Measuring and drawing angles with a protractor
An angle is drawn on a page. How do you find how many degrees it opens, and how do you draw a new angle that opens exactly the number of degrees you want?
Where you are 5 in Measuring angles
When you reach for it
An angle is an amount of turn (an angle is an amount of turn), and Degrees counts that turn in small equal steps: a full turn is 360^\circ, a half turn is 180^\circ and a right angle is 90^\circ. Counting 70 tiny steps by hand would be slow. A protractor has the counting already printed on it.
Reach for a protractor when an angle is drawn and you need its size as a number: the angle between two blades of a fan in a sketch, or between two roads on a map. Reach for it also the other way round, when you know the number and need to draw the angle: a road on a map that leaves a town at 75^\circ from another road, say.
This lesson takes three days. Part 1 is the parts of a protractor and how to measure with it. Part 2 is the two rows of numbers, how to choose the right one, and how to estimate first. Part 3 is drawing an angle of a given size, arms that are too short, and turns of more than a half turn.
The procedure
Part 1: the parts of a protractor, and measuring an angle
A protractor is a half circle, usually clear plastic. Its straight edge carries a line called the base line. Halfway along the base line is the center mark, a small mark or hole. Around the curved edge run tick marks, and beside them two rows of numbers from 0 to 180.
Why a half circle, and why 180? Picture a ray lying along the base line, pointing right, with its endpoint on the center mark. Now swing it about the center mark, all the way round to the left end. It has made a half turn, 180^\circ. Every tick mark on the curved edge is where the swinging ray would point after turning some number of degrees. So the curved edge is a half turn cut into degrees, and every one of those degrees meets at the center mark.
On a real protractor there is a mark for every single degree. The pictures in this lesson show a mark every 5^\circ, a longer mark every 10^\circ, and numbers every 30^\circ, so you have to work out what the unlabeled marks stand for (reading a scale between marks: find what one space is worth before you count spaces). From one number to the next is 30 degrees, and the long marks cut that into 3 equal spaces, so each long mark is 30 \div 3 = 10 degrees. A short mark sits halfway between two long ones, so it adds 5.
To measure an angle, two things must line up:
- The center mark goes on the vertex. The size of the angle is the turn about the vertex, and the protractor counts turn about its center mark. Only when those two points are the same point does the protractor count the right turn.
- The base line goes along one arm. That arm is where the turn starts, so it must sit on a 0. The base line has a 0 at each end, so whichever way the arm runs, it lands on one of them.
Then look at where the other arm crosses the curved edge, and read the number there on the row whose 0 sits on the arm you lined up. That number is how many degrees the other arm is turned from the first one, so it is the measure of the angle. Part 2 says more about the two rows; for now, start counting from the 0 under your first arm.
Part 2: the two scales, choosing one, and estimating first
Look again at the picture of the parts. The inner scale starts at 0 on the right end of the base line and counts up as it goes round to the left. The outer scale starts at 0 on the left end and counts up as it goes round to the right. Two scales mean the protractor works whichever way the first arm points, left or right.
At every place on the curved edge the two numbers add to 180. At the top both rows say 90. One long mark to the right of the top, the inner row says 80 and the outer row says 100, and 80 + 100 = 180. That has to happen: the two rows count the same half turn of 180^\circ from opposite ends, so whatever one row has counted, the other row counts the rest.
So an arm always crosses at two numbers, and you need to know which one is the angle. Read the row that is 0 on the arm you lined up. That row starts counting where the turn starts. Put your finger on the 0 at the end of the base line where your first arm runs, and follow that same row round to the other arm, counting up: 0, 10, 20 and so on. If your first arm runs to the right end, that is the inner row in these pictures; if it runs to the left end, it is the outer row. A real protractor may print its rows the other way about, so always find the 0 under your arm rather than remembering “inner” or “outer”.
Estimate before you measure. A quick estimate tells you which of the two numbers makes sense, before the protractor is even on the page. Four angles make good benchmarks (a size you know by heart, used to judge other sizes):
- 90^\circ is the corner of a sheet of paper, a square corner.
