76\" stroke-width=\"1.4\"/><text x=\"126\" y=\"96\" fill=\"#1f2024\" font-size=\"11\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">5√3</text><text x=\"186\" y=\"92\" 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\">10</text></svg>",
          "diagramCaption": "Apothem to a side midpoint. Not drawn to scale"
        },
        {
          "id": "apm-003",
          "topic": "Quadrilateral Area",
          "difficulty": "medium",
          "stem": "A rhombus has an area of 84 sq units and one diagonal of length 12 units. What is the length of the other diagonal?",
          "choices": [
            "14 units",
            "7 units",
            "28 units",
            "21 units"
          ],
          "correctIndex": 0,
          "explanations": [
            "Correct. A rhombus has area ½d₁d₂, so 84 = ½(12)d₂ = 6d₂, giving d₂ = 14 units.",
            "7 units drops the ½ from the area formula, solving 84 = 12d₂ instead. This is the most common slip on this question — it treats the rhombus as a rectangle on its diagonals.",
            "28 units doubles the correct answer, applying the ½ on the wrong side of the equation.",
            "21 units would follow from 84 = 4d₂ or a similar arithmetic slip. Half of 12 is 6, and 84 ÷ 6 = 14."
          ],
          "hint": "Write the area formula first, substitute both knowns, and check whether the ½ is helping you or being dropped."
        },
        {
          "id": "apm-004",
          "topic": "Quadrilateral Area",
          "difficulty": "easy",
          "stem": "A kite has diagonals measuring 16 inches and 10 inches. What is its area?",
          "choices": [
            "80 sq inches",
            "160 sq inches",
            "100 sq inches",
            "40 sq inches"
          ],
          "correctIndex": 0,
          "explanations": [
            "Correct. A kite's diagonals are perpendicular, so the same formula as the rhombus applies: A = ½d₁d₂ = ½(16)(10) = 80 sq inches.",
            "160 is the full product of the diagonals, which is the area of the RECTANGLE they span — exactly twice the kite.",
            "100 is 10², using only one diagonal. Both are needed.",
            "40 quarters the product rather than halving it."
          ],
          "hint": "A kite's diagonals meet at right angles, so it obeys the same area rule as a rhombus."
        },
        {
          "id": "apm-005",
          "topic": "Circles",
          "difficulty": "medium",
          "stem": "What is the area of a 90° sector of a circle with radius 8 units? (Leave in terms of π.)",
          "choices": [
            "16π sq units",
            "64π sq units",
            "8π sq units",
            "2π sq units"
          ],
          "correctIndex": 0,
          "explanations": [
            "Correct. A sector is the fraction of the circle its angle cuts: (90/360)πr² = ¼ · π(64) = 16π sq units.",
            "64π is the area of the WHOLE circle. The quarter fraction still has to be applied.",
            "8π is the ARC LENGTH plus a slip, or half the correct sector. Arc length uses circumference; area uses πr².",
            "2π is the arc length of this sector, (90/360)(2π·8) = 4π — and even that is not 2π. Check which formula produces square units."
          ],
          "hint": "90° is a quarter turn. Take that fraction of the whole circle's area."
        },
        {
          "id": "apm-006",
          "topic": "Circles",
          "difficulty": "medium",
          "stem": "What is the arc length of a 90° arc in a circle with radius 8 units? (Leave in terms of π.)",
          "choices": [
            "4π units",
            "16π units",
            "2π units",
            "8π units"
          ],
          "correctIndex": 0,
          "explanations": [
            "Correct. Arc length is the same fraction of the CIRCUMFERENCE: (90/360)(2πr) = ¼(16π) = 4π units.",
            "16π is the whole circumference, 2π(8). The quarter fraction has still to be applied.",
            "2π halves the correct answer — perhaps by using πr instead of 2πr for the circumference.",
            "8π is half the circumference, which would be a 180° arc."
          ],
          "hint": "Arc length is to circumference what sector area is to area. Which of the two do you need here — and note the units are linear."
