In this video, we’re going to talk about measuring angles. And to begin, let’s remind ourselves just what an angle is? One way to start thinking about angles is to consider this object we have here: this square object with a couple of diagonal lines across it.
Now, if we were to take this object and translate it some distance after the right, we know that one way to measure that distance would be to put a straight edge ruler to the path that the object has travelled. If we take the difference between the start point of the object and its endpoint, that distance tells us just how much this object has moved. So that’s one type of motion. We call that motion translation.
But what about the motion of rotation, that is, if the object turned around an axis through its centre? In order to measure this type of motion, we could start by putting a dot at the point around which our object turns. And then before our object rotates, we’ll draw a line a ray out from that point. Since our rotating object already has a diagonal line pointing up into the right, let’s follow that with our ray.
So there we have it: the initial position of this line on our object before the object is rotated. Then, when we turn this object, let’s say our object rotates so that that’s same blue line that our green line was on top of is now in this position. What we have then with these green rays that start from this common origin is a measure of the rotation that our object went through.
Just like with our measurement of distance, we have an initial position of our object and a final position. But in this case, those positions are different from one another by an angle rather than a distance. So that’s what an angle is. It’s a representation of some amount of rotation or turning.
Every angle regardless of how big or small has a couple of parts to it. For example, the point where our two rays meet or these two lines come together is called the vertex of the angle. And then, the two sides of the angle — we’ve been calling them rays as well — are also sometimes called the arms of the angle.
One thing to notice about an angle is that the important thing about it is not how long the arms are. In other words, we could extend these arms as long or as short as we want and it wouldn’t change with this angle is. That’s because what characterizes an angle, what characterizes this angle, is the rotation from one arm to the other. Often, we represent that rotation using a symbol in Greek called 𝜃.
So anyway, these are the parts of an angle. And when we talk about measuring an angle, perhaps you can see that the device we’ll use won’t be the same as the device we used to measure distance. In that case, we used a ruler. But in this case, to measure angles, we’ll use something called a protractor.
Here, we have a sketch of a protractor. In real life, this might be something like a bit of plastic or glass marked out this way. The first thing to notice about the protractor is the numbers that are marked on it. Notice that they start at zero although that’s not marked out and then go up by increments of 10: 10, 20, 30, 40, 50 all the way up to the midpoint of 90.
As you might guess, these numbers represent angles relative to these zero point. The numbers continue on for larger and larger angles, getting all the way up to what’s called a straight angle, an angle of 180 degrees.
All of the markings of the protractor are in reference to a very important point. It’s this point right here, called the protractor midpoint. When we want to measure an angle using a protractor, we take the vertex of that angle and we place it right at the protractor midpoint. In fact, if we’re writing out a step-by-step process for measuring angles. That would be step one. So let’s write that down.
Okay, so here’s our recipe for measuring an angle. And like we said, the first step in that process is to place the protractor at midpoint on the vertex of the angle. So here we go, we’ve taken our angle and we’ve located its vertex right at our protractor midpoint.
Clearly, though we could still use some help measuring this angle because the arms of our angle don’t extend far enough to the measure portion of our protractor. Like we said earlier though, the length of the arms of our angle can be anything we want them to be. No matter how long or how short, they don’t affect the actual angle itself. This means we could extend the arms of this angle we want to measure until they reach out to the measure portion of our protractor.
Now at this point, we start to see where we could make progress measuring this angle. We can look at where the first arm crosses our protractor and write down that measurement. That looks to be about 53 degrees and then we can look at where the second arm crosses and write down that angle measurement. That looks to be about 96 degrees.
With these values figured out, we’re now ready to write out an expression for the actual angle we want to measure. Remember we often refer to this angle using the Greek letter 𝜃. We could say then that 𝜃 is equal to the difference between these two measured angles. It’s equal to 96 degrees minus 53 degrees. This is one way to make an angle measurement, to measure out the angle 𝜃.
But notice that there is a disadvantage to the way we’ve done it. Using this process, we’ve had to make two angle measurements: one for the first arm of our angle and one for the second. That means two chances at making a misreading of our protractor or another slight error.
One thing that could make this whole measurement process easier would be after we put the vertex of our angle at the protractor midpoint to rotate our angle so that the one arm is lying along the zero degree direction, in other words, to rotate our angle so it now looks something like this, where now if we extend the lower arm of our angle, it crosses the zero degree marking on our protractor.
The advantage of this — of lining up the lower arm of our angle at zero degrees — is that now we only need to make one angle measurement and that will be at the place where the line extended from the upper arm of our angle crosses the protractor. This means we’ll only have to make one measurement. And therefore, we only have one chance to make a measurement error. Overall then, this measurement process will lead to fewer errors than the other process we tried.
Let’s add that step then of rotating our angle so that one arm lies along zero degrees to our process of angle measurement. So like we said, step one is to place the protractor midpoint on the vertex of the angle and step two is to line up one side of the angle with the zero line of the protractor. That’s what it means to have one of the arms of the angle cross this zero point.