- 45^\circ is half of that corner, the angle you get by folding the corner in half.
- 135^\circ is a square corner and half of another, 90 + 45 = 135.
- 180^\circ is a straight line.
Every angle up to a half turn falls between two of these. An angle narrower than a square corner is less than 90^\circ, and an angle wider than a square corner is more than 90^\circ (acute, right and obtuse angles are named for exactly this). Since the two numbers at a crossing add to 180, one of them is less than 90 and the other more (unless both are 90). The estimate picks the right one.
Part 3: drawing an angle, short arms, and turns past a half turn
Drawing an angle of a given size runs the measuring backward. Draw one arm first, a ray, with a ruler. Put the center mark on its endpoint and the base line along the ray. The ray now lies on a 0. Count up on the row that starts at that 0 until you reach the number you want, and make a small dot at the curved edge beside it. Lift the protractor away and join the endpoint to the dot with a ruler. The new arm points exactly where a ray would point after turning that many degrees from the first arm, so the angle has the size you wanted. Measuring the finished angle should give the number back.
Arms that are too short. Sometimes an angle is drawn with arms that stop before they reach the protractor’s curved edge, so there is nothing to read. Lay a ruler along the short arm and draw it longer, in the same direction, until it passes the scale. That is allowed because an angle’s size does not depend on how long its arms are drawn (longer arms do not make a bigger angle): the longer arm still points the same way, so the turn between the arms is unchanged.
Turns of more than a half turn. A protractor only reaches 180^\circ. A turn the long way round, more than a half turn, cannot be read straight off it. But two arms from one point split a full turn into two openings, the inside one and the outside one, and together they make 360^\circ. So measure the inside opening, and the long way round is the rest of the full turn:
Worked examples
Examples 1 and 2 go with Part 1, Examples 3 and 4 with Part 2, and Examples 5 to 7 with Part 3.
Reading the scale between the numbers
Meabh has drawn \angle ABC, with arm BA pointing right (the figure below). Measure it.
Meabh puts the center mark on the vertex B, because the angle is the turn about B. She lays the base line along arm BA, so BA runs to the right end of the base line, where the inner row has its 0.
Arm BC crosses the curved edge between the inner row’s 60 and 90. Between those two numbers the long marks stand for 10 degrees each, and BC crosses at the first long mark past 60:
So m\angle ABC = 70^\circ. (The notation m\angle ABC means the measure of \angle ABC; see naming angles.)
Check. The angle is narrower than a square corner, so it must be less than 90^\circ, and 70 is. The outer row reads 110 at the same place, 180 - 70 = 110, but the outer row’s 0 is at the left end, where no arm runs.
An angle that is turned on the page
Fionn traces the edges of two fan blades from a picture and gets an angle at V whose arms point up and to the right (VS) and up and to the left (VT). Neither arm runs across the page. Measure \angle SVT.
The base line has to lie along one arm, wherever that arm points. Fionn turns the protractor (or the paper) until the center mark is on V and the base line runs along VS. Turning the protractor does not change the angle: both arms keep their directions, so the turn between them is the same.
Now VS runs to the end of the base line where the inner row starts at 0. Arm VT crosses the inner row past the 90 at the top. Counting on in tens from 90, it crosses at the second long mark:
So m\angle SVT = 110^\circ.
Check. The angle opens wider than a square corner, so it is more than 90^\circ. It is less than 135^\circ, a square corner and a half.
Choosing the scale
Benedita’s map has a compass rose, so west is to the left. A trail leaves the town T heading west, toward W, and a road leaves T toward R. Measure \angle WTR.
Benedita puts the center mark on T and the base line along TW. This time the lined-up arm runs to the left end of the base line. The row that has its 0 at the left end is the outer row, so the outer row is the one that counts the turn from TW.
