M3a Angles and radians

In this chapter

• Say what an angle measures, and why a full turn is 360 degrees.
• Use π and the circumference \(C = 2\pi r\), and see how Archimedes trapped π between polygons.
• Find an arc length as a fraction of the circumference.
• Measure angles in radians, convert between degrees and radians, and use \(s = r\theta\).
• Use angular speed \(\omega\) and \(v = r\omega\) for things that turn.
• Use the small-angle rule, size ≈ distance × angle, to measure the Moon.

What an angle measures

What an angle measures

What an angle measures

Why 360? Nobody knows for certain. It most likely comes from Babylonian astronomy: they counted in sixties, and the Sun moves about 1/360 of the way round its yearly path each day. 360 also divides exactly by 24 different numbers (100 by only 9). The sixties live on in angles (1° = 60 minutes of arc, 1′ = 60 seconds of arc) and in clocks.

What an angle measures

Checkpoint: What an angle measures (1 of 2)

Checkpoint: What an angle measures (2 of 2)

Circumference and pi

Circumference and pi

A hexagon inside a circle has perimeter \(6r = 3d\), and the circle is longer, so π > 3. A square outside has perimeter \(4d\), so π < 4.

Circumference and pi

Circumference and pi

Archimedes doubled the polygons' sides up to 96 and proved \(3\tfrac{10}{71} < \pi < 3\tfrac{1}{7}\). Zu Chongzhi, in 5th-century China, pinned π between 3.1415926 and 3.1415927, and gave the fraction 355/113.

Circumference and pi

Circumference and pi

Checkpoint: Circumference and pi (1 of 2)

Checkpoint: Circumference and pi (2 of 2)

Arc length

Arc length

Arc length

Arc length

Arc length

Arc length

Checkpoint: Arc length (1 of 2)

Checkpoint: Arc length (2 of 2)

Radians

Radians

Radians

A full turn is \(2\pi r \div r = 2\pi\) radians.

Radians

Turned round, \(s = r\theta\): no 360, no 2π, but only with \(\theta\) in radians.

So a quarter turn, 90°, is \(\tfrac{\pi}{2}\) rad, a half turn is \(\pi\) rad, and three quarters of a turn is \(\tfrac{3\pi}{2}\) rad.

Radians

Radians

Radians

A radian is a length ÷ a length, so it's a pure number: metres × radians = metres.

Checkpoint: Radians (1 of 3)

Checkpoint: Radians (2 of 3)

Checkpoint: Radians (3 of 3)

Turning speed

Turning speed

Turning speed

Checkpoint: Turning speed (1 of 3)

Checkpoint: Turning speed (2 of 3)

Checkpoint: Turning speed (3 of 3)

Small angles

Small angles

Small angles

Checkpoint: Small angles (1 of 2)

Checkpoint: Small angles (2 of 2)

Angle mistakes

Checkpoint: Angle mistakes

Key terms

Angle

An amount of turning between two lines that meet at a point, the vertex.

Degree (°)

1/360 of a full turn. A right angle is 90°.

Radius, diameter, circumference

Centre to edge; straight across (\(2r\)); all the way round (\(2\pi r\)).

π (pi)

Circumference ÷ diameter, the same for every circle: 3.14159…

Key terms

Arc, arc length

A curved piece of a circle's edge, and how long it is.

Radian (rad)

The angle whose arc equals the radius; arc length ÷ radius. \(2\pi\) rad = 360°; 1 rad ≈ 57.3°.

Angular speed (\(\omega\))

Angle turned per second, in rad/s. A point at radius \(r\) moves at \(v = r\omega\).

Small-angle rule

For small angles, size ≈ distance × angle (in radians).

Why 360? (Babylon, first millennium BC)

Archimedes squeezes π (Syracuse, about 250 BC)

Zu Chongzhi (Jiankang, now Nanjing, 429–501)

Roger Cotes and James Thomson: the radian (1714–1873)

Tony Hawk's 900 (San Francisco, 27 June 1999)

Data

Eratosthenes (Alexandria, about 240 BC)

The problem

Problem. According to Cleomedes, Eratosthenes knew that at noon on midsummer's day the Sun was straight overhead at Syene (Aswan), while at Alexandria, about 5000 stadia to the north, the Sun's rays made an angle of 1/50 of a full circle with the vertical. How big is the Earth?

Solution

1. Parallel rays: the Sun is so far away that its rays arrive parallel. At Syene a ray points straight at the Earth's centre.
2. Equal angles: extend the pole at Alexandria to the centre. It crosses two parallel rays, and a line crossing two parallel lines makes equal angles with each, so the shadow angle at Alexandria equals the angle at the centre between Syene and Alexandria: \(7.2^\circ = 360^\circ/50\).

Solution

3. Multiply up: the 5000 stadia arc is 1/50 of the circumference: \(C = 50 \times 5000 = 250{,}000\) stadia. (In radians: \(7.2^\circ = 0.1257\) rad and \(R = s/\theta = 5000/0.1257 \approx 40{,}000\) stadia.)
4. Units: a stadion was somewhere between about 157.5 m and 185 m, and we don't know which Eratosthenes used. That gives 39,400 to 46,300 km.

Solution

5. Check: the modern value through the poles is 40,008 km: between almost exactly right and about 15% too big.
6. Caveats: Syene isn't exactly under the midsummer Sun, nor exactly south of Alexandria (about 3° of longitude off), and 5000 stadia is a round figure. Some errors partly cancel.

Solution

Myth: people thought the Earth was flat until Columbus. Educated people had known it was round since ancient Greece; the argument over Columbus's voyage was about its size. The flat-Earth story spread in the 1800s, starting with Washington Irving's romanticised biography of Columbus (1828).

