Degrees
One angle is here and another is on a page across town. How can you say exactly how far each one opens, with a number that anyone could check?
Where you are 5 in Measuring angles
Why this exists
You can often tell which of two angles is bigger by laying one on top of the other (comparing angles by fitting). That only works when the two can be brought together, and it only ever answers “bigger” or “smaller”. To say how much bigger, or to write an angle down so that someone far away can draw the same one, you need a number, and a number needs a unit.
Length already solved this. A centimeter is a small fixed length, and a length measures however many centimeters fit along it end to end. Angles need the same thing: a small fixed amount of turn that can be counted. The unit in everyday use is the degree, and it is built from the one amount of turn that nobody has to measure: the full turn, where an arm swings all the way round and lands back where it started.
Definition
The size of an angle is the amount of turn from one of its arms to the other (an angle is an amount of turn).
Cut a full turn into 360 equal turns. Each one is a one-degree angle, and its measure is one degree, written 1^\circ. An angle that turns through n one-degree angles has a measure of n degrees, written n^\circ.
So a full turn measures 360^\circ, and one degree is \displaystyle \frac{1}{360} of a full turn.
The small raised circle is the degree sign: 90^\circ is read “ninety degrees”. The measure of \angle ABC is written m\angle ABC, so m\angle ABC = 50^\circ says that the angle turns through fifty one-degree angles. Naming angles explains why the angle and its measure get different symbols.
How small a single degree is, and how any number of degrees turns into a fraction of a turn, is the part below called one degree as a fraction of a turn. Why the full turn is cut into 360 parts, rather than 100 or 10, is a choice made long ago; the part called why a full turn is three hundred sixty degrees tells that story and gives the reason the choice has lasted.
Intuition
Turns you already know
Figure 1 puts numbers on four turns. A full turn is 360^\circ. A half turn is half of that, 180^\circ: the moving arm ends up pointing the opposite way, and the two arms make one straight line. A quarter turn is 90^\circ: the arms make a square corner, a right angle. Three quarter turns make 270^\circ.
Counting degrees
One degree is a very thin angle, since 360 of them fit round a single point. Figure 2 groups them in tens: the full turn is cut into 36 equal slices, and each slice is 10^\circ because 36 \times 10 = 360. The five shaded slices together make an angle of 50^\circ, since they hold 5 \times 10 = 50 one-degree angles.
That is what measuring an angle means, exactly as for a length: choose a unit, then count how many copies of it fit into the thing being measured, edge to edge, with no gaps and no overlaps. The count has nothing to do with how long the arms are drawn. A bigger circle in Figure 2 would give longer slices, but still 36 of them (longer arms do not make a bigger angle).
One degree as a fraction of a turn
The degree is the unit that gets counted, so it pays to see one on its own. Figure 3 cuts a right angle, the corner of a page, into one-degree angles. There are 90 of them, because a quarter turn is 360 \div 4 = 90 degrees. The single shaded slice is one degree: it is \displaystyle \frac{1}{90} of a right angle and \displaystyle \frac{1}{360} of a full turn.
An angle of n^\circ is n of those slices laid side by side, each sharing an arm with the next. Since each slice is \displaystyle \frac{1}{360} of a turn, n of them make n lots of \displaystyle \frac{1}{360}:
So, for two angles met already:
The first is the quarter turn of Figure 1, and the second is five of the 36 slices in Figure 2.
The fact also runs backwards: a fraction \displaystyle \frac{a}{b} of a turn is \displaystyle \frac{a}{b} \times 360 degrees.
How wide a degree looks on paper depends on how far out you look, even though the angle itself does not. The gap between its two arms grows as the arms get longer. Take a circle whose rim is 360 centimeters long, which is a circle with a radius of about 57 centimeters, roughly an arm’s length. The 360 one-degree slices cut that rim into 360 equal pieces, each 1 centimeter long. So a thing 1 centimeter wide, held at arm’s length, fills about one degree of what you see. On a circle with a rim half as long, each degree takes half a centimeter of rim.
