An angle is an amount of turn
A weather vane swings from pointing north to pointing east. How much has it moved, and would the answer change if its arrow were twice as long?
Where you are 5 in Measuring angles
Why this exists
An angle is two rays that start at the same point, the vertex, and naming angles gives each angle a name. A name says which angle you mean. It does not say how big the angle is.
So what is the size of an angle? It cannot be the length of its arms. A ray goes on forever, so every arm is endlessly long, and when we draw an angle we just stop the arms wherever the paper runs out. Two pictures of the same corner can have short arms or long ones.
What does stay the same is how far apart the two arms point. A weather vane on a barn roof, the minute hand of a clock and a revolving door all show this: each one turns about a fixed center, and the question is always “how far did it turn?” This lesson answers that question with fractions of a full turn. Later, Degrees gives the same answer as a number of small units.
Definition
Start with a second ray lying right on top of one arm of the angle. Hold the vertex still and swing that ray around the vertex until it lies along the other arm. The size of the angle is the amount the ray turned. It is measured the shorter way around, so the size is never more than a half turn.
A full turn brings the ray all the way around, back to where it started. Smaller turns are fractions of it:
- a half turn leaves the ray pointing the opposite way, so the two arms make one straight line. That angle is a straight angle.
- a quarter turn leaves the arms making a square corner, like the corner of a page. That angle is a right angle.
Cut a full turn into 360 equal tiny turns, and each one is a degree: 1^\circ is \displaystyle \frac{1}{360} of a full turn. Counting angles in degrees is the work of Degrees.
Intuition
Part 1: an angle is a turn
Figure 1 shows three turns. In each panel the moving arm began on top of the fixed arm and swung counterclockwise about the vertex.
The eighth of a turn is a thin angle. The quarter turn makes a square corner. The half turn leaves the arms pointing opposite ways, in one straight line. The bigger the turn, the bigger the angle.
A turn can go either way. The weather vane in Figure 2 swings clockwise from north to east. Turning clockwise or counterclockwise does not change the size: a quarter turn is a quarter turn either way.
The four main directions, north, east, south and west, cut a full turn into 4 equal parts. So each step from one of them to the next is \displaystyle \frac{1}{4} of a turn. The four in-between directions, northeast, southeast, southwest and northwest, sit halfway between. They cut the full turn into 8 equal parts, so each small step is \displaystyle \frac{1}{8} of a turn.
A clock face is already cut up the same way. The 12 numbers are evenly spaced around a full turn, so as the minute hand moves from one number to the next, it turns \displaystyle \frac{1}{12} of a turn. It takes 5 minutes to do that, since 12 steps of 5 minutes make the 60 minutes of an hour.
Part 2: arm length does not matter, and angles can be compared by fitting
Longer arms do not make a bigger angle. Picture a weather vane with a short arrow and a much taller vane next to it with a long arrow. If both swing from north to east, both make a quarter turn. The tip of the long arrow travels farther, but it is pointing exactly where the short one is pointing. The size of the angle is about where the arms point, not how far they reach. Figure 5, in Example 5, shows angle P with long arms and angle Q with short arms, and Q is the bigger angle.
Comparing by fitting. You can tell which of two angles is bigger without measuring either, the same way you compare two pencils by lining up their ends. Trace one angle onto thin paper. Lay it on the other angle with vertex on vertex and one arm along one arm, with both angles opening toward the same side. Now look at the two free arms.
- If they lie on top of each other, the angles are the same size.
- If the traced free arm falls inside the other angle, the traced angle is smaller.
- If it falls outside, the traced angle is bigger.
A handy shape to fit against is the corner of a sheet of paper. It is a quarter turn, so it tells you at once whether an angle is smaller or bigger than a right angle.
Comparing by counting copies. Another way to compare is to see how many copies of an angle fit around one point. If 8 copies fill a full turn exactly, with no gaps and no overlaps, then each copy is \displaystyle \frac{1}{8} of a turn. An angle that needs fewer copies to fill the turn is bigger.
Results
An angle is an amount of turn. The size of an angle is how far one arm has to turn about the vertex, the shorter way around, to land on the other arm.
Two turns everyone knows. A quarter turn is a right angle. A half turn is a straight angle.
Arm length does not change an angle. Making the arms longer or shorter leaves the size of the angle the same.
Fitting compares angles. With vertex on vertex and one arm on one arm, the angle whose free arm lies inside the other is the smaller one.
Equal parts of a turn. If n equal angles fill a full turn around a point, each one is \displaystyle \frac{1}{n} of a turn.
