Angles in a Cuboid | Math Coach Amy

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Key Ideas Easy Medium Hard

AG = square root of l² + w² + h²

tan θ = hbase diagonal

What to Remember

Label the box

ABCD is the bottom rectangle and EFGH is the top. E is above A, F above B, G above C, and H above D.

AB is the length, BC is the width, and AE is the height. Diagonal AC crosses the base. AG is a space diagonal.

A right triangle on a face

A face diagonal is square root of length² + width² on that face.

The angle between that diagonal and an edge uses tangent: opposite over adjacent.

The space diagonal

AG = square root of AB² + BC² + AE²

The angle between AG and the base uses tan θ = heightbase diagonal.

Easy

A face diagonal

ABCD is the base. AC joins opposite corners of the base.

Example

A cuboid has AB = 8 cm, BC = 6 cm, and AE = 5 cm. Find the length of the base diagonal AC and the angle between AC and AB.

AC = square root of AB² + BC²

AC = square root of 8² + 6²

AC = square root of 100

AC = 10 cm

tan (angle between AC and AB) = BCAB

tan θ = 68

θ = 36.9°

Answer: AC = 10 cm, θ = 36.9°

A cuboid has AB = 9 cm, BC = 12 cm, and AE = 4 cm. Find AC and the angle between AC and AB.

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AC = square root of 9² + 12²

AC = 15 cm

tan θ = 129

θ = 53.1°

Answer: AC = 15 cm, θ = 53.1°

A cuboid has AB = 7 cm, BC = 24 cm, and AE = 5 cm. Find AC and the angle between AC and AB.

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AC = square root of 7² + 24²

AC = 25 cm

tan θ = 247

θ = 73.7°

Answer: AC = 25 cm, θ = 73.7°

A cuboid has AB = 15 cm, BC = 8 cm, and AE = 6 cm. Find AC and the angle between AC and AB.

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AC = square root of 15² + 8²

AC = 17 cm

tan θ = 815

θ = 28.1°

Answer: AC = 17 cm, θ = 28.1°

A cuboid has AB = 20 cm, BC = 21 cm, and AE = 9 cm. Find AC and the angle between AC and AB.

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AC = square root of 20² + 21²

AC = 29 cm

tan θ = 2120

θ = 46.4°

Answer: AC = 29 cm, θ = 46.4°

A cuboid has AB = 5 cm, BC = 12 cm, and AE = 3 cm. Find AC and the angle between AC and AB.

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AC = square root of 5² + 12²

AC = 13 cm

tan θ = 125

θ = 67.4°

Answer: AC = 13 cm, θ = 67.4°

Medium

The space diagonal and the base

AG runs from a bottom corner to the opposite top corner.

Example

A cuboid has AB = 6 cm, BC = 8 cm, and AE = 5 cm. Find the length of the space diagonal AG and the angle between AG and the base.

AC = square root of 6² + 8²

AC = 10.0 cm

AG = square root of AC² + AE²

AG = square root of 10.0² + 5²

AG = 11.2 cm

tan θ = AEAC

tan θ = 510.0

θ = 26.6°

Answer: AG = 11.2 cm, θ = 26.6°

A cuboid has AB = 9 cm, BC = 12 cm, and AE = 8 cm. Find the length of the space diagonal AG and the angle between AG and the base.

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AC = square root of 9² + 12²

AC = 15.0 cm

AG = square root of AC² + AE²

AG = square root of 15.0² + 8²

AG = 17.0 cm

tan θ = AEAC

tan θ = 815.0

θ = 28.1°

Answer: AG = 17.0 cm, θ = 28.1°

A cuboid has AB = 7 cm, BC = 4 cm, and AE = 4 cm. Find the length of the space diagonal AG and the angle between AG and the base.

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AC = square root of 7² + 4²

AC = 8.06 cm

AG = square root of AC² + AE²

AG = square root of 8.06² + 4²

AG = 9.00 cm

tan θ = AEAC

tan θ = 48.06

θ = 26.4°

Answer: AG = 9.00 cm, θ = 26.4°

A cuboid has AB = 10 cm, BC = 10 cm, and AE = 7 cm. Find the length of the space diagonal AG and the angle between AG and the base.

