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Amy Ferguson Moncure

IB AA · Unit 2 · 5C Applications of Trigonometry

Angles in a Pyramid and a Cone

Find the angle between a sloping edge and the base, and the angle between a face and the base.

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Key Ideas Easy Medium Hard

tan(edge) = heighthalf diagonal

tan(face) = heighthalf side

What to Remember

Apex above the center

O is the center of the base and M is the apex. OM is the perpendicular height.

The angle between a sloping edge and the base is the angle at the base corner, in the right triangle from the corner to O to M.

Square base

Half the diagonal of the base is 12 × side × square root of 2.

tan(edge angle) = heighthalf diagonal

tan(face angle) = heighthalf the side

The face angle uses the midpoint of a base edge, not the corner.

A cone

The angle between the sloping edge and the base uses tan θ = heightradius.

Easy

Square base

O is the center. The edge angle uses a corner. The face angle uses the midpoint of a side.

Example
A right pyramid has a square base of side 10 cm and height 12 cm. The apex is above the center of the base. Find the angle between a sloping edge and the base.

Half the base diagonal = 12 × 10 × square root of 2

Half the base diagonal = 7.07 cm

tan θ = heighthalf diagonal

tan θ = 127.07

θ = 59.5°

Answer: 59.5°

1
A right pyramid has a square base of side 8 cm and height 9 cm. The apex is above the center of the base. Find the angle between a sloping edge and the base.
Show Answer

Half the base diagonal = 12 × 8 × square root of 2

Half the base diagonal = 5.66 cm

tan θ = heighthalf diagonal

tan θ = 95.66

θ = 57.8°

Answer: 57.8°

2
A right pyramid has a square base of side 6 cm and height 8 cm. The apex is above the center of the base. Find the angle between a sloping edge and the base.
Show Answer

Half the base diagonal = 12 × 6 × square root of 2

Half the base diagonal = 4.24 cm

tan θ = heighthalf diagonal

tan θ = 84.24

θ = 62.1°

Answer: 62.1°

3
A right pyramid has a square base of side 14 cm and height 10 cm. Find the angle between a side face and the base.
Show Answer

Distance from the center to the midpoint of a side = 12 × 14

That distance = 7 cm

tan θ = heightthat distance

tan θ = 107

θ = 55.0°

Answer: 55.0°

4
A right pyramid has a square base of side 12 cm and height 5 cm. Find the angle between a side face and the base.
Show Answer

Distance from the center to the midpoint of a side = 12 × 12

That distance = 6 cm

tan θ = heightthat distance

tan θ = 56

θ = 39.8°

Answer: 39.8°

5
A right pyramid has a square base of side 16 cm and height 15 cm. The apex is above the center of the base. Find the angle between a sloping edge and the base.
Show Answer

Half the base diagonal = 12 × 16 × square root of 2

Half the base diagonal = 11.3 cm

tan θ = heighthalf diagonal

tan θ = 1511.3

θ = 53.0°

Answer: 53.0°

Medium

A cone or a rectangle

For a cone, the right triangle uses the radius and the height.

Example
A cone has diameter 10 cm and perpendicular height 12 cm. Find the angle between a sloping edge and the base.

tan θ = heightradius

tan θ = 125

θ = 67.4°

Answer: 67.4°

1
A cone has base radius 4 cm and height 7 cm. Find the angle between a sloping edge and the base.
Show Answer

tan θ = heightradius

tan θ = 74

θ = 60.3°

Answer: 60.3°

2
A cone has base radius 6 cm and height 8 cm. Find the angle between a sloping edge and the base.
Show Answer

tan θ = heightradius

tan θ = 86

θ = 53.1°

Answer: 53.1°

3
A cone has base radius 9 cm and height 12 cm. Find the angle between a sloping edge and the base.
Show Answer

tan θ = heightradius

tan θ = 129

θ = 53.1°

Answer: 53.1°

4
A right pyramid has a rectangular base 8 cm by 6 cm and height 7 cm. The apex is above the center. Find the angle between a sloping edge and the base.
Show Answer

Half diagonal = square root of (4)² + (3)²

Half diagonal = 5.00 cm

tan θ = 75.00

θ = 54.5°

Answer: 54.5°

5
A right pyramid has a rectangular base 8 cm by 6 cm and height 7 cm. Find the angle between the base and a face whose base edge is 8 cm.
Show Answer

The face along the 8 cm side is 3 cm from the center.

tan θ = 73

θ = 66.8°

Answer: 66.8°

Hard

The angle is given

Rearrange tangent to find the height or the base.

Example
A square-based pyramid has base side 12 cm. A sloping edge makes an angle of 58° with the base. Find the height.

Half diagonal = 12 × 12 × square root of 2

Half diagonal = 8.49 cm

tan 58° = h8.49

h = 8.49 × tan 58°

h = 13.6 cm

Answer: 13.6 cm

1
A square-based pyramid has base side 10 cm. A side face makes an angle of 62° with the base. Find the height.
Show Answer

tan 62° = h5

h = 5 × tan 62°

h = 9.40 cm

Answer: 9.40 cm

2
A right pyramid has an equilateral triangle base of side 9 cm and height 8 cm. Find the angle between a sloping edge and the base.
Show Answer

Distance from the center of an equilateral triangle to a vertex = 9square root of 3

That distance = 5.20 cm

tan θ = 85.20

θ = 57.0°

Answer: 57.0°

3
A right pyramid has an equilateral triangle base of side 6 cm and height 11 cm. Find the angle between a sloping edge and the base.
Show Answer

Distance from the center of an equilateral triangle to a vertex = 6square root of 3

That distance = 3.46 cm

tan θ = 113.46

θ = 72.5°

Answer: 72.5°

4
A square-based pyramid has height 9 cm. The angle between a side face and the base is 48°. Find the side of the base.
Show Answer

tan 48° = 9half side

half side = 9tan 48°

half side = 8.10 cm

side = 16.2 cm

Answer: 16.2 cm

5
A cone has height 15 cm. The sloping edge makes a 40° angle with the base. Find the base radius and the slant height.
Show Answer

tan 40° = 15r

r = 15tan 40°

r = 17.9 cm

sin 40° = 15l

l = 15sin 40°

l = 23.3 cm

Answer: r = 17.9 cm, l = 23.3 cm

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