NCEA Level 1VolumeAS 91947

Volume: 7 exam questions solved step by step

What this lesson covers

  • A pyramid or cone holds one third of the prism or cylinder with the same base and height.
  • The volume of a composite solid is the sum or difference of the volumes of its parts.
Key idea

7 questions, one idea

  • Prism, cylinder: base area × height.
  • Pyramid, cone: ⅓ × base area × height.
  • Sphere: 4/3 πr³. Hemisphere: half.
  • Composite: add or take away the parts.
Your turn

Question 1 · Achieved

  • A triangular prism is 15 cm long.
  • Its end is a right-angled triangle,
  • sides 6 cm, 8 cm and 10 cm.
  • Find its volume.
Question 1 · Achieved
Worked example

Q1: triangular prism, 15 cm

  • Cross-section × length.
  • Use 6 and 8, not 10.
  1. V = A × lcross-section × length
  2. A = ½ × b × h
  3. A = ½ × 6 × 8 = 24 cm²sides at right angle
  4. V = 24 × 15
  5. V = 360 cm³
Your turn

Question 2 · Achieved

  • A square-based pyramid has a base
  • 9 cm by 9 cm, vertical height 10 cm.
  • Find its volume.
Worked example

Q2: square-based pyramid

  • A third of the prism.
  • Base area first.
  1. V = ⅓ × A × ha third of the prism
  2. A = 9 × 9 = 81 cm²
  3. V = ⅓ × 81 × 10
  4. 81 × 10 = 810the matching prism
  5. 810 ÷ 3 = 270
  6. V = 270 cm³
  7. 3 × 270 = 810
Your turn

Question 3 · Merit

  • A cone has base radius 6 cm
  • and slant height 10 cm.
  • Find its volume, to 1 d.p.
  • Careful: 10 is not the height you need.
Question 3 · Merit
Worked example

Q3: cone, r = 6, slant 10

  • Height first, by Pythagoras.
  • Then ⅓πr²h.
  1. h² + 6² = 10²Pythagoras, slant = 10
  2. h² = 100 − 36 = 64
  3. h = 8 cm
  4. V = ⅓πr²ha third of a cylinder
  5. V = ⅓ × π × 6² × 8
  6. = ⅓ × 288π = 96π cm³
  7. 96π = 301.592...
  8. V ≈ 301.6 cm³ (1 d.p.)
Your turn

Question 4 · Merit, in context

  • A marae kitchen bowl is a hemisphere,
  • inside diameter 18 cm.
  • How many litres does it hold, full?
  • Give your answer to 2 d.p.
Worked example

Q4: the bowl, in litres

  • Radius, not diameter.
  • 1 L = 1000 cm³.
  1. r = 18 ÷ 2 = 9 cmhalf the diameter
  2. V = ½ × 4/3 πr³ = ⅔πr³half a sphere
  3. V = ⅔ × π × 9³
  4. 9³ = 729, ⅔ × 729 = 486
  5. V = 486π cm³
  6. 486π = 1526.81... cm³
  7. 486π ÷ 1000 = 1.5268... L1 L = 1000 cm³
  8. The bowl holds about 1.53 L
Your turn

Question 5 · Merit, in context

  • A grain silo is a cylinder, radius 3 m,
  • height 7 m, with a hemisphere on top.
  • Find its volume, to 1 d.p.
Worked example

Q5: the silo

  • Find each part, then add.
  • Keep π in.
  1. V = cylinder + hemisphere
  2. Cylinder: π × 3² × 7πr²h
  3. = 63π m³
  4. Dome: ⅔ × π × 3³⅔πr³
  5. = 18π m³
  6. V = 63π + 18π = 81π m³
  7. 81π = 254.469...
  8. V ≈ 254.5 m³ (1 d.p.)
Your turn

Question 6 · Excellence

  • A cylindrical can is 10 cm tall
  • and must hold exactly 1 litre.
  • Find the radius it needs, to 1 d.p.
Worked example

Q6: working back to r

  • Units first: cm³.
  • Round only at the end.
  1. 1 L = 1000 cm³match the cm
  2. V = πr²h
  3. πr² × 10 = 1000
  4. r² = 1000/(10π) = 100/π÷ 10π both sides
  5. r² = 31.831...
  6. r = ±5.6419...√ both sides
  7. r = −5.6419...a radius is positive
  8. π × 5.6419² × 10 ≈ 1000
  9. r ≈ 5.6 cm (1 d.p.)
Your turn

Question 7 · Excellence

  • A ball fits exactly inside a can,
  • touching the sides, top and bottom.
  • What fraction of the can is empty?
  • Show it is the same for any size ball.
Worked example

Q7: the can and the ball

  • No numbers: use r.
  • Height = diameter = 2r.
  1. Ball radius = r
  2. Can: radius r, height 2rtouches all round
  3. Can: V = πr² × 2rπr²h
  4. = 2πr³
  5. Ball: V = 4/3 πr³sphere
Worked example

Q7: what fraction is empty?

  • Empty = can − ball.
  • r cancels: any size.
  1. Can 2πr³, ball 4/3 πr³
  2. Empty = 2πr³ − 4/3 πr³
  3. = 6/3 πr³ − 4/3 πr³ = ⅔πr³
  4. Fraction = ⅔πr³ ÷ 2πr³empty ÷ whole can
  5. = ⅔ ÷ 2 = ⅓πr³ cancels
  6. r = 3: 18π ÷ 54π = ⅓
  7. ⅓ of the can is empty, any r
Summary

What solved all seven

  • Formula, substitute, keep π, round last.
StepHow
Prismsbase area × height, πr²h
Pyramids⅓ × base area × height
Spheres4/3 πr³, hemisphere ⅔πr³
Compositeadd or take away the parts
Check firstradius, true height, units
Round lastkeep π in, then cm³, m³ or L

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