Why a full turn is three hundred sixty degrees
Nothing about circles forces the number 360. Any number of equal parts could have been chosen, and other units do cut the turn differently (the gradian of Exercise 33 uses 400). The degree goes back more than two thousand years, to the astronomers of ancient Babylon, who counted in sixties rather than in tens. Their sixties still live in the 60 minutes of an hour and the 60 seconds of a minute. Why they cut the turn into 360 parts in particular, nobody knows for certain. One popular guess goes through the sky: the Sun seems to travel once round the sky, against the stars, in a year of about 365 days, so it moves a little under one of these small steps a day, and 360 = 6 \times 60 is a round number in sixties that sits close to 365. That story may be right, but it is a guess; the counting in sixties is not.
Whatever its origin, 360 has stayed because so many whole numbers divide it. Here are all of its factors:
That is 24 factors, and no whole number smaller than 360 has as many. Each factor b means a full turn can be cut into b equal parts that are each a whole number of degrees. Every number from 1 to 10 divides 360 except 7, and so does 12. Halves, thirds, quarters, fifths, sixths, eighths, ninths, tenths and twelfths of a turn all come out whole. Compare a turn cut into 100 equal units instead:
| Equal parts | Each part, with 360 to a turn | Each part, with 100 to a turn |
|---|---|---|
| 2 | 180 | 50 |
| 3 | 120 | \displaystyle 33\frac{1}{3} |
| 4 | 90 | 25 |
| 5 | 72 | 20 |
| 6 | 60 | \displaystyle 16\frac{2}{3} |
| 8 | 45 | \displaystyle 12\frac{1}{2} |
| 9 | 40 | \displaystyle 11\frac{1}{9} |
| 10 | 36 | 10 |
| 12 | 30 | \displaystyle 8\frac{1}{3} |
With 360 every row is a whole number; with 100 only four are. That matters wherever a circle is shared out evenly: the twelve hours on a clock face sit exactly 30^\circ apart, and the corners of a square, a regular hexagon or an equilateral triangle all come out in whole degrees.
Results
A full turn is 360^\circ, and the other familiar turns follow from it:
| Turn | Measure |
|---|---|
| a full turn | 360^\circ |
| three quarters of a turn | 270^\circ |
| a half turn (a straight angle) | 180^\circ |
| a quarter turn (a right angle) | 90^\circ |
| one degree | \displaystyle \frac{1}{360} of a full turn |
Part of a turn. When a full turn is cut into b equal parts, each part measures 360^\circ \div b, and a of those parts together measure a \times (360^\circ \div b).
Degrees as a fraction of a turn. An angle of n^\circ is \displaystyle \frac{n}{360} of a full turn.
Whole-degree parts. A full turn cuts into b equal parts of a whole number of degrees exactly when b is a factor of 360. There are 24 such b; among 2 to 12, only 7 and 11 fail.
Why it is true
That a full turn is 360^\circ needs no proof: it is how the degree was defined.
The quarter turn follows. Four quarter turns, all the same size, make one full turn, so together they hold 360 one-degree angles and each holds a quarter of them:
Two quarter turns make a half turn, 90 + 90 = 180, and three make 3 \times 90 = 270. The same reasoning works for any number of equal parts: if b equal turns fill a full turn, each holds 360 \div b one-degree angles, and a of them hold a times as many.
Degrees as a fraction of a turn. Each one-degree angle is one of 360 equal parts of a full turn, so it is \displaystyle \frac{1}{360} of the turn. An angle of n^\circ is made of n of them with no gaps and no overlaps, so it is
of the turn.
Whole-degree parts. Each of b equal parts is 360 \div b degrees, and that is a whole number exactly when b divides 360. Break 360 into primes:
A factor of 360 is built from some of these: none to three 2s, none to two 3s, and the 5 or not (every whole number breaks into primes in only one way). That is 4 \times 3 \times 2 = 24 choices, which is why there are 24 factors. The primes 7 and 11 are not in the list, so neither divides 360, and neither does any number with 7 or 11 among its primes.