        },
        {
          "id": "apm-007",
          "topic": "Quadrilateral Measurement",
          "difficulty": "medium",
          "stem": "A rhombus has diagonals of length 6 and 8. What is the length of one of its sides?",
          "choices": [
            "5 units",
            "25 units",
            "7 units",
            "10 units"
          ],
          "correctIndex": 0,
          "explanations": [
            "Correct. The diagonals of a rhombus are perpendicular bisectors of each other, so they cut it into four congruent right triangles with legs 3 and 4 — half of each diagonal. The side is the hypotenuse: √(9 + 16) = 5 units.",
            "25 is the square of the side, 3² + 4². The root is the last step.",
            "7 adds the half-diagonals, 3 + 4. They are perpendicular legs, so they combine by Pythagoras.",
            "10 is the average of the diagonals, or half their sum. The side is shorter than either full diagonal."
          ],
          "hint": "The diagonals bisect each other at right angles. Look at just one of the four right triangles they create.",
          "diagram": "<svg viewBox=\"0 0 240 168\" width=\"100%\" style=\"max-width:240px; height:auto; display:block; margin:0 auto;\" role=\"img\"><polygon points=\"120,20 190,80 120,140 50,80\" fill=\"none\" stroke=\"#7c3aed\" stroke-width=\"2\" stroke-linejoin=\"round\"/><line x1=\"120\" y1=\"20\" x2=\"120\" y2=\"140\" stroke=\"#6b6b76\" stroke-width=\"1.4\" stroke-linecap=\"round\"/><line x1=\"50\" y1=\"80\" x2=\"190\" y2=\"80\" stroke=\"#6b6b76\" stroke-width=\"1.4\" stroke-linecap=\"round\"/><path d=\"M 120,68 L 132,68 L 132,80\" fill=\"none\" stroke=\"#6b6b76\" stroke-width=\"1.4\"/><text x=\"126\" y=\"52\" fill=\"#1f2024\" font-size=\"11\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">3</text><text x=\"152\" y=\"74\" fill=\"#1f2024\" font-size=\"11\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">4</text><text x=\"160\" y=\"44\" fill=\"#c0392b\" font-size=\"13\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">?</text><text x=\"84\" y=\"158\" 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\">diagonals 6 and 8</text></svg>",
          "diagramCaption": "Four congruent right triangles. Not drawn to scale"
        },
        {
          "id": "apm-008",
          "topic": "Regular Polygons",
          "difficulty": "easy",
          "stem": "What is the apothem of a regular polygon?",
          "choices": [
            "The perpendicular distance from the centre to the midpoint of any side",
            "The total length around the polygon boundary",
            "The distance from the centre to any vertex",
            "The length of one side of the polygon"
          ],
          "correctIndex": 0,
          "explanations": [
            "Correct. The apothem runs from the centre perpendicular to a side, landing on that side's midpoint. It is also the radius of the inscribed circle, which is why it appears as the height in the ½ × apothem × perimeter area formula.",
            "That is the perimeter.",
            "That is the circumradius — the radius of the CIRCUMSCRIBED circle, which reaches the corners. The apothem is shorter, reaching only to the edges.",
            "The side length is a separate measurement. In a regular hexagon of side 10 the apothem is 5√3 ≈ 8.66, not 10."
          ],
          "hint": "Two radii can be drawn in a regular polygon: one to a corner and one to an edge. This is the shorter of the two."
        },
        {
          "id": "apm-009",
          "topic": "Right Triangle Measurement",
          "difficulty": "medium",
          "stem": "In a right triangle, an altitude drawn to the hypotenuse divides it into segments of 2 and 8 units. What is the length of the altitude?",
          "choices": [
            "4 units",
            "16 units",
            "5 units",
            "6 units"
          ],
          "correctIndex": 0,
          "explanations": [
            "Correct. The altitude to the hypotenuse is the geometric mean of the two segments it creates: h = √(2 × 8) = √16 = 4 units.",
            "16 is h², one square root short.",
            "5 is the arithmetic mean, (2 + 8)/2. The relationship here is geometric — multiply and take the root.",
            "6 would follow from segments of 4 and 9. Here they are 2 and 8, whose product is 16."
          ],
          "hint": "Multiply the two pieces of the hypotenuse and take the square root."