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AC = square root of 10² + 10²

AC = 14.1 cm

AG = square root of AC² + AE²

AG = square root of 14.1² + 7²

AG = 15.8 cm

tan θ = AEAC

tan θ = 714.1

θ = 26.3°

Answer: AG = 15.8 cm, θ = 26.3°

A cuboid has AB = 15 cm, BC = 20 cm, and AE = 12 cm. Find the length of the space diagonal AG and the angle between AG and the base.

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AC = square root of 15² + 20²

AC = 25.0 cm

AG = square root of AC² + AE²

AG = square root of 25.0² + 12²

AG = 27.7 cm

tan θ = AEAC

tan θ = 1225.0

θ = 25.6°

Answer: AG = 27.7 cm, θ = 25.6°

A cuboid has AB = 6 cm, BC = 8 cm, and AE = 5 cm. Find the angle between AG and AB.

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AG = square root of 6² + 8² + 5²

AG = 11.2 cm

cos θ = ABAG

cos θ = 611.2

θ = 57.5°

Answer: 57.5°

Hard

A longer path through the box

Find the third edge of the right triangle before you use tangent or cosine.

Example

In a cuboid, AB = 4 cm, BC = 7 cm, and AE = 9 cm. Find the length of AG and the angle between AG and the base.

AC = square root of 4² + 7²

AC = 8.06 cm

AG = square root of AC² + AE²

AG = square root of 8.06² + 9²

AG = 12.1 cm

tan θ = AEAC

tan θ = 98.06

θ = 48.1°

Answer: AG = 12.1 cm, θ = 48.1°

Three face diagonals of a cuboid meet at one corner and measure 10.0 cm, 12.0 cm, and 10.8 cm. The base uses the diagonal of length 10.0 cm. Find the space diagonal and the angle it makes with the base.

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Adding the three face-diagonal equations counts every edge twice.

2 AG² = (10.0)² + (12.0)² + (10.8)²

AG = 13.5 cm

The base diagonal is the face diagonal of the base.

Height² = 12 × ((12.0)² + (10.8)² − (10.0)²)

Height = 9.00 cm

tan θ = 9.0010.0

θ = 42.0°

Answer: AG = 13.5 cm, θ = 42.0°

A cuboid has AB = 5 cm, BC = 7 cm, and AE = 8 cm. Find the angle between AG and the base, and the angle between AG and AB.

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AC = square root of 5² + 7²

AC = 8.60 cm

AG = square root of AC² + AE²

AG = square root of 8.60² + 8²

AG = 11.7 cm

tan θ = AEAC

tan θ = 88.60

θ = 42.9°

AG = square root of 5² + 7² + 8²

AG = 11.7 cm

cos θ = ABAG

cos θ = 511.7

θ = 64.8°

Answer: with the base 42.9°, with AB 64.8°

A cuboid has AB = 12 cm, BC = 5 cm, and AE = 16 cm. Find the length of the space diagonal AG and the angle between AG and the base.

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AC = square root of 12² + 5²

AC = 13.0 cm

AG = square root of AC² + AE²

AG = square root of 13.0² + 16²

AG = 20.6 cm

tan θ = AEAC

tan θ = 1613.0

θ = 50.9°

Answer: AG = 20.6 cm, θ = 50.9°

A cuboid has AB = 9 cm, BC = 12 cm, and AE = 20 cm. Find the angle between AG and AB.

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AG = square root of 9² + 12² + 20²

AG = 25.0 cm

cos θ = ABAG

cos θ = 925.0

θ = 68.9°

Answer: 68.9°

A cuboid has AB = 11 cm, BC = 7 cm, and AE = 5 cm. Find the angle between the space diagonal AG and the edge AB.

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AG = square root of 11² + 7² + 5²

AG = 14.0 cm

cos θ = ABAG

cos θ = 1114.0

θ = 38.0°

Answer: 38.0°

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