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Measures and solids · Ages 14–15

Cones & spheres worksheet

Covers: Volume of a pyramid, Volume of a cone, Surface area of pyramids and cones, Volume of a sphere, Surface area of a sphere, Similar figures and solids, Solids at work, Folding a box

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Cones & spheres worksheet

Measures and solids · Ages 14–15

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Name:Date:Score:／48

Write each answer in the space. For word problems, show your working and write a sentence answer.

Part A: Working out and filling in (2 marks each)

  1. 1.The pyramid in the picture stands on a 6 cm × 8 cm rectangle and is 15 cm high. What is its volume, in cm³?

    Answer:

  2. 2.A cube of side 7 cm is split, by joining its centre to its eight corners, into 6 identical pyramids, each standing on one face of the cube. What is each pyramid's volume, in cm³?

    Answer in simplest form

    Answer:

  3. 3.The cone in the picture has a base radius of 15 cm and a height of 10 cm. Correct to 1 decimal place, what is its volume, in cm³?

    Answer:

  4. 4.The cone in the picture has a base radius of 36 cm and a slant height of 39 cm. What is its curved surface area, in terms of π?

    Answer:

  5. 5.Taking π as 3.14, a cone has a base radius of 2 cm and a slant height of 28 cm. What is its curved surface area, in cm²?

    Answer:

  6. 6.Taking π as 3.14, what is the volume of the sphere in the picture (radius 3 cm), in cm³?

    Answer:

  7. 7.The sphere in the picture is 30 cm across. What is its surface area, in terms of π?

    Answer:

  8. 8.Two similar cylinders have volumes in the ratio 27 : 64. The small one is 6 cm high. How high is the big one, in cm?

    Answer:

  9. 9.The frustum in the picture has a base radius of 12 cm, a top radius of 6 cm and a height of 4 cm. Put back the cone it was cut from: the small cone cut off is cm high, and the frustum's volume, in terms of π, is π cm³.

    Fill in each blank

    Answer:

Part B: Choose, or true or false (2 marks each)

  1. 10.The cone in the picture is 10 cm across its base and 9 cm high. What is its volume?

    • A. 300π cm³
    • B. 225π cm³
    • C. 15π cm³
    • D. 75π cm³

    Choose the right answer

    Answer:

  2. 11.The sphere in the picture is 30 cm across. What is its volume?

    • A. 36000π cm³
    • B. 300π cm³
    • C. 4500π cm³
    • D. 900π cm³

    Choose the right answer

    Answer:

  3. 12.A square sheet of side 13 cm has squares of side 2 cm cut from its corners and is folded into a box with no lid. How long is a side of the box's base?

    • A. 11 cm
    • B. 13 cm
    • C. 9 cm
    • D. 5 cm

    Choose the right answer

    Answer:

Part C: Word problems (4 marks each)

  1. 13.A chocolate is a pyramid on a square of side 4 cm, its volume 112 cm³. Find its height: put the steps in order.

    • A. Height = 336 ÷ 16 = 21 cm
    • B. Base area = 4 × 4 = 16
    • C. 13 × 16 × height = 112, so 16 × height = 336

    Write the letters in the right order

    Working:

    Answer:

  2. 14.A cone-shaped ice cream cone is 3 cm in radius at its mouth and 9 cm deep. It holds π cm³ in terms of π, and cm³ taking π as 3.14.

    Fill in each blank

    Working:

    Answer:

  3. 15.A round fish bowl, a sphere, holds 2304π cm³. Find its radius: put the steps in order.

    • A. 4 × r3 = 6912, so r3 = 1728
    • B. r = 12 cm
    • C. 43 × π × r3 = 2304π

    Write the letters in the right order

    Working:

    Answer:

  4. 16.A workshop makes 20 balls of radius 7 cm, each covered in leather. In terms of π, how many cm² of leather does it need in all?

    Working:

    Answer:

  5. 17.A model car is made to a scale of 1 : 50, and the model has 140 cm² to paint. How many m² does the real car have to paint?

    Working:

    Answer:

  6. 18.A cone-shaped container full of water (3 cm in radius at the top, 12 cm deep) is emptied into a cylindrical container of base radius 6 cm. How deep is the water, in cm?

    Working:

    Answer:

Show the answers and working

Cones & spheres worksheet: Answers and working

Part A: Working out and filling in

  1. 1. 240 cm³

    Base 6 × 8 = 48 cm², volume = 13 × 48 × 15 = 240 cm³.

  2. 2. 3436 cm³

    The cube 343 ÷ 6 = 3436 cm³; or base 72, height 3.5 (centre to face): 13 × 72 × 3.5 = 3436.

  3. 3. 2356.2 cm³

    13 × π × 152 × 10 with the calculator's π, rounded at the end: about 2356.2 cm³.

  4. 4. 1404 π cm²

    Curved surface = πrl = π × 36 × 39 = 1404π cm².

  5. 5. 175.84 cm²

    πrl = 3.14 × 2 × 28 = 175.84 cm².

  6. 6. 113.04 cm³

    43 × π × 33 = 36π; with 3.14, 113.04 cm³.

  7. 7. 900 π cm²

    Radius 15: 4 × π × 152 = 900π cm² (that is, π × diameter2).

  8. 8. 8 cm

    The cube root of the volume ratio gives the lengths', 3 : 4: 6 × 4 ÷ 3 = 8 cm.

  9. 9. 4, 336

    Similar: small height : big height = 6 : 12, and the big height = the small + 4, so the small cone is 4 high and the big 8; volume 13 × π × (122 × 8 − 62 × 4) = 336π.

Part B: Choose, or true or false

  1. 10. D. 75π cm³

    Radius 5: 13 × π × 52 × 9 = 75π cm³.

  2. 11. C. 4500π cm³

    Radius 15: 43 × π × 153 = 4500π cm³.

  3. 12. C. 9 cm

    13 − 2 × 2 = 9 cm.

Part C: Word problems

Marking: 2 for the working, 1 for the answer, 1 for the sentence.

  1. 13. B → C → A

    Height = 3 × volume ÷ base = 3 × 112 ÷ 16 = 21 cm.

  2. 14. 27, 84.78

    13 × π × 32 × 9 = 27π; × 3.14 = 84.78.

  3. 15. C → A → B

    Divide by π and multiply by 3: 4r³ = 6912, r³ = 1728, and the cube root is 12 cm.

  4. 16. 3920 π cm²

    Each is 4 × π × 72; 20 of them, 3920π cm².

  5. 17. 35 m²

    Areas grow 502 = 2500 times: 140 × 2500 cm²; 1 m² = 10 000 cm²: 35 m².

  6. 18. 1 cm

    The water is 13 × π × 32 × 12; ÷ the cylinder's base π × 62 = 1 cm.

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