Chapter 14 · V · Mensuration

Math of Space: Surface Area and Volume Worksheet

13 questions Cuboids and cubes Cylinders and cones Pyramids Spheres and hemispheres Scaling and recasting

Try each question before opening its solution.

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Foundation

Cone dimensions and formulas · 1 practice mark

Question 1

For a right circular cone with radius 3 cm and height 4 cm, the slant height is:

  1. 1 cm
  2. 5 cm
  3. 7 cm
  4. 12 cm
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(b) 5 cm

  1. The radius, height and slant height form a right triangle.
  2. l = √(3² + 4²) = 5 cm.

Cuboids, cubes and their relationship · Scaling and systematic comparisons · 1 practice mark

Question 2

When all dimensions of a cuboid are multiplied by 2, its volume is multiplied by:

  1. 2
  2. 4
  3. 6
  4. 8
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(d) 8

  1. V = lwh.
  2. Each of the three factors doubles, so V becomes 2³V = 8V.

Sphere and hemisphere area and volume · Scaling and systematic comparisons · 1 practice mark

Question 3

Increasing a sphere’s radius by 10% increases its volume by:

  1. 10%
  2. 21%
  3. 30%
  4. 33.1%
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(d) 33.1%

  1. Volume is proportional to r³.
  2. The new factor is 1.1³ = 1.331; the increase is 0.331 = 33.1%.

Cone dimensions and formulas · 1 practice mark

Question 4

A conical cup is filled to half its vertical depth, measured from its apex. The water occupies what fraction of its capacity?

  1. 1/2
  2. 1/4
  3. 1/8
  4. 3/4
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(c) 1/8

  1. The water makes a similar cone with every linear dimension half the original.
  2. Its volume factor is (1/2)³ = 1/8.

Open hemisphere · 1 practice mark

Question 5

The inner area of an ideal thin open hemispherical bowl of radius r, excluding the rim, is:

  1. πr²
  2. 2πr²
  3. 3πr²
  4. 4πr²
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(b) 2πr²

  1. An open bowl has only its curved hemispherical surface, not a flat circular lid.
  2. Half a sphere’s surface area is 2πr².

Practice

Cuboids, cubes and their relationship · Sphere and hemisphere area and volume · 3 practice marks

Question 6

A solid sphere of radius 6 cm is melted without loss and recast as spheres of radius 2 cm. Find the number and compare the combined surface area before and after recasting.

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27 spheres; combined surface area becomes three times the original.

  1. Conserve volume: number = 6³/2³ = 27.
  2. Original area is 4π × 6² = 144π cm².
  3. New combined area is 27 × 4π × 2² = 432π cm². Volume is conserved, not surface area.

Cone dimensions and volume · 3 practice marks

Question 7

A right circular cone has radius 5 cm and vertical height 12 cm. Find its slant height, curved surface area and volume, leaving π exact.

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Slant height 13 cm; CSA 65π cm²; volume 100π cm³.

  1. Pythagoras gives l = √(5² + 12²) = 13 cm.
  2. CSA = πrl = π × 5 × 13 = 65π cm².
  3. V = (1/3)πr²h = (1/3)π × 25 × 12 = 100π cm³.

Challenge

Cuboids, cubes and their relationship · Cylinder area and volume · 4 practice marks

Question 8

Compare a cube of side 6 cm and a cuboid 9 cm × 6 cm × 4 cm. Find their volumes and total surface areas. State the cylinder formulas for radius r and height h.

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Both volumes 216 cm³; cube TSA 216 cm², cuboid TSA 228 cm². Cylinder CSA 2πrh, TSA 2πr(h + r), V = πr²h.

  1. Cube: V = 6³ = 216 and TSA = 6 × 6² = 216.
  2. Cuboid: V = 9 × 6 × 4 = 216 and TSA = 2(54 + 24 + 36) = 228.
  3. A cube is a cuboid whose length, width and height are equal. Equal volumes need not give equal surface areas.
  4. Unrolling a cylinder gives curved area circumference × height, 2πrh; add two bases for TSA. Volume is base area × height.