Worked examples
Degrees in part of a turn
Find the measure of (a) a third of a turn, (b) a twelfth of a turn, (c) two thirds of a turn.
Solution(a) Three equal thirds make a full turn of 360^\circ, so each third holds a third of the degrees: 360 \div 3 = 120. A third of a turn is 120^\circ.
(b) Twelve equal parts fill 360^\circ, so each is 360 \div 12 = 30 degrees: 30^\circ. This is the angle between neighboring numbers on a clock face, which has twelve of them evenly spaced round a full turn.
(c) Two thirds of a turn is two of the thirds from (a): 2 \times 120 = 240, so 240^\circ.
Check. Twelve angles of 30^\circ make 12 \times 30 = 360 degrees, a full turn.
What part of a turn
What fraction of a full turn is (a) 45^\circ, (b) 60^\circ?
SolutionAsk how many copies of the angle fill a full turn.
(a) 360 \div 45 = 8, so eight angles of 45^\circ fill a full turn exactly. Each is one of eight equal parts: 45^\circ is \displaystyle \frac{1}{8} of a turn.
(b) 360 \div 60 = 6, so 60^\circ is \displaystyle \frac{1}{6} of a turn.
Check. 45^\circ is half of a right angle. Eight halves of a right angle are four right angles, and four right angles make a full turn.
Any angle as a fraction of a turn
(a) What fraction of a full turn is 1^\circ? (b) What fraction is 135^\circ? (c) What fraction is 25^\circ? (d) How many degrees is \displaystyle \frac{7}{20} of a full turn?
SolutionCopies of 135^\circ or 25^\circ do not fill a full turn a whole number of times, so the method of Example 2 stops working. Count one-degree angles instead.
(a) One degree is one of the 360 equal parts of a full turn, so it is \displaystyle \frac{1}{360} of a turn.
(b) 135^\circ is 135 one-degree angles, each \displaystyle \frac{1}{360} of a turn, so it is \displaystyle \frac{135}{360} of a turn. Both numbers are multiples of 45, since 135 = 3 \times 45 and 360 = 8 \times 45, so the fraction simplifies to \displaystyle \frac{3}{8}.
(c) In the same way 25^\circ is \displaystyle \frac{25}{360} of a turn. Dividing top and bottom by 5 gives \displaystyle \frac{5}{72}.
(d) A twentieth of a turn is 360 \div 20 = 18 degrees, and seven of them make 7 \times 18 = 126, so 126^\circ.
Check. In (b), three eighths of a turn is three of the 45^\circ parts of Example 2, and 3 \times 45 = 135. In (d), \displaystyle \frac{126}{360} simplifies, dividing by 18, back to \displaystyle \frac{7}{20}.
Which equal parts come out whole
For which numbers b from 2 to 12 is a bth of a full turn a whole number of degrees? What is a seventh of a turn? How would the answer to the first question change if a full turn were cut into 100 equal units instead of 360?
SolutionA bth of a turn is 360 \div b degrees, which is a whole number exactly when b divides 360. Working through: 2, 3, 4, 5, 6 divide 360 (giving 180, 120, 90, 72, 60), 7 does not, 8, 9, 10 do (giving 45, 40, 36), 11 does not, and 12 does (giving 30). So every b from 2 to 12 works except 7 and 11.
A seventh of a turn: 7 \times 51 = 357, which leaves 3 over, so \displaystyle 360 \div 7 = 51\frac{3}{7}. A seventh of a turn is \displaystyle 51\frac{3}{7} degrees, not a whole number.
With 100 units to a turn, b has to divide 100. Of 2 to 12 only 2, 4, 5 and 10 do. A third of a turn alone would be \displaystyle 33\frac{1}{3} units. This is the reason given in the part on why a full turn is three hundred sixty degrees: 360 shares out evenly in many more ways.
