Angles at work and play

You now know what angles are, how they pair up, and why the rules are true. This layer takes angles out of the textbook: into clocks and compasses, airports and cricket grounds, ramps, ladders, roofs and pie charts, and into the puzzles that appear in mathematics olympiads.

You will meet a formula that finds the angle between clock hands at any time, the exterior angle property of triangles, the bearings that pilots and sailors use, a completely different unit for angles called the radian, and a set of puzzles that need every tool you have. The layer ends with projects to try, careers that use angles every day, and questions nobody has fully answered.

At h hours and m minutes, measure both hands clockwise from 12.

• The minute hand turns 6° per minute, so it is at 6m degrees.
• The hour hand turns 30° per hour plus 0.5° per minute, so it is at 30h + 0.5m degrees.

The angle between them is the difference: (30h + 0.5m) − 6m = 30h − 5.5m. Since we only care about the size, we take the value without its sign:

angle = |30h − 5.5m|

The bars | | mean “ignore any minus sign” (the absolute value). If the answer is more than 180°, subtract it from 360° to get the smaller angle. Use h from 0 to 11, writing 12 o'clock as h = 0.

Extend one side of a triangle beyond a corner. The angle between the extended side and the next side is called an exterior angle of the triangle. The two angles of the triangle at the other corners are the interior opposite angles.

Exterior angle property: an exterior angle of a triangle equals the sum of the two interior opposite angles.

Proof. Let the triangle be ABC, and extend BC beyond C to D. Call the exterior angle ∠ACD.

1. ∠ACB + ∠ACD = 180° (linear pair on line BD).
2. ∠A + ∠B + ∠ACB = 180° (angle sum of a triangle).
3. Both expressions equal 180°, so ∠ACB + ∠ACD = ∠A + ∠B + ∠ACB.
4. Take ∠ACB away from both sides: ∠ACD = ∠A + ∠B. ∎

A handy consequence: an exterior angle is always bigger than each interior opposite angle.

Now take every exterior angle of a shape, one at each corner, all going the same way round. Remember the robot from Investigate, walking round a shape and turning at each corner? Its turns are exactly the exterior angles. It ends facing the way it started, so:

The exterior angles of any convex polygon add up to 360°.

For a triangle, check: each exterior angle is 180° minus the interior angle, so the three exterior angles total 3 × 180° − 180° = 360°. ✓ For a regular polygon with n sides, each exterior angle is 360° ÷ n, which gives a fast way to find the interior angle: 180° − 360° ÷ n. A regular 20-gon has exterior angles of 18° and interior angles of 162°.

Navigators give directions as bearings: an angle measured clockwise from north, always written with three figures. East is 090°, south is 180°, west is 270°, and north-east is 045°.

The back bearing (the way back) differs by exactly 180°: add 180° if the bearing is less than 180°, otherwise subtract 180°. If a ship sails from Kochi on a bearing of 250°, the bearing from the ship back to Kochi is 250° − 180° = 070°.

A real example: airport runways are numbered from their direction measured from magnetic north. ICAO's rule (Annex 14, §5.2.2.4) is that the two-digit number is the whole number nearest one tenth of that bearing: round the bearing to the nearest 10° and drop the final zero. A runway pointing roughly east, about 090°, is called runway 09. Its other end points roughly west, 270°, so it is runway 27. The two numbers at the ends of any runway always differ by 18, because the two directions differ by 180°. Many runways carry pairs like 09/27, 14/32 or 10/28.

Degrees are a human choice. Is there a more natural way to measure angles, one that does not depend on the Babylonians? Mathematicians found one.

Draw a circle. Take a piece of string as long as the radius and lay it along the edge of the circle. The angle at the centre made by that piece of arc is called one radian.

How many radians fit in a full turn? The distance round a circle (its circumference) is 2π × radius, where π ≈ 3.14159. So exactly 2π radians fit in a full turn: about 6.283 radians. That means:

• 360° = 2π radians, so 180° = π radians.
• 1 radian = 180° ÷ π ≈ 57.3°, a bit less than 60°.

Why bother? In higher mathematics and physics (the motion of pendulums, waves and planets), formulas become much simpler in radians. For example, the length of an arc is just radius × angle in radians. Your calculator has a “RAD” mode for exactly this.

These puzzles combine several angle facts. The key skill is angle chasing: finding one angle after another, each with a reason, until you reach the target. When stuck, try drawing an extra line: a parallel, a diagonal or an extension.