Bigger turn past a half. If one arm turns more than a half turn to reach the other, the angle between them is the rest of the full turn, the part needed to finish it.
Why it is true
Why arm length does not matter. When a ray turns about its endpoint, every point on it turns together, near ones and far ones, because the ray stays straight. Making the arm longer only adds points farther out along the same direction. The direction is unchanged, so the turn needed to reach it is unchanged.
Why fitting works. Suppose angle Y is laid on angle X, sharing the vertex and one arm, and Y’s free arm falls inside X. Swing a ray from the shared arm toward X’s free arm. On the way, it passes Y’s free arm. So the turn for Y is only part of the turn for X, and Y is smaller.
Why n equal angles are each \displaystyle \frac{1}{n} of a turn. This is the same idea as cutting a whole into equal parts. The full turn is the whole. If n equal angles fill it with no gaps and no overlaps, each is one of n equal parts, which is \displaystyle \frac{1}{n} of the whole.
Why a turn past a half gives the rest. Two rays from one point cut the full turn into two openings that add up to the whole turn. The angle is the smaller opening. So if one way around is \displaystyle \frac{3}{4} of a turn, the other way is the \displaystyle \frac{1}{4} that finishes the turn, and the angle is \displaystyle \frac{1}{4} of a turn.
Worked examples
Examples 1 to 4 go with Part 1, and Examples 5 to 7 with Part 2.
The weather vane on the barn
Allira watches the weather vane in Figure 2. (a) In the morning it swings clockwise from north to east. What fraction of a turn is that? (b) Later it swings from north all the way to south. What fraction of a turn, and what kind of angle do the two positions make? (c) One evening it swings clockwise from east, past south and west, to north. How far did it turn, and what is the angle between where it started and where it stopped?
Solution(a) North, east, south and west cut a full turn into 4 equal parts, and north to east is one of them. That is \displaystyle \frac{1}{4} of a turn, a right angle.
(b) North to south is two of those parts, \displaystyle \frac{2}{4} of a turn. Two quarters make one half, so it is \displaystyle \frac{1}{2} of a turn. North and south point opposite ways, so the two positions make a straight angle.
(c) East to south, south to west and west to north are three steps, so the vane turned \displaystyle \frac{3}{4} of a turn. But the angle between east and north is measured the shorter way around, and the other way from east to north is just one step:
The vane turned \displaystyle \frac{3}{4} of a turn, and the angle between its start and end is \displaystyle \frac{1}{4} of a turn, a right angle.
The minute hand during homework
Marlee starts her homework at ten past four and checks the clock at twenty-five past four (Figure 3). Then she works until a quarter to five. (a) What fraction of a turn does the minute hand make in the first stretch? (b) In the second stretch? (c) From the start to the end, how far does the hand turn, and what is the angle between its first and last positions?
The 12 numbers cut a full turn into 12 equal steps, so each step is \displaystyle \frac{1}{12} of a turn.
(a) At ten past four the minute hand points at the 2, and at twenty-five past four it points at the 5. That is 3 steps, so \displaystyle \frac{3}{12} of a turn. Three steps out of twelve is the same as one group out of four equal groups of three:
The hand made a quarter turn, which is why its two positions in Figure 3 make a square corner.
(b) At a quarter to five the hand points at the 9. From the 5 to the 9 is 4 steps: \displaystyle \frac{4}{12} = \frac{1}{3} of a turn.
(c) Altogether the hand went from the 2 to the 9, which is 3 + 4 = 7 steps, or \displaystyle \frac{7}{12} of a turn. That is more than a half turn (\displaystyle \frac{6}{12}), so the angle between the first and last positions is the other way around:
The hand turned \displaystyle \frac{7}{12} of a turn, and the angle between its first and last positions is \displaystyle \frac{5}{12} of a turn.
The revolving door
Aurora goes into the library through a revolving door with four evenly spaced wings (Figure 4, left). (a) What is the angle between two neighboring wings? (b) She pushes the door through a half turn. How many wings pass the entrance? (c) The museum next door has a revolving door with only three evenly spaced wings (Figure 4, right). Is the angle between its neighboring wings bigger or smaller than a right angle?
(a) The four wings cut the full turn into 4 equal angles, so each is \displaystyle \frac{1}{4} of a turn, a right angle.
(b) A half turn is \displaystyle \frac{2}{4} of a turn, which is two of the quarter turns between wings. So the door moves on by two wing spaces, and two wings pass the entrance.