Check. \displaystyle 7 \times 51\frac{3}{7} = 357 + 3 = 360.
The minute hand
The minute hand of a clock goes once round the face in 60 minutes. (a) Through how many degrees does it turn in one minute? (b) In 20 minutes? (c) How many minutes does it take to turn 90^\circ?
Solution(a) The hour is sixty equal one-minute turns that together make a full turn, so each is 360 \div 60 = 6 degrees. The minute hand turns 6^\circ a minute.
(b) Twenty minutes is twenty of those turns: 20 \times 6 = 120, so 120^\circ. That agrees with Example 1: 20 minutes is a third of an hour, and a third of a turn is 120^\circ.
(c) Each minute gives 6^\circ, so 90^\circ takes 90 \div 6 = 15 minutes. A quarter of an hour gives a quarter turn, as it should.
The angle between the hands
The hour hand of a clock goes once round the face in 12 hours. Find the angle between the hour hand and the minute hand at (a) 4 o’clock, (b) 9 o’clock, (c) half past two.
SolutionThe twelve numbers cut the face into twelve equal parts, so neighboring numbers are 30^\circ apart (Example 1).
(a) At 4 o’clock the minute hand points at the 12 and the hour hand at the 4. Between them are four of the twelfth parts: 4 \times 30 = 120, so the angle is 120^\circ.
(b) At 9 o’clock the minute hand points at the 12 and the hour hand at the 9. Going clockwise from the 12 to the 9 passes nine parts, 9 \times 30 = 270 degrees, but going the other way passes only three, 3 \times 30 = 90 degrees. The two hands are two rays from the center, and the angle between two rays is by agreement the opening that is at most a half turn (see Boundaries). So the angle between the hands is 90^\circ: they make a square corner.
(c) At half past two the minute hand has gone half way round, so it points at the 6, which is 6 \times 30 = 180 degrees clockwise from the 12. The hour hand is not on the 2. In one hour it moves from one number to the next, 30^\circ, so in half an hour it has moved half of that, 30 \div 2 = 15 degrees, past the 2. It is 2 \times 30 + 15 = 75 degrees clockwise from the 12. The angle between the hands is 180 - 75 = 105 degrees, so 105^\circ (Figure 4).
Check. In (b), the two ways round make up the whole face: 270 + 90 = 360. In (c), the other way round between the hands is 360 - 105 = 255 degrees, more than a half turn, so 105^\circ is the right one.
How far a wheel rolls
A bicycle wheel has a rim 216 centimeters long all the way round. As the wheel rolls without slipping along a straight road, each bit of rim touches the road once in each turn, so the bicycle moves forward by the length of rim that has come down onto the road. (a) How far does the bicycle move while the wheel turns 1^\circ? (b) While it turns 150^\circ? (c) Through how many degrees does the wheel turn while the bicycle moves 30 centimeters? (d) And while it moves 540 centimeters?
Solution(a) In one full turn the whole rim, 216 centimeters, comes down onto the road. One degree is \displaystyle \frac{1}{360} of a turn, so it brings down \displaystyle \frac{1}{360} of the rim: 216 \div 360 = 0.6 centimeters.
(b) 150^\circ is 150 one-degree turns, each moving the bicycle 0.6 centimeters: 150 \times 0.6 = 90 centimeters.
(c) Each degree gives 0.6 centimeters, so 30 centimeters takes 30 \div 0.6 = 50 degrees: 50^\circ.
(d) 540 \div 0.6 = 900, so the wheel turns through 900^\circ. That is more than a full turn: 900 = 2 \times 360 + 180, two and a half turns. As a turn it is 900^\circ, even though the spoke that started pointing down ends pointing straight up, at 180^\circ from where it began (see Boundaries).
Check. In (d), two and a half turns of a 216 centimeter rim is 2.5 \times 216 = 540 centimeters.
