edubun 한국어
Home›Geometry and drawings›Angles

Measuring angles with a protractor

A full turn is 360 degrees, so one degree is a 360th of a turn. A school protractor marks 180 of them along its curved edge.

What an angle is

Open a pair of scissors a little, then a lot. The blades stay the same length, yet something clearly changes: the amount of opening. That amount is what an angle measures. OpenStax describes an angle as two rays that share a common endpoint. The shared point is the vertex (plural vertices) and the two rays are the sides of the angle. Because the rays go on forever, the length you happen to draw them has nothing to do with the size of the angle. A tiny angle drawn with long arms is still tiny.

Another way to picture it: keep one ray still and turn the other one about the vertex. The angle is the amount of turning. Turn the moving ray all the way round until it lands back on the first one and it has swept out one full rotation, which is 360 degrees, written 360°. Half a turn is 180°, a quarter turn is 90°.

Naming an angle

An angle is named with three points: one on each side and the vertex in the middle. The angle below is ∠ABC, which is the same angle as ∠CBA. When only one angle sits at a vertex you can shorten the name to ∠B. To talk about its size we write m∠ABC = 38°, read "the measure of angle ABC is 38 degrees". The vertex letter always goes in the middle; ∠BAC would be a different angle, the one at A.

Vertex B in the middle of the name. The arc marks which opening is meant.

Why 360?

BackgroundThe MacTutor history archive at the University of St Andrews notes that the astronomer Ptolemy, writing in Alexandria in the second century, divided the circle into 360 degrees and in doing so showed the same Babylonian influence as Hipparchus before him. Babylonian arithmetic counted in sixties, and the habit survives in the small units: half a degree is 30 minutes, written 30′, so one degree is 60 minutes. Clocks and maps still split hours and degrees the same way.

Reading a protractor

A protractor has a small centre mark on its straight edge and two rows of numbers on its curved edge, one running 0 to 180 from the right and the other 0 to 180 from the left. To measure an angle, put the centre mark exactly on the vertex and line the straight edge up with one side. Then follow the other side out to the scale and read the mark it crosses. Use the row that reads 0 on the side you lined up. OpenStax adds that the most common error is a centre mark that is not on the vertex; the reading is then simply wrong.

Loading

Drag to turn. Scroll or pinch to zoom.

Drag anywhere to swing the orange arm. Press New angle to get a hidden angle, read it off the scale, then press Show.

Before you read the scale, decide roughly how big the angle is. If it is narrower than the corner of a sheet of paper it is under 90°, so a reading of 140° must come from the wrong row; the answer is 40°. This one habit catches most protractor mistakes. For an angle wider than 180°, measure the smaller opening on the other side and subtract it from 360°.

  1. The vertex of ∠PQR is:
    • Q
    • P
    • R
    • the arc
  2. How many degrees in half a turn?
  3. How many degrees in a quarter turn?
  4. A wheel turns one eighth of a full turn. How many degrees?
  5. You draw the sides of an angle twice as long. The angle:
    • stays the same
    • doubles
    • halves
    • becomes 180°
  6. An angle is clearly narrower than a right angle, but you read 125° on the protractor. The angle is:
    • 55°
    • 125°
    • 35°
    • 235°
  7. The small opening on one side of an angle measures 70°. What is the large opening on the other side?
  8. How many minutes (′) are in 1.5°?
  9. Where must the centre mark of the protractor go?
    • on the vertex
    • on the end of a side
    • anywhere on a side
    • on the arc
  10. The minute hand of a clock turns a full circle in 60 minutes. How many degrees does it turn in one minute?
Key points
  • An angle: two rays from one vertex. Its size is the turning between them.
  • Full turn 360°, half turn 180°, quarter turn 90°; 1° = 60′.
  • Name with the vertex in the middle: ∠ABC; size m∠ABC.
  • Protractor: centre mark on the vertex, base on one side, read the row that starts at 0 there.
  • Estimate first; over 180°, subtract the small side from 360°.