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Amy Ferguson Moncure

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

Pyramids

Find the volume of a pyramid from the base area and the height, then use a right triangle on a square base.

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Key Ideas Easy Medium Hard

V = 13 A h

h = square root of slant² − (half side)²

What to Remember

Volume

V = 13 A h

A is the area of the base. h is the perpendicular height from the apex to the base.

Square base and slant height

The slant height here is the height of a triangular face, measured from the midpoint of a base edge up to the apex.

Half the base side, the pyramid height, and that slant height make a right triangle.

h = square root of slant² − (half the side)²

Area of a face

Each side face is a triangle.

Area of one face = 12 × base edge × slant height of that face.

Given the volume

For a square base of side a, V = 13 a² h.

Rearrange before you substitute.

Easy

Base area given

Any base shape works once you know its area.

Example
A pyramid has base area 48 cm² and perpendicular height 9 cm. Find the volume.

V = 13 A h

V = 13 (48) (9)

V = 144 cm³

Answer: 144 cm³

1
A square-based pyramid has base area 36 cm² and height 15 cm. Find the volume.
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V = 13 A h

V = 13 (36) (15)

V = 180 cm³

Answer: 180 cm³

2
A triangular-based pyramid has base area 20 m² and height 12 m. Find the volume.
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V = 13 A h

V = 13 (20) (12)

V = 80 m³

Answer: 80 m³

3
A rectangular-based pyramid has base area 63 cm² and height 8 cm. Find the volume.
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V = 13 A h

V = 13 (63) (8)

V = 168 cm³

Answer: 168 cm³

4
A triangular-based pyramid has base area 45 cm² and height 6 cm. Find the volume.
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V = 13 A h

V = 13 (45) (6)

V = 90 cm³

Answer: 90 cm³

5
A square-based pyramid has base area 100 m² and height 21 m. Find the volume.
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V = 13 A h

V = 13 (100) (21)

V = 700 m³

Answer: 700 m³

Medium

Square base

The right triangle sits from the center of the base to the midpoint of a side.

Example
A square-based pyramid has base side 10 cm. The slant height of each side face is 13 cm. Find the volume and the total surface area.

h = square root of 13² − 5²

h = square root of 169 − 25

h = square root of 144

h = 12 cm

V = 13 A h

V = 13 (100) (12)

V = 400 cm³

Area of one face = 12 × 10 × 13

Area of one face = 65 cm²

Four faces = 260 cm²

Base = 100 cm²

Total surface = 360 cm²

Answer: V = 400 cm³, total surface = 360 cm²

1
A square-based pyramid has base side 6 cm. The slant height of each side face is 5 cm. Find the volume and the total surface area.
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h = square root of 5² − 3²

h = square root of 16

h = 4 cm

V = 13 A h

V = 13 (36) (4)

V = 48 cm³

Area of one face = 12 × 6 × 5

Area of one face = 15 cm²

Total surface = 96 cm²

Answer: V = 48 cm³, total surface = 96 cm²

2
A square-based pyramid has base side 8 cm. The slant height of each side face is 5 cm. Find the volume and the total surface area.
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h = square root of 5² − 4²

h = square root of 9

h = 3 cm

V = 13 A h

V = 13 (64) (3)

V = 64 cm³

Area of one face = 12 × 8 × 5

Area of one face = 20 cm²

Total surface = 144 cm²

Answer: V = 64 cm³, total surface = 144 cm²

3
A square-based pyramid has base side 18 cm. The slant height of each side face is 15 cm. Find the volume and the total surface area.
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h = square root of 15² − 9²

h = square root of 144

h = 12 cm

V = 13 A h

V = 13 (324) (12)

V = 1296 cm³

Area of one face = 12 × 18 × 15

Area of one face = 135 cm²

Total surface = 864 cm²

Answer: V = 1296 cm³, total surface = 864 cm²

4
A square-based pyramid has base side 16 cm. The slant height of each side face is 17 cm. Find the volume and the total surface area.
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h = square root of 17² − 8²

h = square root of 225

h = 15 cm

V = 13 A h

V = 13 (256) (15)

V = 1280 cm³

Area of one face = 12 × 16 × 17

Area of one face = 136 cm²

Total surface = 800 cm²

Answer: V = 1280 cm³, total surface = 800 cm²

5
A square-based pyramid has base side 14 cm. The slant height of each side face is 25 cm. Find the volume and the total surface area.
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h = square root of 25² − 7²

h = square root of 576

h = 24 cm

V = 13 A h

V = 13 (196) (24)

V = 1568 cm³

Area of one face = 12 × 14 × 25

Area of one face = 175 cm²

Total surface = 896 cm²

Answer: V = 1568 cm³, total surface = 896 cm²

Hard

Find a missing length

Start from V = 13 A h and rearrange.

Example
A square-based pyramid has height 9 cm and volume 432 cm³. Find the side of the base.

V = 13 a² h

432 = 13 a² (9)

432 = 3 a²

a² = 144

a = 12 cm

Answer: 12 cm

1
A square-based pyramid has height 15 cm and volume 500 cm³. Find the side of the base.
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V = 13 a² h

500 = 13 a² (15)

a² = 500 × 315

a² = 100

a = 10 cm

Answer: 10 cm

2
A square-based pyramid has base side 12 cm and height 8 cm. Find the volume.
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V = 13 a² h

a² = 144

V = 13 (144) (8)

V = 384 cm³

Answer: 384 cm³

3
A square-based pyramid has base side 12 cm. Each side face has area 60 cm². Find the volume and the total surface area.
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Area of one face = 12 × base × slant

60 = 12 × 12 × slant

60 = 6 slant

slant = 10 cm

h = square root of 10² − 6²

h = square root of 64

h = 8 cm

V = 13 (144) (8)

V = 384 cm³

Total surface = 144 + 4(60) = 384 cm²

Answer: V = 384 cm³, total surface = 384 cm²

4
A square-based pyramid has base side 14 cm and perpendicular height 24 cm. Find the volume and the slant height of a side face.
Show Answer

V = 13 a² h

V = 13 (196) (24)

V = 1568 cm³

slant² = h² + (half the side)²

slant² = 24² + 7²

slant² = 576 + 49

slant² = 625

slant = 25 cm

Answer: V = 1568 cm³, slant height = 25 cm

5
A square-based pyramid has volume 384 cm³ and base side 12 cm. Find the height, then the total area of the four side faces.
Show Answer

V = 13 a² h

384 = 13 (144) h

384 = 48 h

h = 8 cm

slant = square root of 8² + 6²

slant = square root of 100

slant = 10 cm

Four faces = 4 × 12 × 12 × 10

Four faces = 240 cm²

Answer: height = 8 cm, side faces = 240 cm²

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