Pyramid base, apex, area and volume · Cone dimensions and formulas · 4 practice marks

Question 9

A right square pyramid has base side 6 cm, vertical height 4 cm, and triangular-face slant height 5 cm. Find its volume and total surface area. Identify its base and apex and compare its volume rule with a cone.

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Volume 48 cm³; total surface area 96 cm².

  1. The base is the square; the apex is the common vertex of the four triangular faces above it.
  2. Volume = (1/3) × base area × vertical height = (1/3) × 36 × 4 = 48 cm³.
  3. Each triangular face has area (1/2) × 6 × 5 = 15 cm². TSA = 36 + 4 × 15 = 96 cm².
  4. Both pyramid and cone volumes are one third of base area × vertical height; the cone uses circular base area πr².
Right square pyramid with vertical height four, face slant height five and half-base length threeh = 4slant = 5half side = 3apex

Sphere and hemisphere area and volume · Scaling and systematic comparisons · 4 practice marks

Question 10

For a sphere and a hemisphere each of radius 3 cm, find sphere surface area and volume, hemisphere curved area, total area and volume.

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Sphere: 36π cm² and 36π cm³. Hemisphere: CSA 18π cm², TSA 27π cm², volume 18π cm³.

  1. Sphere SA = 4πr² = 36π and V = (4/3)πr³ = 36π.
  2. Hemisphere CSA is half the sphere area, 2πr² = 18π.
  3. Include the circular base for hemisphere TSA: 3πr² = 27π.
  4. Hemisphere volume is half the sphere volume: (2/3)πr³ = 18π. Distinguish square and cubic units.

Cylinder area and volume · Cone dimensions and formulas · 4 practice marks

Question 11

A cylinder and cone have radius 3 cm and height 4 cm. Find each volume, cylinder total surface area and cone curved surface area. Compare their volumes.

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Cylinder V = 36π cm³, TSA = 42π cm²; cone V = 12π cm³, CSA = 15π cm²; ratio 3:1.

  1. Cylinder volume = π × 3² × 4 = 36π.
  2. Cylinder TSA = 2π × 3 × (4 + 3) = 42π.
  3. Cone slant height = √(3² + 4²) = 5; CSA = π × 3 × 5 = 15π.
  4. Cone volume = (1/3) × π × 3² × 4 = 12π; the volumes are in ratio 3:1.

Pyramid base, apex, area and volume · Scaling and systematic comparisons · 4 practice marks

Question 12

A regular square pyramid has base side 8 cm and face slant height 5 cm. Find its vertical height, volume and lateral surface area. What happens to volume and surface area when every length doubles?

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Height 3 cm; volume 64 cm³; lateral area 80 cm²; doubled size gives volume ×8 and area ×4.

  1. The face slant height, vertical height and half-base side form a right triangle: h² + 4² = 5², giving h = 3.
  2. Volume = (1/3) × 8² × 3 = 64 cm³.
  3. Four triangular faces give lateral area 4 × (1/2) × 8 × 5 = 80 cm².
  4. Uniform scale factor k multiplies area by k² and volume by k³; for k = 2 the factors are 4 and 8.

Regular square pyramid · 4 practice marks

Question 13

A regular square pyramid has base side 10 cm and vertical height 12 cm. Find its triangular-face slant height, total surface area and volume. Distinguish slant height from sloping edge length.

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Face slant height 13 cm; TSA 360 cm²; volume 400 cm³.

  1. From base centre to the midpoint of a side is 5 cm. Face slant height = √(12² + 5²) = 13 cm.
  2. Four triangular faces have combined area 4 × (1/2) × 10 × 13 = 260 cm². Add base 100 to obtain TSA 360 cm².
  3. Volume = (1/3) × 100 × 12 = 400 cm³.
  4. A sloping edge joins the apex to a base corner, not its side midpoint. Its length is √(12² + 5² + 5²) = √194 cm, different from 13 cm.