(c) Three wings cut the full turn into 3 equal angles, so each is \displaystyle \frac{1}{3} of a turn. A whole cut into 3 equal parts has bigger parts than the same whole cut into 4, so \displaystyle \frac{1}{3} is more than \displaystyle \frac{1}{4}. The angle between neighboring wings is bigger than a right angle.
Check. Three angles of \displaystyle \frac{1}{3} make \displaystyle 3 \times \frac{1}{3} = 1 full turn.
Eighths of a turn
Allira’s vane points north. The wind pushes it \displaystyle \frac{3}{8} of a turn clockwise. (a) Which way does it point now? (b) What is the angle between the start and the new position? (c) It then turns another \displaystyle \frac{3}{8} of a turn clockwise. Where does it point, and what is the angle between this position and north?
SolutionThe eight directions cut a full turn into 8 equal steps of \displaystyle \frac{1}{8} of a turn.
(a) Clockwise from north the directions go northeast, east, southeast. Three steps lands on southeast.
(b) \displaystyle \frac{3}{8} is less than half a turn, \displaystyle \frac{4}{8}, so the turn is already the shorter way. The angle is \displaystyle \frac{3}{8} of a turn.
(c) Three more steps from southeast go south, southwest, west. The vane points west. In all it has turned
of a turn, which is more than a half. The other way around from north to west is \displaystyle \frac{8}{8} - \frac{6}{8} = \frac{2}{8}, which is \displaystyle \frac{1}{4} of a turn. So the angle between west and north is a right angle, as it should be: west and north are neighboring main directions.
Long arms, short arms
Figure 5 shows angle P, drawn with arms 6 centimeters long, and angle Q, drawn with arms 2 centimeters long. P is \displaystyle \frac{1}{8} of a turn and Q is \displaystyle \frac{1}{6} of a turn. Allira says P looks bigger. Which angle is bigger? Would stretching Q’s arms to 6 centimeters change the answer?
The size of an angle is its turn, so compare \displaystyle \frac{1}{8} with \displaystyle \frac{1}{6}. Both are one equal part of the same full turn. Cutting the turn into 6 parts gives bigger parts than cutting it into 8, so \displaystyle \frac{1}{6} is more than \displaystyle \frac{1}{8}. Q is the bigger angle.
P looks bigger only because its arms are longer, and arm length is not part of an angle’s size. Stretching Q’s arms to 6 centimeters keeps both arms pointing the same way, so Q is still \displaystyle \frac{1}{6} of a turn and still the bigger angle.
Fitting one angle on another
Aurora has angles X and Y drawn on different pages (Figure 6, left), pointing different ways. Without measuring, decide which is bigger. Then decide whether X is bigger or smaller than a right angle.
The arms point different ways, so the angles cannot be judged where they are. Aurora traces Y onto thin paper and slides and turns the tracing until its vertex sits on X’s vertex and one arm of Y lies along one arm of X, with Y opening toward X’s other arm (Figure 6, right). Sliding and turning the tracing does not stretch or bend it, so Y keeps its size.
Y’s free arm, dashed, falls inside X, between X’s arms. Turning from the shared arm, a ray reaches Y’s free arm before it reaches X’s. So Y is smaller than X.
To compare X with a right angle, she lays the corner of a sheet of paper on X the same way. That corner is a quarter turn. X’s free arm falls just inside the paper’s corner, so X is a little smaller than a right angle.
Copies around a point
Marlee cuts out many copies of angle J and fits them around one point, edge to edge. Exactly 8 copies fill a full turn. Copies of angle K fill a full turn with exactly 5. (a) What fraction of a turn is each angle, and which is bigger? (b) How many copies of J make a right angle? (c) Can some number of copies of K make a right angle exactly?
Solution(a) 8 equal copies fill the turn, so J is \displaystyle \frac{1}{8} of a turn. In the same way K is \displaystyle \frac{1}{5} of a turn. K needs fewer copies to fill the same turn, so each copy of K is bigger.
(b) A right angle is \displaystyle \frac{1}{4} of a turn, which is \displaystyle \frac{2}{8}. So 2 copies of J make a right angle.
(c) Write both as twentieths: a right angle is \displaystyle \frac{1}{4} = \frac{5}{20}, and K is \displaystyle \frac{1}{5} = \frac{4}{20}. One copy of K is \displaystyle \frac{4}{20}, a little less than a right angle, and two copies are \displaystyle \frac{8}{20}, more than a right angle. No whole number of copies lands exactly on \displaystyle \frac{5}{20}, so no.