Arm TR crosses where the outer row reads between 30 and 60. Counting up from the outer row’s 30: 40 at the first long mark, 50 at the second. So m\angle WTR = 50^\circ.
The inner row reads 130 at the same place. That number counts the turn from the right end of the base line, where neither arm of this angle runs, so it is not the angle.
Check. The two numbers add to 180: 50 + 130 = 180. And the angle is clearly narrower than a square corner, so it is less than 90^\circ: it is 50^\circ, not 130^\circ.
Estimating first, and letting the estimate choose
Estimate each angle in the figure using the benchmarks 45^\circ, 90^\circ, 135^\circ and 180^\circ. Then a protractor is laid on each one, with the center mark on the vertex and the base line along one arm. On angle 1 the other arm crosses where the two rows read 20 and 160; on angle 2, 40 and 140; on angle 3, 80 and 100. Give each angle’s measure.
Angle 1 is nearly a straight line, but not quite. It is wider than a square corner and a half, so it lies between 135^\circ and 180^\circ. Of 20 and 160, only 160 is in that range: the angle is 160^\circ.
Angle 2 is narrow. Folding a square corner in half gives 45^\circ, and this angle is a little narrower than that, so it is a little less than 45^\circ. Of 40 and 140, it is 40^\circ.
Angle 3 is a little wider than a square corner, so a little more than 90^\circ and well less than 135^\circ. Of 80 and 100, it is 100^\circ.
Check. In each pair the two numbers add to 180: 20 + 160, 40 + 140 and 80 + 100. That is why one reading is always on each side of 90, and why a rough estimate is enough to choose.
Drawing an angle of a given size
(a) Meabh needs an angle of 75^\circ. She draws ray YX pointing right. Draw the angle. (b) Fionn needs an angle of 140^\circ on a ray that points left. Which row does he use, and where does the dot go?
(a) Meabh puts the center mark on Y, the endpoint of the ray, and the base line along YX. Ray YX runs to the right end, where the inner row starts at 0, so she counts on the inner row. She wants 75: past the label 60 comes 70 at the first long mark, then 75 at the short mark after it. She makes a dot at the curved edge there, takes the protractor away, and joins Y to the dot with a ruler, labeling the far end Z. Then m\angle XYZ = 75^\circ, because
(b) Fionn’s ray runs to the left end of the base line, so he uses the row whose 0 is at the left end, the outer row in these pictures. He counts up to the label 120, then on two long marks, 130 and 140, and puts his dot at 140:
Check. In (a), 75 is between the benchmarks 45 and 90 and much closer to 90, so the new arm should stand a little short of straight up, tipped toward X, and it does. In (b), 140 is more than 90, so the new arm should lean past the top of the protractor, away from the ray it starts on, and 140 is just past 135, a square corner and a half.
Arms that are too short to reach the scale
Fionn’s sketch of a fan has an angle at F with arms only 1 centimeter long. With the center mark on F, neither arm reaches the protractor’s numbers. Measure the angle.
Fionn lays a ruler along each short arm in turn and extends it, keeping its direction, until it runs past the curved edge (the dashed lines). Extending an arm leaves it pointing the same way, so the turn between the two arms, which is the angle’s size, does not change.
Now he lines up as usual: center mark on F, base line along the arm that points right. That arm runs to the right end, where the inner row starts at 0. The other extended arm crosses the inner row at the first long mark past 120:
The angle is 130^\circ.
Check. It opens wider than a square corner but less than a square corner and a half, so it is between 90^\circ and 135^\circ, close to 135^\circ.
A turn of more than a half turn
Benedita sticks a dot on one blade of a fan and switches the fan on for a moment. The blade starts at arm OA, pointing right, and turns clockwise the long way round, past the bottom, until it stops at arm OB, pointing up and a little left. The protractor reads 100^\circ for the inside opening \angle AOB. How far did the blade turn?
The blade’s turn is the outside opening, more than a half turn, which the protractor cannot reach. The inside and outside openings together fill one full turn about O, 360^\circ, so the outside one is what is left after the inside one:
The blade turned 260^\circ.
