The diagram shows a protractor being used to measure the angle 𝜃. What is the value of 𝜃, in degrees?
Looking at the diagram, we see that the protractor and the angle 𝜃 are both already lined up, so that the centre or midpoint of the protractor and the vertex of the angle 𝜃 are at the same location. This means that the angle is positioned relative to the protractor, so that we can make an accurate measurement of it.
The second thing we can notice is that one of the sides of this angle is already pointing at a horizontal direction. We can call this zero degrees. And notice by calling it zero degrees, we’re saying that we measure the angle 𝜃 according to this top scale on our protractor, from zero, 10, 20, 30, 40, 50 up to 60. This is the scale we use when we measure an angle starting at the left side. But we are here and moving to the right. This means that, to solve for the value of 𝜃, this angle here will simply want to read out the place, where this arm of 𝜃, the top arm, crosses our measurement scale on our protractor.
The question then becomes where does it cross that scale? To find out, we want to know just what is the angular distance between the larger hash marks we see, the ones that are marked out in tens of degrees and the smaller ones that appear in between those. If we take the two large hash marks of 70 and 80 degrees as our boundaries, then we can count the number of smaller hash marks we see that appear between them.
Starting at 70 degrees, we have one hash mark here then two, three, four, five, six, seven, eight, and then nine. And the 10th hash mark is the larger one corresponding to 80 degrees. So the angular interval between 70 and 80 degrees on our protractor is divided up into 10 even parts. This tells us that each one of the parts is equal to one degree, one-tenth of 10.
We could write that this way. We could say that 𝛥𝜃, what we call the angular resolution of our protractor, is equal to one degree. That’s the smallest change in angle that these markings will let us measure. So back to the arm of our angle that we want to measure, this one which will help us solve for 𝜃.
Looking carefully, we see that this arm crosses our measurement scale one small tick mark above the larger tick mark, which corresponds to 60 degrees. We’ve just seen that that angular difference corresponds to one degree. And this means that this arm of our angle crosses the protractor at 61 degrees. And because the other arm of our angle is positioned at zero degrees, we don’t need to add or subtract anything to this value to solve for 𝜃.
The value of the angle 𝜃 is 61 degrees.