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Amy Ferguson Moncure

IB AA · Unit 2 · 5A Volumes and Surface Area

Compound Solids

Add volumes of pieces that are joined, and leave every hidden face out of the surface area.

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Key Ideas Easy Medium Hard

V = π r² h

Frustum = large cone − small cone

Leave the join out of the surface area

What to Remember

Add or subtract the volumes

A solid built from two pieces uses the sum of the volumes.

A hole uses the difference.

Do not paint the join

Where two pieces meet, that face is inside the solid.

Leave it out of the surface area. Keep every face that is still on the outside.

A cap on a cylinder

If a hemisphere or a cone covers one end of a cylinder, drop that circular end from the cylinder’s surface.

Read the height carefully when the problem gives the total height of the whole solid.

A frustum

A frustum is a cone with a smaller cone cut off the top, parallel to the base.

The two cones are similar, so the radii are in the same ratio as the heights.

Volume of the frustum = large volume − small volume.

Easy

Volumes of two pieces

Add a cone or a hemisphere onto a cylinder.

Example
A solid is a cylinder of radius 4 cm and height 10 cm with a solid hemisphere of radius 4 cm on one end. Find the volume.

Cylinder: V = π r² h

V = π (4)² (10)

V = 160π cm³

Hemisphere: V = 23 π r³

V = 23 π (4)³

V = 128π3 cm³

Total volume = cylinder + hemisphere

Total volume = 608π3 cm³

Answer: 608π3 cm³

1
A solid is a cylinder of radius 3 cm and height 8 cm with a cone of the same radius and height 6 cm on one end. Find the volume.
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Cylinder: V = π r² h

V = π (3)² (8)

V = 72π cm³

Cone: V = 13 π r² h

V = 13 π (3)² (6)

V = 18π cm³

Total = 90π cm³

Answer: 90π cm³

2
A solid is a cylinder of radius 5 cm and height 12 cm with a hemisphere of radius 5 cm on one end. Find the volume.
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Cylinder: V = π r² h

V = π (5)² (12)

V = 300π cm³

Hemisphere: V = 23 π r³

V = 23 π (5)³

V = 250π3 cm³

Total = 1150π3 cm³

Answer: 1150π3 cm³

3
A solid is a cylinder of radius 6 cm and height 10 cm with a cone of the same radius and height 8 cm on one end. Find the volume.
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Cylinder: V = π r² h

V = π (6)² (10)

V = 360π cm³

Cone: V = 13 π r² h

V = 13 π (6)² (8)

V = 96π cm³

Total = 456π cm³

Answer: 456π cm³

4
A solid is a cylinder of radius 2 cm and height 15 cm with a hemisphere of radius 2 cm on one end. Find the volume.
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Cylinder: V = π r² h

V = π (2)² (15)

V = 60π cm³

Hemisphere: V = 23 π r³

V = 23 π (2)³

V = 16π3 cm³

Total = 196π3 cm³

Answer: 196π3 cm³

5
Two solid hemispheres, of radii 4 cm and 7 cm, are joined at their flat faces. Find the volume.
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V = 23 π r³ for each hemisphere

First = 128π3 cm³

Second = 686π3 cm³

Total = 814π3 cm³

Answer: 814π3 cm³

Medium

Outside surface

A covered end is not part of the outside.

Example
A cylinder has radius 4 cm and height 10 cm. A hemisphere of the same radius covers one end. Find the total surface area of the outside.

Cylinder curved surface = 2 π r h

Cylinder curved surface = 2 π (4) (10)

Cylinder curved surface = 80π cm²

Open bottom of the cylinder = π r²

Bottom = 16π cm²

Hemisphere curved surface = 2 π r²

Hemisphere curved surface = 32π cm²

The covered end is inside the solid, so leave it out.

Total surface = 128π cm²

Answer: 128π cm²

1
A post is a cylinder with a hemisphere on top. The diameter is 14 cm and the total height of the post is 50 cm. Find the volume.
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Radius = 12 × 14 = 7 cm

Cylinder height = 50 − 7 = 43 cm

Cylinder: V = π (7)² (43)

V = 2107π cm³

Hemisphere: V = 23 π (7)³

V = 686π3 cm³

Total = 7007π3 cm³

As a decimal, the total is about 7340 cm³

Answer: 7007π3 cm³, about 7340 cm³

2
A tablet is a cylinder of radius 3 cm and length 10 cm with a hemisphere of the same radius on each end. Find the volume and the outside surface area.
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Two hemispheres make one sphere.