Boundaries
By agreement, the measure of an angle is at most a half turn. A turn and an angle between two rays are not quite the same thing. A wheel that goes round twice turns through 360^\circ + 360^\circ = 720^\circ, but it ends pointing the way it started, so as two rays it shows no opening at all. Two rays from one point also split the full turn into two openings, like the 270^\circ and 90^\circ in Example 6. The usual convention is that m\angle ABC means the opening between \overrightarrow{BA} and \overrightarrow{BC} that is no bigger than a half turn, so m\angle ABC is never more than 180^\circ. Bigger amounts are described as turns.
Not every angle is a whole number of degrees. The hour hand turns 30^\circ in 60 minutes, so it turns \displaystyle 30 \div 60 = \frac{1}{2} of a degree each minute, and a seventh of a turn is \displaystyle 51\frac{3}{7} degrees (Example 4). Parts of a degree are as real as parts of a centimeter, and they have smaller units of their own (degrees, minutes and seconds). An angle read off a drawing is also only as exact as the drawing; a measure stated in a problem is exact.
The degree is a choice, not a fact about circles. Any fixed amount of turn could serve as the unit, and 360 is a convention, kept for its many factors rather than forced by anything about a circle. Later mathematics mostly uses the radian, a unit tied to the length of an arc of a circle, with a full turn of 2\pi radians.
Where this goes next
To measure an angle that is already drawn, the counting comes printed on a protractor, a half turn marked off in single degrees: reading a protractor learns the tool, the two scales on a protractor explains its two rows of numbers, and drawing an angle of a given size runs it the other way.
Once an angle has a number, angles can be sorted by size into acute, right, obtuse and straight, judged by eye before they are measured (estimating an angle), and put together: angle measures add when an angle is split into parts, and that gives a missing angle from a known total such as a right angle or a straight line. The wheel of Example 7, where a fraction of a turn brings down the same fraction of the rim, grows into the radian. Much further on, Bearings use degrees to name a direction, counting clockwise from north.
Exercises
Turns and degrees
Find the measure, in degrees, of each turn.
- A half turn.
- A tenth of a turn.
- A fifth of a turn.
- Three quarters of a turn.
- Five sixths of a turn.
What fraction of a full turn is each angle?
- 90^\circ
- 20^\circ
- 150^\circ
- How many angles of 15^\circ fit round one point, edge to edge with no gaps, to make a full turn?
- A wheel turns round exactly twice. Through how many degrees has it turned?
One degree as a fraction of a turn
- What fraction of a right angle is 1^\circ?
- What fraction of a full turn is 225^\circ? Give it in simplest form.
- What fraction of a full turn is 162^\circ? Give it in simplest form.
- A circle has a rim 360 centimeters long. How much of the rim lies between the arms of a 1^\circ angle at its center? Of a 40^\circ angle?
- A wheel has a rim 90 centimeters long and rolls along the ground without slipping. How far does it roll while it turns 1^\circ? While it turns 100^\circ?
- The distance round the Earth through both poles is about 40{,}000 kilometers. Two towns lie on that circle, and the turn at the Earth’s center from one to the other is 1^\circ. About how far apart are they, to the nearest kilometer?
Why three hundred sixty
- Which of these is a whole number of degrees: a fifteenth of a turn, a sixteenth, an eighteenth, a twenty-fifth, a forty-fifth? Give each one’s size.
- Find a seventh of a turn in degrees, as a mixed number.
- Suppose a full turn were cut into 100 equal units instead. How many units would a third of a turn be? A sixth?
- A full turn is to be cut into two or more equal parts, each a whole number of degrees and each at least 10^\circ. In how many ways can this be done?
- The Sun seems to travel once round the sky in about 365 days. About how many degrees does it move in one day, to two decimal places?
The clock
The minute hand of a clock goes once round in 60 minutes.
