Bearings and three-dimensional triangles

Bearings start at north and turn clockwise

A three-figure bearing of 065° is a clockwise turn of 65° from north. The reverse bearing is 245°, a half-turn away. A bearing of 300° points northwest; the smaller angle west of north is 60°, but the bearing remains 300°.

For a 10 km path on bearing 030°, the northward component is 10 cos 30° ≈ 8.66 km and the eastward component is 10 sin 30° = 5 km. The reference angle is from north, so the adjacent horizontal component is northward, not eastward.

Draw the angle between the actual paths

Two paths from a station have bearings 040° and 115° and lengths 6 km and 8 km. Their included angle at the station is 75°. The distance between their ends follows the cosine rule: d² = 6² + 8²-2(6)(8) cos 75°, so d ≈ 8.67 km.

For routes beginning at different points, first draw their north lines as parallel. The bearing difference alone may not be the triangle’s required interior angle.

A three-dimensional angle needs its plane

A rectangular room is 6 m by 8 m with height 4 m. The floor diagonal is square root of (6² + 8²) = 10 m. A space diagonal from one lower corner to the opposite upper corner then has length square root of (10² + 4²) = square root of 116 m.

The angle θ of that space diagonal to the horizontal floor plane lies between the diagonal and its floor projection. Hence tan θ = 4/10 and θ ≈ 21.8°. Using 6 m alone would choose a different vertical triangle and a different line.

Check your understanding 1 Name the projection first

A cuboid is 3 cm by 4 cm on its base and 12 cm high. Find its space diagonal and the angle it makes with the base plane.

Your answer or drawing on paper

Show answer

The base projection has length 5 cm. The space diagonal is square root of (5² + 12²) = 13 cm. The angle to the base satisfies tan θ = 12/5, giving about 67.4°. The angle is not measured from a vertical edge.

Check an elevation model

A tower’s top is observed from two ground points. A calculation needs not only both elevation angles, but also their ground separation and relative positions. Two angles alone do not determine the tower height; draw and label the known data before selecting a rule.