Check. 260 is more than 180, a half turn, as a turn past the bottom must be, and less than 270, three quarters of a turn. Three quarters of a turn from pointing right would end pointing straight up; the blade stopped just short of that, which matches OB in the figure.
Boundaries
A protractor reaches only a half turn. Its scale stops at 180^\circ. The measure of an angle between two rays is never more than 180^\circ anyway, by agreement (Degrees explains why). A turn of more than a half turn is found as 360^\circ minus the inside angle, as in Example 7.
Both line-ups are needed. The center mark must be on the vertex and the base line along one arm. If the center mark is off the vertex, the protractor counts turn about the wrong point. If the base line lies along neither arm, the first arm is not on a 0 and the number at the second arm is not the angle (Exercise 25 shows what can still be done).
A reading is as exact as the marks and the drawing. A real protractor has one-degree marks, so it measures to the nearest degree, and a pencil line has some width. An angle stated in a problem is exact; an angle read off a drawing is close.
The arms must reach the scale. If they do not, they can be extended along their own direction. Bending or shifting an arm while extending it would change the angle.
Where this goes next
With a protractor in hand, angles can be put together and taken apart: angle measures add, so an angle split into parts measures the parts added together, and a missing part can be found from the whole. Measured angles are then sorted and paired in angle types and pairs. The compass rose on a map grows into bearings, where a direction is given as a number of degrees turned from north. A protractor also helps measure what cannot be reached, in measuring with shadows. And Compass and straightedge constructions draws some exact angles, a right angle or half of any angle, with no scale at all, using only a compass and a straight edge.
Exercises
Part 1: measuring an angle
Use the figure of five rays for Exercises 1 to 3. The base line runs along OA and OE.
- Find m\angle AOB.
- Find m\angle AOC.
- Find m\angle AOD.
- A protractor has numbers every 30^\circ, with two long marks between each pair of numbers and a short mark halfway between long ones. How many degrees does a long mark stand for? A short mark?
- Meabh lines up an angle correctly, with its first arm on the inner row’s 0. Its other arm crosses two long marks and then one short mark past the inner row’s 90. What is the angle?
- The other arm of an angle crosses one short mark before the label 60, on the row whose 0 is on the first arm. What is the angle?
Part 2: choosing the scale and estimating
Use the figure of five rays again for Exercises 7 to 9.
- Find m\angle EOD.
- Find m\angle EOB.
- Find m\angle EOC.
- An arm crosses where the two rows read 35 and 145. The angle is wider than a square corner. What is it?
- An arm crosses where one row reads 62. What does the other row read there?
- Between which two benchmarks does an angle of 112^\circ lie, and which is it closer to?
- An angle opens just a little wider than the corner of a sheet of paper. The protractor shows 85 and 95 where its arm crosses. What is the angle?
- An angle is narrower than the corner of a sheet of paper. One row of the protractor reads 150 where its arm crosses. What is the angle?
- A compass rose shows eight directions, evenly spaced. What is the angle between north and northeast? Between north and southeast?
Part 3: drawing, short arms, and long turns
- Benedita draws a ray pointing right and wants an angle of 25^\circ. Which row does she count on, and where does her dot go?
- Fionn draws a ray pointing left and wants an angle of 105^\circ. Which row does he count on, and where does his dot go?
- Meabh draws a ray, then draws a 45^\circ angle and a 135^\circ angle on it, both on the same side and counted from the same 0. What is the angle between the two new arms?
- A ceiling fan has three blades, evenly spaced. Fionn wants to draw the angle between two neighboring blades. What size is it, and where does his dot go?
- The inside angle between two arms measures 65^\circ. How far is the turn from one arm to the other the long way round?
- A fan blade turns 210^\circ counterclockwise and stops. What will a protractor read for the inside angle between where it started and where it stopped?
- On a compass rose, an arrow turns clockwise from north to west. How far does it turn, and what is the angle between north and west?
