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Amy Ferguson Moncure

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

Cones

Find the volume and surface area of a cone. The slant height comes from a right triangle.

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Key Ideas Easy Medium Hard

V = 13 π r² h

l = square root of r² + h²

Curved surface = π r l

What to Remember

Volume

V = 13 π r² h

r is the base radius and h is the perpendicular height.

Slant height

The radius, the height, and the slant height l make a right triangle.

l = square root of r² + h²

Surface area

Curved surface = π r l

Base = π r²

Total surface = π r l + π r²

Working backwards

If the volume and one measurement are known, rearrange for the missing one.

If a cone is melted into a sphere, the volume stays the same.

Easy

Volume

The height is perpendicular to the base.

Example
A cone has base radius 6 cm and height 9 cm. Find the volume.

V = 13 π r² h

V = 13 π (6)² (9)

V = 108π cm³

As a decimal, V is about 339 cm³

Answer: 108π cm³, about 339 cm³

1
A cone has base radius 4 cm and height 12 cm. Find the volume.
Show Answer

V = 13 π r² h

V = 13 π (4)² (12)

V = 64π cm³

As a decimal, V is about 201 cm³

Answer: 64π cm³, about 201 cm³

2
A cone has base radius 5 cm and height 6 cm. Find the volume.
Show Answer

V = 13 π r² h

V = 13 π (5)² (6)

V = 50π cm³

As a decimal, V is about 157 cm³

Answer: 50π cm³, about 157 cm³

3
A cone has base radius 8 m and height 15 m. Find the volume.
Show Answer

V = 13 π r² h

V = 13 π (8)² (15)

V = 320π m³

As a decimal, V is about 1010 m³

Answer: 320π m³, about 1010 m³

4
A cone has base radius 3 cm and height 10 cm. Find the volume.
Show Answer

V = 13 π r² h

V = 13 π (3)² (10)

V = 30π cm³

As a decimal, V is about 94.2 cm³

Answer: 30π cm³, about 94.2 cm³

5
A cone has base radius 7 cm and height 9 cm. Find the volume.
Show Answer

V = 13 π r² h

V = 13 π (7)² (9)

V = 147π cm³

As a decimal, V is about 462 cm³

Answer: 147π cm³, about 462 cm³

Medium

Slant height and surface area

Curved surface uses the slant height. Total surface adds the base.

Example
A cone has base radius 8 cm and height 15 cm. Find the slant height and the total surface area.

l = square root of r² + h²

l = square root of 8² + 15²

l = 17 cm

Curved surface = π r l

Curved surface = π (8) (17)

Curved surface = 136π cm²

Base = π r²

Base = 64π cm²

Total surface = curved surface + base

Total surface = 200π cm²

Answer: l = 17 cm, total surface = 200π cm²

1
A cone has base radius 5 cm and height 12 cm. Find the slant height and the total surface area.
Show Answer

l = square root of r² + h²

l = square root of 5² + 12²

l = 13 cm

Curved surface = π r l

Curved surface = π (5) (13)

Curved surface = 65π cm²

Base = π r²

Base = 25π cm²

Total surface = curved surface + base

Total surface = 90π cm²

Answer: l = 13 cm, total surface = 90π cm²

2
A cone has base radius 7 cm and height 24 cm. Find the slant height and the curved surface area.
Show Answer

l = square root of r² + h²

l = square root of 7² + 24²

l = 25 cm

Curved surface = π r l

Curved surface = π (7) (25)

Curved surface = 175π cm²

Answer: l = 25 cm, curved surface = 175π cm²

3
A cone has base radius 9 m and height 40 m. Find the slant height and the total surface area.
Show Answer

l = square root of r² + h²

l = square root of 9² + 40²

l = 41 m

Curved surface = π r l

Curved surface = π (9) (41)

Curved surface = 369π m²

Base = π r²

Base = 81π m²

Total surface = curved surface + base

Total surface = 450π m²

Answer: l = 41 m, total surface = 450π m²

4
A cone has base radius 20 cm and height 21 cm. Find the slant height and the curved surface area.
Show Answer

l = square root of r² + h²

l = square root of 20² + 21²

l = 29 cm

Curved surface = π r l

Curved surface = π (20) (29)

Curved surface = 580π cm²

Answer: l = 29 cm, curved surface = 580π cm²

5
A sculpture is a cone with base diameter 8.4 m and height 5.6 m. Find the volume to 3 significant figures.
Show Answer

V = 13 π r² h

V = 13 π (4.20)² (5.60)

V = 103 m³

Answer: 103 m³

Hard

Rearrange or melt

The volume does not change when a solid is melted and recast.

Example
A cone has slant height 25 cm and perpendicular height 24 cm. Find the base radius and the volume.

l² = r² + h²

25² = r² + 24²

r² = 625 − 576

r² = 49

r = 7 cm

V = 13 π r² h

V = 13 π (7)² (24)

V = 392π cm³

Answer: r = 7 cm, V = 392π cm³

1
A cone has volume 100π cm³ and base radius 5 cm. Find the height.
Show Answer

V = 13 π r² h

100 π = 13 π (5)² h

100 = 13 × 25 h

100 = 25 h3

h = 12 cm

Answer: 12 cm

2
A cone has base radius 4 cm and total surface area 80π cm². Find the height.
Show Answer

Total surface = π r l + π r²

80 π = π (4) l + π (4)²

80 = 4 l + 16

4 l = 64

l = 16 cm

l² = r² + h²

16² = 4² + h²

h² = 256 − 16

h² = 240

h = square root of 240 cm

h = 15.5 cm

Answer: 15.5 cm

3
A metal ornament is a cone with radius 6 cm and height 8 cm. It is melted into a sphere. Find the radius of the sphere to 3 significant figures.
Show Answer

V = 13 π r² h

V = 13 π (6)² (8)

V = 96π cm³

43 π R³ = 96 π

R³ = 96 × 34

R³ = 72

R = 4.16 cm

Answer: 4.16 cm

4
A cone has base radius 12 cm and height 5 cm. Find the volume.
Show Answer

V = 13 π r² h

V = 13 π (12)² (5)

V = 240π cm³

As a decimal, V is about 754 cm³

Answer: 240π cm³, about 754 cm³

5
A cone has base radius 15 cm and slant height 39 cm. Find the perpendicular height and the volume.
Show Answer

l² = r² + h²

39² = 15² + h²

h² = 1521 − 225

h² = 1296

h = 36 cm

V = 13 π r² h

V = 13 π (15)² (36)

V = 2700π cm³

As a decimal, V is about 8480 cm³

Answer: h = 36 cm, V = 2700π cm³, about 8480 cm³

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