        },
        {
          "id": "apm-010",
          "topic": "Right Triangle Measurement",
          "difficulty": "hard",
          "stem": "In a right triangle, an altitude to the hypotenuse splits it into segments of 3 and 12 units. What is the length of the shorter leg?",
          "choices": [
            "3√5 units",
            "6 units",
            "9 units",
            "6√5 units"
          ],
          "correctIndex": 0,
          "explanations": [
            "Correct. Each leg is the geometric mean of the whole hypotenuse and the segment ADJACENT to it. The hypotenuse is 3 + 12 = 15, so the shorter leg (adjacent to the 3) is √(15 × 3) = √45 = 3√5 ≈ 6.7 units.",
            "6 is the ALTITUDE, √(3 × 12) = √36. That is the geometric mean of the two segments, not of the hypotenuse and a segment.",
            "9 does not follow from any of the three geometric-mean relationships in this figure.",
            "6√5 = √180 = √(15 × 12) is the LONGER leg, the one adjacent to the 12-unit segment."
          ],
          "hint": "Three geometric means live in this figure: one for the altitude and one for each leg. The leg ones use the full hypotenuse.",
          "diagram": "<svg viewBox=\"0 0 240 140\" width=\"100%\" style=\"max-width:240px; height:auto; display:block; margin:0 auto;\" role=\"img\"><polygon points=\"25,110 215,110 63,32\" fill=\"none\" stroke=\"#7c3aed\" stroke-width=\"2\" stroke-linejoin=\"round\"/><line x1=\"63\" y1=\"32\" x2=\"63\" y2=\"110\" stroke=\"#1f2024\" stroke-width=\"1.8\" stroke-linecap=\"round\"/><path d=\"M 63,98 L 75,98 L 75,110\" fill=\"none\" stroke=\"#6b6b76\" stroke-width=\"1.4\"/><path d=\"M 55,40 L 65,46 L 59,54\" fill=\"none\" stroke=\"#6b6b76\" stroke-width=\"1.4\"/><text x=\"38\" y=\"128\" fill=\"#6b6b76\" 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><text x=\"134\" y=\"128\" fill=\"#6b6b76\" font-size=\"12\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">12</text><text x=\"30\" y=\"74\" fill=\"#c0392b\" font-size=\"13\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">?</text><text x=\"68\" y=\"76\" fill=\"#1f2024\" font-size=\"11\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">h</text></svg>",
          "diagramCaption": "The altitude falls to the hypotenuse. Not drawn to scale"
        },
        {
          "id": "apm-011",
          "topic": "Measurement and Units",
          "difficulty": "medium",
          "stem": "A central angle of 90° in a circle of radius 6 subtends an arc. What is that arc's length in radians-based form, using s = rθ?",
          "choices": [
            "3π units",
            "6π units",
            "1.5π units",
            "270 units"
          ],
          "correctIndex": 0,
          "explanations": [
            "Correct. Converting first: 90° = π/2 radians. Then s = rθ = 6(π/2) = 3π units. The degree-based route agrees: (90/360)(2π·6) = ¼(12π) = 3π.",
            "6π is half the circumference, which would be a 180° arc.",
            "1.5π halves the correct answer, which happens if the radius is taken as 3 or the angle as 45°.",
            "270 multiplies 90 by 3 and keeps degrees. Arc length is a distance, so it cannot be measured in degrees — the units alone rule this out."
          ],
          "hint": "s = rθ only works when θ is in radians. Convert the 90° first, then multiply."
        },
        {
          "id": "apm-012",
          "topic": "Composite Area",
          "difficulty": "medium",
          "stem": "A running track is a rectangle 100 m by 60 m with a semicircle of diameter 60 m attached to each of the two shorter ends. What is the total enclosed area?",
          "choices": [
            "6000 + 900π m²",
            "6000 + 1800π m²",
            "6000 + 3600π m²",
            "6000 + 450π m²"
          ],
          "correctIndex": 0,
          "explanations": [
            "Correct. The rectangle contributes 100 × 60 = 6000 m². The two semicircles of diameter 60 (radius 30) together make one full circle of area π(30²) = 900π m². Total: 6000 + 900π ≈ 8827 m².",
            "1800π doubles the circle — it treats each semicircle as a whole circle rather than two halves making one.",
            "3600π uses the DIAMETER as the radius: π(60²). The radius is half the diameter, so 30.",
            "450π halves the circle's area, counting only one of the two semicircles."
