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Measuring angles with a protractor

Geometry · level 3, lesson 2 of 6

A question to start

Skateboarders and snowboarders name their tricks by how far they spin. "A 180" means turning to face backwards. "A 360" means spinning all the way round. The very best riders land a "1080". How many times do they spin round in a 1080?

Those numbers are degrees, the units we use to measure angles. In this lesson you will learn what a degree is, how to measure one with a tool called a protractor, and why a full turn is 360 of them and not 100 or 10.

(Answer: 1080 is 360 three times, so a 1080 is three full spins.)

The idea

Why we need a unit

In the last lesson you sorted angles into families: acute, right, obtuse, straight, reflex. But "acute" covers a lot of angles. The tip of a needle and the corner of a slice of cake are both acute, and they are very different. To say exactly how big an angle is, we need a unit: a small standard turn that we can count, the way centimetres count length.

What a degree is

Cut a full turn into 360 equal tiny turns. Each one is called one degree, written 1°. The small circle is the sign for degrees, so 90° is read "ninety degrees".

One degree is a very small turn. If you turned one degree at a time, you would need 360 little steps to spin all the way round.

Now the turns you know get numbers:

Turn How many degrees Why
full turn 360° that is how we chose the unit
half turn (straight angle) 180° half of 360
quarter turn (right angle) 90° half of 180, or a quarter of 360
three quarter turn 270° 3 × 90

So the families have number ranges too:

  • acute: more than 0° and less than 90°
  • right: exactly 90°
  • obtuse: more than 90° and less than 180°
  • straight: exactly 180°
  • reflex: more than 180° and less than 360°

Why 360?

Nobody chose 360 last week. The number goes back thousands of years to the astronomers of ancient Babylon (in what is now Iraq). They counted in groups of 60 instead of groups of 10, and we still use their 60s every day: 60 seconds in a minute, 60 minutes in an hour. A full circle of 360 is six groups of 60. The story often told is that 360 also sits close to the number of days in a year (about 365), so the Sun seems to move round the sky by about one degree a day. Historians are not sure that was the reason, but it is a nice way to remember it.

Whatever the reason, 360 turned out to be a wonderful choice, because it can be shared out evenly in many ways. 360 can be divided exactly by 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120 and 180. That is 22 ways, not counting 1 and 360 themselves. So a half, a third, a quarter, a fifth, a sixth, an eighth, a tenth, a twelfth of a turn are all whole numbers of degrees.

90° 90° 90° 90° 4 × 90° = 360° 60° 60° 60° 60° 60° 60° 6 × 60° = 360° 45° 45° 45° 45° 45° 45° 45° 45° 8 × 45° = 360°

Compare 100. If a full turn were 100 units, a third of a turn would be 33.333... units, a number that never ends, and an eighth of a turn would be 12.5. With 360 you never meet that trouble for the common fractions of a turn.

Meet the protractor

A protractor is a see-through half circle (sometimes a full circle) with degrees marked round its curved edge. Along the straight edge is a small mark or hole at the middle: the centre. A line from the centre to each end is the zero line (also called the base line).

Most protractors have two scales, one on the inside and one on the outside. One scale counts from 0 on the right to 180 on the left; the other counts from 0 on the left to 180 on the right. That is so you can measure an angle that opens either way. It is also the cause of the most common mistake, so read the steps carefully.

180 0 170 10 160 20 150 30 140 40 130 50 120 60 110 70 100 80 90 90 80 100 70 110 60 120 50 130 40 140 30 150 20 160 10 170 0 180 centre on the vertex zero line on one arm

How to measure an angle, step by step

  1. Estimate first. Is the angle acute or obtuse? Roughly how big? Write your guess down. This is your safety net.
  2. Put the centre of the protractor exactly on the vertex.
  3. Turn the protractor so the zero line lies along one arm. The arm should pass through the 0 mark.
  4. Find the scale that starts at 0 on that arm. Follow that scale round, counting up from 0.
  5. Read the number where the other arm crosses the scale. If the arm is too short to reach the scale, extend it carefully with a ruler first. Making the arm longer does not change the angle.
  6. Check against your estimate. In the drawing, the arm crosses 50 on one scale and 130 on the other. The angle is clearly acute, so it is 50°, not 130°.

How to draw an angle of a given size

To draw an angle of 65°:

  1. Draw a straight line with a ruler: this is the first arm. Mark one end as the vertex.
  2. Put the protractor's centre on the vertex and its zero line along the arm.
  3. Using the scale that starts at 0 on your arm, count round to 65 and make a small dot.
  4. Take the protractor away and join the vertex to the dot with a ruler.
  5. Label the angle 65° and check: it should look a little less than three quarters of a right angle.