Boundaries
The shorter way around. Two rays from one point make two openings that together fill a full turn. The angle between them means the smaller one, never more than a half turn. A turn can be bigger than that, as in Example 4, but then the angle between the start and the end is the rest of the full turn.
A full turn. A vane that spins all the way around has made a whole turn, yet its arrow points where it started, so there is no opening between the start and the end. The turn and the angle are different things once the turn passes a half.
Fitting needs a shared vertex and a shared arm. Two angles laid down with their vertices apart, or with no arm lined up, cannot be compared by eye. Fitting also says only “bigger”, “smaller” or “the same”. To say how much bigger, the angles need a unit, which is what Degrees gives.
A drawing is only as good as its lines. An angle read off a picture is as exact as the picture. When a problem states that an angle is \displaystyle \frac{1}{8} of a turn, that is exact, whatever the drawing looks like.
Where this goes next
Once an angle is a turn, a unit of turn can be counted: Degrees cuts the full turn into 360 equal parts, and a protractor is the tool that counts them on a drawing. Angles are sorted by how far they turn compared with a quarter and a half turn in acute, right, obtuse and straight angles, and two lines that meet in quarter turns are perpendicular. Much later, turning a whole shape about a point uses the same idea, in the center and angle of a rotation.
Exercises
Part 1: turns as fractions
- A weather vane turns from pointing east to pointing west. What fraction of a turn is that, and what kind of angle do the two positions make?
- A weather vane turns clockwise from south to west. What fraction of a turn is that?
- What fraction of a turn does the minute hand make from five past nine to twenty-five to ten?
- What fraction of a turn does the minute hand make from twenty past two to half past two?
- The minute hand moves from a quarter past seven to five to eight. How far does it turn? What is the angle between its first and last positions?
- How many minutes does the minute hand take to make \displaystyle \frac{2}{3} of a turn?
- A revolving door has 6 evenly spaced wings. What fraction of a turn is the angle between neighboring wings? Is it bigger or smaller than a right angle?
- A revolving door with 4 evenly spaced wings turns until the third wing after Aurora’s is where hers started. What fraction of a turn did it make?
- A vane points north and turns \displaystyle \frac{5}{8} of a turn clockwise. Which way does it point? What is the angle between its start and end positions?
- A vane points east. It turns \displaystyle \frac{1}{4} of a turn clockwise, then \displaystyle \frac{1}{2} of a turn more, clockwise. Which way does it point, how far did it turn in all, and what is the angle between its start and end positions?
- Which is the bigger angle, \displaystyle \frac{1}{3} of a turn or \displaystyle \frac{1}{4} of a turn?
- An angle is \displaystyle \frac{2}{8} of a turn. What simpler fraction of a turn is it, and what kind of angle is it?
Part 2: arm length and comparing
- Allira draws an angle of \displaystyle \frac{1}{5} of a turn with arms 3 centimeters long. She then extends both arms to 9 centimeters. What fraction of a turn is the angle now?
- Order the angles in Figure 7 from smallest to biggest.
- Which angles in Figure 7 are smaller than the corner of a sheet of paper?
- Marlee lays angle A on angle B with vertex on vertex and one arm along one arm. A’s free arm falls outside B. Which angle is bigger?
- Exactly 10 copies of an angle fill a full turn. What fraction of a turn is it? Is it bigger or smaller than an angle that needs 12 copies?
- 3 copies of an angle fit exactly into a right angle. What fraction of a full turn is the angle?
- 2 copies of an angle fit exactly into a straight angle. What fraction of a turn is the angle, and what kind of angle is it?
- Angle K fills a full turn with 6 copies, and angle L fills one with 12. Which is bigger, and how many copies of L fit exactly into K?
- One vane turns \displaystyle \frac{1}{4} of a turn clockwise from north, and another turns \displaystyle \frac{1}{4} of a turn counterclockwise from north. Where does each point, and are the two angles the same size?
Reasoning
- Explain why making both arms of an angle longer does not change the size of the angle.
- Explain why, when comparing two angles by fitting, the vertices must sit on top of each other and one arm must lie along one arm.
- A weather vane makes one full turn. How far did it turn, and what angle is there between where it started and where it stopped?
- Can two copies of an angle that is bigger than a right angle fit side by side inside a straight angle, without overlapping? Explain.
Answers
- \displaystyle \frac{1}{2} of a turn, a straight angle. East and west are two of the four equal steps apart, and \displaystyle \frac{2}{4} = \frac{1}{2}; they point opposite ways.
- \displaystyle \frac{1}{4} of a turn. South to west is one of the four equal steps between main directions.