          ],
          "hint": "Two semicircles of the same diameter make exactly one circle. And check whether 60 is the radius or the diameter.",
          "diagram": "<svg viewBox=\"0 0 240 132\" width=\"100%\" style=\"max-width:240px; height:auto; display:block; margin:0 auto;\" role=\"img\"><line x1=\"60\" y1=\"40\" x2=\"180\" y2=\"40\" stroke=\"#7c3aed\" stroke-width=\"2\" stroke-linecap=\"round\"/><line x1=\"60\" y1=\"110\" x2=\"180\" y2=\"110\" stroke=\"#7c3aed\" stroke-width=\"2\" stroke-linecap=\"round\"/><path d=\"M 60,40 A 35 35 0 0 0 60,110\" fill=\"none\" stroke=\"#7c3aed\" stroke-width=\"2\"/><path d=\"M 180,40 A 35 35 0 0 1 180,110\" fill=\"none\" stroke=\"#7c3aed\" stroke-width=\"2\"/><line x1=\"60\" y1=\"40\" x2=\"60\" y2=\"110\" stroke=\"#6b6b76\" stroke-width=\"1.5\" stroke-dasharray=\"5 4\" stroke-linecap=\"round\"/><line x1=\"180\" y1=\"40\" x2=\"180\" y2=\"110\" stroke=\"#6b6b76\" stroke-width=\"1.5\" stroke-dasharray=\"5 4\" stroke-linecap=\"round\"/><text x=\"104\" y=\"80\" fill=\"#1f2024\" font-size=\"11\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">100 m</text><text x=\"186\" y=\"78\" 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\">60 m</text></svg>",
          "diagramCaption": "A rectangle capped by two semicircles. Not drawn to scale"
        },
        {
          "id": "apm-013",
          "topic": "Heron's Formula",
          "difficulty": "hard",
          "stem": "Find the area of a triangle with sides 5, 6 and 7 using Heron's formula.",
          "choices": [
            "6√6",
            "9",
            "15",
            "18"
          ],
          "correctIndex": 0,
          "explanations": [
            "Correct. The semi-perimeter is s = (5 + 6 + 7)/2 = 9. Then A = √(s(s−a)(s−b)(s−c)) = √(9 · 4 · 3 · 2) = √216 = 6√6 ≈ 14.7.",
            "9 is the semi-perimeter s, an intermediate value rather than the area.",
            "15 comes from ½ · 5 · 6, treating two of the sides as base and height. That only works when those sides are perpendicular, which they are not here.",
            "18 is the full perimeter, 5 + 6 + 7. Perimeter and area are different measurements."
          ],
          "hint": "Compute the semi-perimeter first, then subtract each side from it in turn before multiplying."
        },
        {
          "id": "apm-014",
          "topic": "Trapezoid Area",
          "difficulty": "medium",
          "stem": "A trapezoid has area 48, height 6, and one base of length 10. Find the other base.",
          "choices": [
            "6",
            "8",
            "16",
            "4"
          ],
          "correctIndex": 0,
          "explanations": [
            "Correct. From A = ½(b₁ + b₂)h we get 48 = ½(10 + b)(6) = 3(10 + b). So 16 = 10 + b and b = 6.",
            "8 is the average of the two bases, (10 + 6)/2. That average is an intermediate quantity, not the missing base.",
            "16 is the SUM of the two bases, b₁ + b₂. The sum still has to have the known base removed from it.",
            "4 comes from an arithmetic slip in isolating b. Work carefully: 48 divided by 3 gives 16, then subtract 10."
          ],
          "hint": "Substitute everything known into the area formula, then unwind it one operation at a time."