- Through how many degrees does the minute hand turn in 5 minutes?
- Through how many degrees does the minute hand turn in 45 minutes?
- How many minutes does the minute hand take to turn through 150^\circ?
The hour hand goes once round in 12 hours.
- Through how many degrees does the hour hand turn in one hour?
- Through how many degrees does the hour hand turn in 20 minutes?
Find the angle between the hour hand and the minute hand at each time.
- 3 o’clock
- 5 o’clock
- 8 o’clock
- Half past six
- Twenty past three
Reasoning
- One angle of 30^\circ is drawn with arms 2 centimeters long and another with arms 20 centimeters long. Explain why they have the same measure.
- The gradian is another unit of angle, used in some surveying. It cuts a full turn into 400 equal parts. How many gradians make a right angle? A right angle is then described by a different number from 90; explain why it is still the same angle.
- Someone says the angle between the hands of a clock at 9 o’clock is 270^\circ. What have they counted, and what is the angle between the hands?
- An arm turns 400^\circ counterclockwise from where it started. Where does it end up, and what angle does it make with its starting position?
- Without dividing, explain why a fifth of a turn is a whole number of degrees but a seventh of a turn is not.
- Someone suggests cutting a full turn into 100 degrees, because hundreds are easier to count in. Give one thing that would become easier and one that would become harder.
Answers
- 180^\circ. Two equal halves fill 360^\circ, and 360 \div 2 = 180.
- 36^\circ. 360 \div 10 = 36.
- 72^\circ. 360 \div 5 = 72.
- 270^\circ. A quarter turn is 360 \div 4 = 90 degrees, and three of them make 3 \times 90 = 270.
- 300^\circ. A sixth of a turn is 360 \div 6 = 60 degrees, and five of them make 5 \times 60 = 300.
- \displaystyle \frac{1}{4}. 360 \div 90 = 4, so four such angles fill a full turn.
- \displaystyle \frac{1}{18}. 360 \div 20 = 18, so eighteen such angles fill a full turn.
- \displaystyle \frac{5}{12}. One twelfth of a turn is 360 \div 12 = 30 degrees, and 150 \div 30 = 5, so 150^\circ is five twelfths.
- 24. 360 \div 15 = 24.
- 720^\circ. Each full turn is 360^\circ, and 2 \times 360 = 720.
- \displaystyle \frac{1}{90}. A right angle holds 90 one-degree angles, as in Figure 3.
- \displaystyle \frac{5}{8}. 225^\circ is \displaystyle \frac{225}{360} of a turn, and dividing top and bottom by 45 gives \displaystyle \frac{5}{8}.
- \displaystyle \frac{9}{20}. 162^\circ is \displaystyle \frac{162}{360} of a turn; divide top and bottom by 18.
- 1 centimeter and 40 centimeters. The 360 one-degree slices cut the rim into 360 equal pieces of 360 \div 360 = 1 centimeter, and 40^\circ takes 40 of them.
- 0.25 centimeters and 25 centimeters. Each degree brings \displaystyle \frac{1}{360} of the rim down onto the ground: 90 \div 360 = 0.25, and 100 \times 0.25 = 25.
- About 111 kilometers. One degree takes \displaystyle \frac{1}{360} of the circle: 40000 \div 360 = 111.1\ldots
- A fifteenth is 24^\circ, an eighteenth is 20^\circ and a forty-fifth is 8^\circ, all whole. A sixteenth is \displaystyle 22\frac{1}{2} degrees and a twenty-fifth is \displaystyle 14\frac{2}{5} degrees, not whole, since 16 and 25 do not divide 360.
- \displaystyle 51\frac{3}{7} degrees. 7 \times 51 = 357, leaving 3, so \displaystyle 360 \div 7 = 51\frac{3}{7}.
- \displaystyle 33\frac{1}{3} units and \displaystyle 16\frac{2}{3} units. \displaystyle 100 \div 3 = 33\frac{1}{3} and \displaystyle 100 \div 6 = 16\frac{2}{3}; with 360 to a turn they would be 120 and 60.
