Surface Areas and Volumes · Learner pages

Lesson 1 · Count only exposed surfaces

Build a surface-area sum after identifying hidden joins.

Name or initials: ____________   Date: ____________

A joined face inside a composite solid is not part of its exposed surface.

1. Identify the pieces. A cylinder with radius 3 cm and height 5 cm has a hemisphere of radius 3 cm attached on top. The bottom remains exposed.
2. Remove hidden faces. The circular join is internal. Neither the cylinder top nor the hemisphere base should be counted there.
3. Add the visible curved areas. Cylinder curve = 2πrh = 30π. Hemisphere curve = 2πr² = 18π.
4. Add the exposed bottom. Bottom area = πr² = 9π. Total exposed area = 30π + 18π + 9π = 57π cm².

Worked example: For a cone attached to a hemisphere with shared radius 4 cm and cone slant height 5 cm, the join is hidden. Exposed area = πrl + 2πr² = 20π + 32π = 52π cm².

1. A solid consists of a cylinder of radius 2 and height 5, with a hemisphere of the same radius on top. Its flat bottom is exposed. Find total exposed area.

   1. 32π · 2. 28π · 3. 36π

2. A solid consists of a cylinder of radius 3 and height 4, with a hemisphere of the same radius on top. Its flat bottom is exposed. Find total exposed area.

   1. 42π · 2. 60π · 3. 51π

3. A solid consists of a cylinder of radius 4 and height 6, with a hemisphere of the same radius on top. Its flat bottom is exposed. Find total exposed area.

   1. 112π · 2. 96π · 3. 80π

Discuss: Sketch a cylinder with both ends capped by hemispheres. List the surfaces that remain exposed and derive an area expression.

Lesson 2 · Add volumes of solid parts

Use each part’s formula and one consistent unit.

Name or initials: ____________   Date: ____________

Volume measures occupied three-dimensional space and uses cubic units.

1. Partition the object. A solid is a cylinder of radius 3 cm and height 6 cm joined to a hemisphere of radius 3 cm.
2. Compute the cylinder volume. V = πr²h = π × 9 × 6 = 54π cm³.
3. Compute the hemisphere volume. V = (2/3)πr³ = (2/3)π × 27 = 18π cm³.
4. Add and label cubic units. The interiors do not overlap, so total volume = 54π + 18π = 72π cm³. No face-area subtraction is needed for this zero-thickness join.

Worked example: A cylinder of radius 2 cm and height 9 cm has a cone of the same radius and height 3 cm attached. Total volume = 36π + (1/3)π×4×3 = 40π cm³.

1. A solid cylinder (radius 3, height 6) is joined to a hemisphere of radius 3 with no overlap. What is the combined volume?

   1. 72π · 2. 54π · 3. 90π

2. A solid cylinder (radius 6, height 3) is joined to a hemisphere of radius 6 with no overlap. What is the combined volume?

   1. 108π · 2. 396π · 3. 252π

3. A solid cylinder (radius 3, height 9) is joined to a hemisphere of radius 3 with no overlap. What is the combined volume?

   1. 117π · 2. 99π · 3. 81π

Discuss: Draw a composite object, label radius versus diameter and convert every dimension to centimetres before calculating volume.

Lesson 3 · Use volume conservation in an extension

Distinguish conserved volume from surface area during remoulding.

Name or initials: ____________   Date: ____________

Changing a shape without losing material preserves its volume, not generally its surface area.

1. State the modelling assumption. A sphere of radius 6 cm is remoulded into spheres of radius 2 cm without material loss. This is an extension application.
2. Write both volumes. Large sphere volume = (4/3)π×6³ = 288π. One small sphere volume = (4/3)π×2³ = 32π/3.
3. Take the volume ratio. Number = 288π ÷ (32π/3) = 27, equivalently (6/2)³.
4. Avoid an area assumption. The small spheres have more total exposed surface. Volume conservation does not mean equal surface areas. Real manufacturing losses would change the model.

Worked example: A cylinder of radius 3 cm and height 8 cm contains volume 72π cm³. If remoulded without loss into cones each with radius 3 cm and height 4 cm, each cone has 12π cm³, so 6 cones are made.

1. A sphere of radius 4 cm is remoulded into spheres of radius 2 cm with no loss. How many small spheres can be made?

   1. 8 · 2. 4 · 3. 2

2. A sphere of radius 6 cm is remoulded into spheres of radius 2 cm with no loss. How many small spheres can be made?

   1. 9 · 2. 3 · 3. 27

3. A sphere of radius 9 cm is remoulded into spheres of radius 3 cm with no loss. How many small spheres can be made?

   1. 3 · 2. 27 · 3. 9

Discuss: Compare total area before and after splitting one radius-4 sphere into radius-2 spheres. Explain why the area grows although volume stays equal.

Final check · Set 1

Try one set after the lessons. Use the other set after reviewing any errors.

1. A solid consists of a cylinder of radius 2 and height 9, with a hemisphere of the same radius on top. Its flat bottom is exposed. Find total exposed area.

   44π / 52π / 48π

2. A solid consists of a cylinder of radius 6 and height 7, with a hemisphere of the same radius on top. Its flat bottom is exposed. Find total exposed area.

   228π / 192π / 156π

3. A solid cylinder (radius 9, height 3) is joined to a hemisphere of radius 9 with no overlap. What is the combined volume?

   243π / 1215π / 729π

4. A solid cylinder (radius 3, height 12) is joined to a hemisphere of radius 3 with no overlap. What is the combined volume?

   144π / 126π / 108π

5. A sphere of radius 10 cm is remoulded into spheres of radius 2 cm with no loss. How many small spheres can be made?

   25 / 5 / 125

6. A sphere of radius 12 cm is remoulded into spheres of radius 4 cm with no loss. How many small spheres can be made?

   3 / 27 / 9

Final check · Set 2

Try one set after the lessons. Use the other set after reviewing any errors.

1. A solid consists of a cylinder of radius 3 and height 10, with a hemisphere of the same radius on top. Its flat bottom is exposed. Find total exposed area.

   87π / 78π / 96π

2. A solid consists of a cylinder of radius 4 and height 12, with a hemisphere of the same radius on top. Its flat bottom is exposed. Find total exposed area.

   128π / 160π / 144π

3. A solid cylinder (radius 6, height 9) is joined to a hemisphere of radius 6 with no overlap. What is the combined volume?

   468π / 324π / 612π

4. A solid cylinder (radius 9, height 6) is joined to a hemisphere of radius 9 with no overlap. What is the combined volume?

   486π / 1458π / 972π

5. A sphere of radius 15 cm is remoulded into spheres of radius 3 cm with no loss. How many small spheres can be made?

   125 / 25 / 5

6. A sphere of radius 6 cm is remoulded into spheres of radius 3 cm with no loss. How many small spheres can be made?

   4 / 2 / 8