        },
        {
          "id": "apm-015",
          "topic": "Rhombus Area",
          "difficulty": "medium",
          "stem": "A rhombus has area 84 and one diagonal of length 12. Find the other diagonal.",
          "choices": [
            "7",
            "14",
            "12",
            "28"
          ],
          "correctIndex": 1,
          "explanations": [
            "7 is 84/12, which divides the area by the diagonal but skips the ½ in the formula. The ½ must be undone as well.",
            "Correct. For a rhombus (or kite), A = ½d₁d₂. So 84 = ½(12)d = 6d, giving d = 14.",
            "12 repeats the given diagonal. The two diagonals of a rhombus are generally different lengths — equal diagonals would make it a square.",
            "28 doubles the correct answer, applying the ½ in the wrong direction. Dividing 84 by 6 gives 14."
          ],
          "hint": "The diagonal formula carries a factor of ½. Account for it before dividing."
        },
        {
          "id": "apm-016",
          "topic": "Regular Polygon Area",
          "difficulty": "hard",
          "stem": "A regular hexagon has apothem 5√3 and perimeter 60. Find its area.",
          "choices": [
            "150√3",
            "300√3",
            "75√3",
            "60√3"
          ],
          "correctIndex": 0,
          "explanations": [
            "Correct. The area of a regular polygon is A = ½ · apothem · perimeter = ½(5√3)(60) = 150√3 ≈ 259.8.",
            "300√3 is apothem × perimeter with the ½ omitted. That doubles the true area.",
            "75√3 halves the correct answer, applying the ½ twice.",
            "60√3 multiplies the apothem by 12 rather than by half the perimeter. Substitute the given perimeter of 60 directly into the formula."
          ],
          "hint": "The regular-polygon area formula mirrors the triangle formula: half of one length times another."
        },
        {
          "id": "apm-017",
          "topic": "Equilateral Triangle Area",
          "difficulty": "medium",
          "stem": "Find the area of an equilateral triangle with side 6. (Use A = (√3/4)s².)",
          "choices": [
            "9√3",
            "36√3",
            "18√3",
            "6√3"
          ],
          "correctIndex": 0,
          "explanations": [
            "Correct. A = (√3/4)s² = (√3/4)(36) = 9√3 ≈ 15.6.",
            "36√3 substitutes s² = 36 but drops the /4. The denominator matters.",
            "18√3 divides by 2 instead of 4. The formula's denominator is 4, which comes from combining the ½ of the triangle area formula with the height (√3/2)s.",
            "6√3 uses s rather than s². The formula squares the side."
          ],
          "hint": "Square the side first, then apply the full coefficient including its denominator."
        },
        {
          "id": "apm-018",
          "topic": "Reasoning: Error Analysis",
          "difficulty": "medium",
          "stem": "A rectangle's dimensions are doubled. A student says the area doubles. What actually happens, and why?",
          "choices": [
            "The area doubles, since both dimensions changed together.",
            "The area quadruples, because both length and width are doubled and area is their product.",
            "The area is 8 times bigger.",
            "The area stays the same."
          ],
          "correctIndex": 1,
          "explanations": [
            "Doubling ONE dimension doubles the area. Doubling both multiplies it twice over.",
            "Correct. Area = length × width, so doubling each gives (2l)(2w) = 4lw — four times the original.",
            "8× is the volume effect for a three-dimensional figure, where three dimensions each double.",
            "Both dimensions grew, so the area cannot be unchanged."
          ],
          "hint": "Write the new area as a product and see how many factors of 2 appear.",
          "kind": "reasoning"
        },
        {
          "id": "apm-019",
          "topic": "Reasoning: What's Missing",
          "difficulty": "medium",
          "stem": "A student is asked for the area of a trapezoid given only the two parallel sides, and says it is impossible. Are they right?",
          "choices": [
            "No — the two parallel sides are enough.",
            "Yes. The height is also needed, since the formula averages the parallel sides and multiplies by the perpendicular distance between them.",
            "Yes, but only because the angles are missing.",
            "No — the perimeter can be used instead."
          ],
          "correctIndex": 1,
          "explanations": [
            "Two trapezoids can share both parallel sides and have completely different areas if one is taller.",
            "Correct. A = ½(b₁ + b₂)h. Without h there is no way to pin the area down, because the trapezoid could be any height.",
            "The angles are one way to RECOVER the height if a slant side is known, but the missing ingredient is the height itself.",
            "Perimeter does not determine area — shapes with the same perimeter can enclose very different amounts."
          ],
          "kind": "reasoning"
        }
      ]
    }
  }
}