Reasoning
- Explain why the protractor’s center mark has to sit exactly on the vertex of the angle.
- Explain why the two numbers at any place on the curved edge always add to 180.
- Meabh puts the center mark on the vertex but lays the base line along neither arm. The two arms cross the inner row at 20 and at 105. Explain how she can still find the angle, and find it.
Answers
- 15^\circ. OA runs to the right end, the inner row’s 0; OB crosses the inner row at the short mark after the first long mark past 0: 10 + 5 = 15.
- 80^\circ. Counting on the inner row, OC crosses one long mark before the 90 at the top.
- 135^\circ. On the inner row, OD crosses at the short mark just past the long mark 130: 120 + 10 + 5 = 135.
- 10^\circ and 5^\circ. Two long marks cut the 30 degrees between numbers into 3 equal spaces, 30 \div 3 = 10, and a short mark is halfway, 10 \div 2 = 5.
- 115^\circ. 90 + 10 + 10 + 5 = 115.
- 55^\circ. A short mark is 5 degrees, so 60 - 5 = 55.
- 45^\circ. OE runs to the left end, the outer row’s 0, so read the outer row. OD crosses where the inner row reads 135, and the outer row reads 180 - 135 = 45 there.
- 165^\circ. On the outer row, counted from OE, ray OB is at 180 - 15 = 165. It is a wide angle, between 135^\circ and 180^\circ, as it looks.
- 100^\circ. OC is at 80 on the inner row, so at 180 - 80 = 100 on the outer row, counted from OE. It opens a little wider than a square corner.
- 145^\circ. Wider than a square corner means more than 90^\circ, and of the two numbers only 145 is. Check: 35 + 145 = 180.
- 118. The two rows always add to 180: 180 - 62 = 118.
- Between 90^\circ and 135^\circ, just closer to 90^\circ. 112 - 90 = 22 and 135 - 112 = 23.
- 95^\circ. Wider than a square corner means more than 90^\circ.
- 30^\circ. The other row reads 180 - 150 = 30, and an angle narrower than a square corner is less than 90^\circ.
- 45^\circ and 135^\circ. Eight equal steps fill a full turn, so each is 360 \div 8 = 45 degrees. North to southeast is three steps: 3 \times 45 = 135.
- The row whose 0 is at the right end (the inner row in these pictures). The dot goes between 0 and 30, two long marks and one short mark along: 10 + 10 + 5 = 25.
- The row whose 0 is at the left end (the outer row in these pictures). The dot goes one long mark and one short mark past 90: 90 + 10 + 5 = 105.
- 90^\circ. Both arms are counted from the same first arm, so the second is 135 - 45 = 90 degrees farther round than the first.
- 120^\circ, with the dot on the label 120 of the row whose 0 is on the first blade’s line. Three equal angles fill a full turn: 360 \div 3 = 120.
- 295^\circ. 360 - 65 = 295.
- 150^\circ. The two openings fill a full turn, so the inside one is 360 - 210 = 150.
- It turns 270^\circ, three quarter turns of 90^\circ each, 3 \times 90 = 270. The angle between north and west is the inside opening, 360 - 270 = 90, so 90^\circ.
- The size of an angle is the turn about its vertex, and the protractor’s marks count turn about the center mark: every tick is a direction seen from that one point. If the center mark sits somewhere else, the arms are seen from the wrong point, so the tick an arm crosses is not the turn about the vertex.
- Both rows count the same half turn, 180^\circ, one from each end of the base line. At any place, one row has counted the turn from the right end to that place and the other the turn from the left end to it. Those two turns together make the whole half turn, so they add to 180.
- 85^\circ. Both arms are counted from the same 0, so the inner row says how far each arm is turned from the right end of the base line. The second arm is turned 105^\circ and the first 20^\circ, so the turn from the first arm to the second is the difference, 105 - 20 = 85: the 20^\circ turn and the 85^\circ turn together make the 105^\circ turn.