- 15 ways. The number of parts b must be a factor of 360, and each part, 360 \div b degrees, is at least 10 exactly when b is at most 36. The factors from 2 to 36 are 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30 and 36, giving parts of 180^\circ, 120^\circ, 90^\circ, 72^\circ, 60^\circ, 45^\circ, 40^\circ, 36^\circ, 30^\circ, 24^\circ, 20^\circ, 18^\circ, 15^\circ, 12^\circ and 10^\circ.
- About 0.99 degrees, just under one degree a day. 360 \div 365 = 0.986\ldots
- 30^\circ. The minute hand turns 360 \div 60 = 6 degrees a minute, and 5 \times 6 = 30.
- 270^\circ. 45 \times 6 = 270. Forty-five minutes is three quarters of an hour, and three quarters of a turn is 270^\circ.
- 25 minutes. At 6^\circ a minute, 150 \div 6 = 25.
- 30^\circ. One hour is a twelfth of the full turn: 360 \div 12 = 30.
- 10^\circ. The hour hand turns 30^\circ in 60 minutes, so \displaystyle \frac{1}{2} of a degree a minute, and \displaystyle 20 \times \frac{1}{2} = 10.
- 90^\circ. The minute hand is on the 12 and the hour hand on the 3, three twelfths apart: 3 \times 30 = 90.
- 150^\circ. Five twelfths apart: 5 \times 30 = 150, which is less than 180.
- 120^\circ. Clockwise from the 12 to the 8 is 8 \times 30 = 240 degrees, more than a half turn, so the angle is the other way round: 360 - 240 = 120, or four twelfths, 4 \times 30 = 120.
- 15^\circ. The minute hand points at the 6, 180^\circ clockwise from the 12. The hour hand has gone half of the 30^\circ from the 6 to the 7, so it is at 6 \times 30 + 15 = 195 degrees. The angle between them is 195 - 180 = 15.
- 20^\circ. The minute hand points at the 4, 20 \times 6 = 120 degrees clockwise from the 12. The hour hand has moved 10^\circ past the 3 (Exercise 26), so it is at 3 \times 30 + 10 = 100 degrees. The angle between them is 120 - 100 = 20.
- The measure counts the one-degree angles between the two arms, which is the amount of turn from one arm’s direction to the other’s. Lengthening the arms leaves both directions as they were, so the turn between them, and so the measure, is unchanged.
- 100 gradians, since a right angle is a quarter turn and 400 \div 4 = 100. The angle itself is a quarter turn whatever unit counts it, just as a length stays the same whether it is measured in centimeters or in inches. Only the size of the unit changed, so only the count changed.
- They counted clockwise from the minute hand on the 12 to the hour hand on the 9: 9 \times 30 = 270 degrees. Counted the other way round the gap is 3 \times 30 = 90 degrees. By agreement the angle between two rays is the opening that is at most a half turn, so the angle between the hands is 90^\circ.
- It makes a full turn and then 400 - 360 = 40 degrees more, so it ends 40^\circ counterclockwise from its start. The angle between the finishing position and the starting position is 40^\circ.
- 360 = 2 \times 2 \times 2 \times 3 \times 3 \times 5. The 5 is one of its primes, so 5 divides 360 and five equal parts are each a whole number of degrees. The prime 7 is not there, and a whole number breaks into primes in only one way, so 7 does not divide 360 and a seventh of a turn leaves a fraction of a degree.
- Easier: a quarter turn would be 25, a fifth 20, a tenth 10, all simple numbers, and the count would fit counting in tens. Harder: 3 does not divide 100, so a third of a turn would be \displaystyle 33\frac{1}{3} and a twelfth \displaystyle 8\frac{1}{3}, and the hours on a clock face, the corners of an equilateral triangle and of a regular hexagon would no longer be whole numbers.