Sphere: V = 43 π (3)³

V = 36π cm³

Cylinder: V = π (3)² (10)

V = 90π cm³

Total volume = 126π cm³

Outside surface = cylinder curved + sphere surface

Outside surface = 2 π (3) (10) + 4 π (3)²

Outside surface = 96π cm²

Answer: V = 126π cm³, A = 96π cm²

3
A cylinder of radius 5 cm and height 8 cm has a cone of height 12 cm on top. The cone has the same radius. Find the outside surface area.
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l = square root of 5² + 12²

l = 13 cm

Cylinder curved = 2 π (5) (8) = 80π cm²

Bottom = π (5)² = 25π cm²

Cone curved = π (5) (13) = 65π cm²

The shared circle is inside the solid.

Total surface = 170π cm²

Answer: 170π cm²

4
Two solid hemispheres of radii 5 cm and 8 cm are joined at their flat faces. Find the outside surface area.
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Each flat face is hidden.

Outside = 2 π (5)² + 2 π (8)²

Outside = 178π cm²

Answer: 178π cm²

5
A cone has height 8 cm and base radius 6 cm. A hemispherical hole of radius 2 cm is bored into the center of the base. Find the volume and the outside surface area.
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Cone slant = square root of 6² + 8² = 10 cm

Cone volume = 13 π (6)² (8) = 96π cm³

Hole volume = 23 π (2)³ = 16π3 cm³

Volume of the solid = 272π3 cm³

Cone total surface before the hole = π (6) (10) + π (6)² = 96π cm²

The hole removes a circle of area 4π cm² from the base.

The curved wall of the hole adds 2 π (2)² = 8π cm²

Outside surface = 100π cm²

Answer: V = 272π3 cm³, A = 100π cm²

Hard

Frustum and a change of size

Similar cones share the ratio of their heights.

Example
A cone has base radius 9 cm and height 12 cm. A smaller cone of height 4 cm is cut off the top by a slice parallel to the base. Find the volume of the frustum that remains.

The small cone is similar to the large cone.

Small radius = 9 × 412

Small radius = 3 cm

Large volume = 13 π (9)² (12)

Large volume = 324π cm³

Small volume = 13 π (3)² (4)

Small volume = 12π cm³

Frustum = 312π cm³

Answer: 312π cm³

1
A cone has base radius 6 cm and height 8 cm. A smaller cone of height 4 cm is cut off the top, parallel to the base. Find the volume and the total surface area of the solid frustum.
Show Answer

The small cone is similar to the large cone.

Small radius = 6 × 48

Small radius = 3 cm

Large volume = 13 π (6)² (8)

Large volume = 96π cm³

Small volume = 13 π (3)² (4)

Small volume = 12π cm³

Frustum = 84π cm³

Large slant = square root of 6² + 8² = 10 cm

Small slant = square root of 3² + 4² = 5 cm

Frustum slant = 10 − 5 = 5 cm

Curved surface = π (6 + 3) (5)

Curved surface = 45π cm²

Two bases = 45π cm²

Total surface = 90π cm²

Answer: V = 84π cm³, A = 90π cm²

2
A cylinder has height 12 cm and volume 192π cm³. A cone has the same radius and the same volume. Find the height of the cone.
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Cylinder: π r² h = 192 π

π r² (12) = 192 π

12 r² = 192

r² = 16

r = 4 cm

Cone: 13 π r² h = 192 π

13 (16) h = 192

16 h3 = 192

16 h = 576

h = 36 cm

Answer: 36 cm

3
A capsule is a cylinder of radius 3 cm and length 10 cm with a hemisphere on each end. A new capsule has the same shape and 90% of the volume. Find the total length of the new capsule to 3 significant figures.
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The original total length is the cylinder plus two radii.

Total length = 10 + 3 + 3 = 16 cm

Volume scale = 0.90

Length scale³ = 0.90

Length scale = 0.965

New length = 16 × 0.965

New length = 15.4 cm

Answer: 15.4 cm

4
Two solid hemispheres are joined at their flat faces. The radii are 6 cm and 9 cm. Find the volume and the outside surface area.
Show Answer

Volume = 23 π (6)³ + 23 π (9)³

Volume = 630π cm³

The flat faces are hidden.

Surface = 2 π (6)² + 2 π (9)²

Surface = 234π cm²

Answer: V = 630π cm³, A = 234π cm²

5
A fence cap is a cylinder of radius 5 cm and length 40 cm, with a hemisphere of the same radius on one end and a cone of height 12 cm on the other. Find the volume of metal in 200 caps.
Show Answer

Cylinder = π (5)² (40) = 1000π cm³

Hemisphere = 23 π (5)³ = 250π3 cm³

Cone = 13 π (5)² (12) = 100π cm³

One post = 3550π3 cm³

200 posts = 710000π3 cm³

As a decimal, 200 posts measure about 744000 cm³

Answer: 710000π3 cm³, about 744000 cm³

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