Measuring a reflex angle

A half-circle protractor only goes up to 180°. To measure a reflex angle, measure the other angle, the small one on the other side, and subtract it from 360°. The two together make a full turn, so

reflex angle = 360° − the small angle.

reflex: ? 130° measure the small angle: 130° the two make a full turn, 360° reflex angle = 360° − 130° = 230°

Estimating without a protractor

Good estimators keep a few benchmark angles in their heads: 90° (a square corner), 45° (half a square corner, the diagonal of a square), 30° and 60° (a right angle split into three, with 60° being two of the three parts), and 180° (a straight line). To estimate an angle, compare it with the nearest benchmark.

30° 45° 60° 90° 120° 150°

Worked examples

1. Clock angles in degrees. A clock face is a full turn, 360°, split into 12 hour marks. From one hour mark to the next is \(360 \div 12 = 30\) degrees. So at 4 o'clock the hands (the smaller angle) make \(4 \times 30 = 120\) degrees. At 2 o'clock they make \(2 \times 30 = 60\) degrees. At 6 o'clock they make \(6 \times 30 = 180\) degrees, a straight angle, which matches what you saw in the last lesson.

(This uses whole hours only. At half past, the hour hand has moved halfway to the next number, which changes the answer. Try it in the "Try it" questions.)

2. Reading the right scale. An angle opens to the left. Its first arm points left, along the left end of the zero line, and the other arm crosses the scales at 35 and 145. Which is it?

  1. The first arm passes through 0 on the scale that starts on the left. Use that scale.
  2. Counting up from that 0, the other arm is reached at 35.
  3. Check: the angle looks sharp, less than a right angle. So it is 35°, not 145°.

Notice that the two numbers 35 and 145 add to 180. That always happens, because the two scales run in opposite directions round the same half turn.

3. A reflex angle. You measure the small angle between two arms as 75°. What is the reflex angle the other way round? The two make a full turn, so it is \(360 - 75 = 285\) degrees.

4. The spins. A snowboarder lands a "540". How many full turns is that, and which way does she face at the end? \(540 = 360 + 180\), so it is one full turn and a half turn more: one and a half spins. She ends up facing the opposite way from where she started.

Try it

  1. How many degrees is: (a) a half turn; (b) a quarter turn; (c) three quarter turns; (d) two full turns?
  2. Name the family for each angle: 35°, 90°, 137°, 180°, 200°, 359°.
  3. On a protractor, an arm crosses the scales at 70 and 110. The angle is clearly obtuse. Which is the right reading?
  4. The small angle between two arms is 110°. What is the reflex angle?
  5. How many degrees between the hands of a clock at 5 o'clock? At 1 o'clock?
  6. Which of these is the angle a third of a full turn: 100°, 120° or 130°?
  7. A rider does a "720". How many full turns? Does she end up facing the same way?
  8. At half past 3, the minute hand points at 6 and the hour hand is halfway between 3 and 4. What angle do the hands make?
Answers
  1. (a) \(360 \div 2 = 180°\). (b) \(360 \div 4 = 90°\). (c) \(3 \times 90 = 270°\). (d) \(2 \times 360 = 720°\).
  2. 35°: acute. 90°: right. 137°: obtuse. 180°: straight. 200°: reflex. 359°: reflex (just one degree short of a full turn).
  3. 110°. Obtuse means more than 90°, and only 110 fits. (70 and 110 add to 180, as always.)
  4. \(360 - 110 = 250\) degrees.
  5. Each hour mark is 30°. At 5 o'clock: \(5 \times 30 = 150°\). At 1 o'clock: \(30°\).
  6. \(360 \div 3 = 120\), so 120°.
  7. \(720 = 2 \times 360\): two full turns. Yes, after whole turns you face the way you started.
  8. From the 6 back to the 4 is 2 hour marks, \(2 \times 30 = 60°\). The hour hand is halfway between 3 and 4, so it is half a mark (15°) short of the 4. The angle from the hour hand to the 6 is \(60 + 15 = 75\) degrees.

Common mistake

Reading the wrong scale. The arm crosses two numbers, and they always add to 180. Pick the scale that starts at 0 on the first arm, and then check against your estimate: an acute angle must read less than 90, an obtuse angle more than 90.

Not putting the centre on the vertex. If the centre mark is even a little off the vertex, the reading is wrong. Line up the centre first, then turn the protractor about that point until the zero line lies along an arm.

Where it leads