Try it yourself: with a friend a few hundred kilometres north or south of you, mark the tip of an upright stick's shadow at local noon on the same day. Draw each stick and shadow to scale and measure the angle of the Sun's rays from the vertical with a protractor (after M3b you'll be able to calculate it). The difference of your angles is the angle at the Earth's centre; the circumference is your distance apart × 360° ÷ that difference.

★ Warm-up

1. Convert 60° and 270° to radians, and 1.2 rad to degrees.

   Your prediction or attempt, before you look

   Show solution

   Solution

   \(60^\circ = \pi/3 \approx 1.05\) rad; \(270^\circ = 3\pi/2 \approx 4.71\) rad; \(1.2 \times 180/\pi \approx 68.8^\circ\).

Problem 2 (★ Warm-up)

2. How far does a clock's hour hand turn between 9:00 and 13:30? Give the answer in degrees and radians.

   Your prediction or attempt, before you look

   Show solution

   Solution

   4.5 hours × 30° = 135°; \(135 \times \pi/180 = 3\pi/4 \approx 2.36\) rad.

Problem 3 (★ Warm-up)

3. A round table is 1.2 m across. What is its circumference? How long is the edge of one of six equal sectors?

   Your prediction or attempt, before you look

   Show solution

   Solution

   \(C = \pi d = \pi \times 1.2 \approx 3.77\) m. One sixth: \(3.77/6 \approx 0.63\) m (equal to the radius, 0.6 m, times \(\pi/3\)).

★★ Standard

4. Wheels. A wheel of radius 0.35 m turns three full turns without slipping. How far does it roll? What angle (in radians) must it turn through to roll 10 m?

   Your answer mCheck

   Solution

   \(\theta = 3 \times 2\pi \approx 18.85\) rad; \(s = r\theta = 0.35 \times 18.85 \approx 6.6\) m. For 10 m: \(\theta = s/r = 10/0.35 \approx 28.6\) rad, about 4.5 turns.

Problem 5 (★★ Standard)

5. A car's tyres. A car travels at 30 m/s on tyres of radius 0.31 m. What is the tyres' angular speed? How many turns a minute is that?

   Hint

   Rolling without slipping: the edge of the tyre moves at the car's speed, so \(\omega = v/r\).

   Your answer rad/sCheck

   Solution

   \(\omega = 30/0.31 \approx 96.8\) rad/s. Turns a second: \(96.8/2\pi \approx 15.4\); a minute: about 924.

Problem 6 (★★ Standard)

6. The London Eye. How far along its arc does a capsule travel in 12 minutes (the wheel is 120 m across and turns once in about 30 minutes)? What angle is that in radians?

   Your answer mCheck

   Solution

   12/30 = 0.4 of a turn: \(0.4 \times 2\pi \approx 2.51\) rad. Arc \(= r\theta = 60 \times 2.51 \approx 151\) m (or \(0.4 \times \pi \times 120\)).

Problem 7 (★★ Standard)

7. A pendulum. A pendulum 1.2 m long swings so its bob moves along a 0.30 m arc. What angle does the string turn through, in radians and degrees?

   Your answer radCheck

   Solution

   \(\theta = s/r = 0.30/1.2 = 0.25\) rad \(\approx 14.3^\circ\).

★★★ Stretch

8. The Sun's size. The Sun looks 0.53° across from 149.6 million km away. How wide is it?

   Your answer kmCheck

   Solution

   \(0.53^\circ = 0.00925\) rad; size \(\approx 149.6 \times 10^6 \times 0.00925 \approx 1.38\) million km (NASA: 1.39 million km).

Problem 9 (★★★ Stretch)

9. Hiding the Moon. How far from your eye must you hold a 2.3 cm coin so that it just covers the full Moon (0.52° across)?

   Your answer cmCheck

   Solution

   \(0.52^\circ = 0.00908\) rad; distance \(\approx 2.3/0.00908 \approx 253\) cm: about 2.5 m, much further than arm's length.

Problem 10 (★★★ Stretch)

10. An airliner overhead. An airliner about 70 m long flies 10 km overhead. What angle does it cover? Does it look bigger or smaller than the Moon?

    Your answer °Check

    Solution

    \(70/10{,}000 = 0.007\) rad \(\approx 0.40^\circ\): a little smaller than the Moon's 0.52°.

Problem 11 (★★★ Stretch)

11. The Earth's orbit. The Earth goes round the Sun (radius 149.6 million km) once in 365.25 days. What is its angular speed, and how fast does it move?

    Your answer rad/sCheck

    Solution

    \(\omega = 2\pi/(365.25 \times 86{,}400\ \text{s}) \approx 1.99 \times 10^{-7}\) rad/s. \(v = r\omega \approx 1.496 \times 10^{11} \times 1.99 \times 10^{-7} \approx 29{,}800\) m/s: about 30 km/s.

Problem 12 (★★★ Stretch)

12. A turntable. A record on a turntable spins at 45 turns a minute. What is its angular speed? How fast does a point 0.10 m from the centre move?

    Your answer rad/sCheck

    Solution

    45 turns a minute is 0.75 turns a second, so \(\omega = 0.75 \times 2\pi \approx 4.71\) rad/s. Then \(v = r\omega = 0.10 \times 4.71 \approx 0.47\) m/s.

Answer these out loud, from memory. Then try again in a few days.

Check yourself

Check yourself

Check yourself

Check yourself

Check yourself

Check yourself

Check yourself

Check yourself

Check yourself

Summary