- \displaystyle \frac{1}{2} of a turn. The hand goes from the 1 to the 7, which is 6 steps: \displaystyle \frac{6}{12} = \frac{1}{2}.
- \displaystyle \frac{1}{6} of a turn. The hand goes from the 4 to the 6, 2 steps: \displaystyle \frac{2}{12} = \frac{1}{6}.
- It turns \displaystyle \frac{2}{3} of a turn; the angle is \displaystyle \frac{1}{3} of a turn. From the 3 to the 11 is 8 steps, \displaystyle \frac{8}{12} = \frac{2}{3}. That is more than a half, so the angle is the rest: \displaystyle \frac{12}{12} - \frac{8}{12} = \frac{4}{12} = \frac{1}{3}.
- 40 minutes. A full turn takes 60 minutes, a third of it 60 \div 3 = 20 minutes, and two thirds 2 \times 20 = 40 minutes.
- \displaystyle \frac{1}{6} of a turn, smaller than a right angle. Six equal parts are smaller than four equal parts of the same turn, so \displaystyle \frac{1}{6} is less than \displaystyle \frac{1}{4}.
- \displaystyle \frac{3}{4} of a turn. Each wing space is \displaystyle \frac{1}{4} of a turn, and three of them make \displaystyle \frac{3}{4}.
- Southwest; the angle is \displaystyle \frac{3}{8} of a turn. Five steps clockwise from north go northeast, east, southeast, south, southwest. The turn, \displaystyle \frac{5}{8}, is more than a half, so the angle is \displaystyle \frac{8}{8} - \frac{5}{8} = \frac{3}{8}.
- North; \displaystyle \frac{3}{4} of a turn in all; the angle is \displaystyle \frac{1}{4} of a turn. A quarter turn clockwise from east is south, and a half turn more is north. \displaystyle \frac{1}{4} + \frac{2}{4} = \frac{3}{4}, and the other way around is \displaystyle 1 - \frac{3}{4} = \frac{1}{4}.
- \displaystyle \frac{1}{3} of a turn. A whole cut into 3 equal parts has bigger parts than one cut into 4.
- \displaystyle \frac{1}{4} of a turn, a right angle. Two eighths make one quarter: \displaystyle \frac{2}{8} = \frac{1}{4}.
- Still \displaystyle \frac{1}{5} of a turn. Longer arms point the same way, so the turn between them is unchanged.
- \angle 2, then \angle 3, then \angle 1. Traced and fitted, \angle 2 is the narrowest even though its arms are the longest, and \angle 1 opens widest even though its arms are the shortest.
- \angle 2 and \angle 3. With the paper’s corner fitted on each, their free arms fall inside it; the free arm of \angle 1 falls outside, so \angle 1 is bigger than a right angle.
- A. Its free arm is outside B, so the turn for A goes past B’s free arm, and B is only part of A.
- \displaystyle \frac{1}{10} of a turn, bigger. It needs fewer copies to fill the same turn, so each copy is bigger than \displaystyle \frac{1}{12}.
- \displaystyle \frac{1}{12} of a turn. A right angle is \displaystyle \frac{1}{4} of a turn, so 4 right angles fill the turn and need 4 \times 3 = 12 copies.
- \displaystyle \frac{1}{4} of a turn, a right angle. A straight angle is \displaystyle \frac{1}{2} of a turn, so 2 straight angles fill the turn with 2 \times 2 = 4 copies.
- K is bigger, and 2 copies of L fit into K. K is \displaystyle \frac{1}{6} = \frac{2}{12} of a turn and L is \displaystyle \frac{1}{12}.
- The first points east and the second points west. Both angles are \displaystyle \frac{1}{4} of a turn, the same size: the direction of a turn does not change how far it goes.
- The size of an angle is the turn from one arm’s direction to the other’s. Making an arm longer adds points farther out in the same direction, so neither direction changes, and so the turn between them does not change.
- The size is the turn from one arm to the other about the vertex. With vertex on vertex and one arm on one arm, both turns start from the same place, so whichever free arm is reached first marks the smaller turn. If the vertices or the arms are apart, the two turns start from different places and the picture shows nothing about which is bigger.
- It turned one whole turn, but there is no angle between the start and the end: the arrow points where it started. Past a half turn, the turn and the angle are different things.
- No. Each copy is more than \displaystyle \frac{1}{4} of a turn, so two copies together are more than \displaystyle \frac{1}{4} + \frac{1}{4} = \frac{1}{2} of a turn, and a straight angle is exactly a half turn.