As we said, we now only have one measurement to make, that’s the measurement of where this line from the upper arm crosses our protractor. That looks like that’s about at 43 degrees. And so that will be the measurement of this angle. And we’ve just performed our third and final step in this process: reading the degrees where the other arm crosses the number scale on the protractor. So that is our three-step process for measuring any angle using a protractor.
Speaking of angles, there are several different types or classes of angles that we can measure using a protractor. Let’s clear some space to talk about those types. The different kinds of angle that we may encounter are called acute, right, obtuse, straight, and reflex angles.
Looking at the first type, acute angles, this is an angle of less than 90 degrees. So for example, our angle from earlier which was about 43 degrees was an acute angle. When we go to measure an acute angle with a protractor, we’d measure it very much like we did our earlier angle, where we align the vertex of the angle with the midpoint of our protractor and then one of the arms of the angle with zero degrees. We then extend the other arm of the angle as needed in order to cross the number scale of our protractor and read off that value. So that’s how we’d measure this type of angle.
Moving on, a right angle is different from an acute angle. As the name implies, a right angle is an angle of 90 degrees exactly. If we took a right angle and positioned it on our protractor, then with the angle vertex at our protractor midpoint and one arm pointing along zero degrees, the other arm points directly at 90 degrees. That’s what it means to be a right angle.
Moving on to the next angle type: an obtuse angle goes even further than an acute or a right angle. An obtuse angle is always greater than 90 degrees and always less than 180 degrees. When we measure an obtuse angle on a protractor, that angle will always be somewhere between these two boundaries of 90 and 180 degrees. Because our protractor goes up to 180 degrees, obtuse angles are readily measurable using them.
This leads us to the next angle type and that is straight angles. By its name, you may be able to guess what a straight angle looks like. At first, it just looks like a single ray. But actually, a straight angle is two rays: one pointing in one direction and one pointing in 180 degrees away from that original direction. A straight angle then is simply a straight line.
If we laid out a straight angle on our protractor, we could still measure it, but just barely. Notice that this is the maximum range that our protractor allows: zero to 180 degrees. That raises a question though as we move on to our last angle type: reflex angles.
At first, a reflex angle may look like some other angle type until we notice that 𝜃 is actually greater than 180 degrees in this case. In fact, for reflex angle, 𝜃 is greater than 180 degrees and less than 360 degrees that is less than one complete revolution.
But this may seem to be a problem because remember our protractor only goes up to 180 degrees. How would we use it then to measure a reflex angle? As a side note, there are some much less common protractors which are called two-sided protractors. These are protractors which start at zero degrees, go to 90 and then 180, and then keep going because of their second side all the way around to 360 degrees.
If we have a protractor like this, measuring a reflex angle would not be an issue. But it turns out it’s still possible to do it with some clever arranging using our one-sided or regular protractor. Here’s, how we can do that. Recall that for a reflex angle, the angle 𝜃 is always greater than 180 and less than 360. This means that the angle always falls within a range of 180 degrees starting at 180 and ending at 360 the difference between those is 180. And our protractor we notice has a range of 180 degrees as well, starting at zero and ending at 180.
So one way to measure a reflex angle using our protractor is to follow our process as usual, putting its vertex at the midpoint of the protractor and aligning one arm with zero degrees. If we then extend out the other arm and record the value where that line crosses the number scale, we record that value which looks to be about 75 degrees.
Now if this were an acute angle we were working with, that would be the end of the story. 75 degrees would be the angle measure. But remember it’s not really this angle that we’re measuring. But rather, it’s this angle here, much greater than 75 degrees. Here’s how we can figure out the measure of this reflex angle.
First of all, notice that there is an arc of 180 degrees down here below our protractor. And that arc those 180 degrees are included in our angle 𝜃. So we’ll write that the angle of our reflex angle 𝜃 is equal to 180 degrees plus. And here, we see we want to add this angle to 𝜃, that is, the angle between 180 degrees on our protractor and where the upper arm ends.
And what is that angle that we’ve marked out in blue? Well, we can see that it’s this angle here on our protractor, that is between 180 and 75 degrees. So to get that angle, we would take 180 degrees and subtract from it the measured value of 75 degrees. If we added this to 180 degrees, then that would give us the measure of our reflex angle 𝜃. All that to say, regardless of which angle type we’re working with, it’s possible to measure that angle using a regular protractor.
Okay, let’s take a quick moment to summarize all this. The first thing we learned is that there’s a three-step process to measuring an angle using a protractor. First, we place the protractor midpoint on the vertex of the angle or the vertex of the angle on the protractor midpoint, whichever movement is easier. That will look something like this.
Then, as a next step, we go on to line up one side or arm of the angle with the zero line of the protractor. And finally, we read off the degrees where the other side crosses the number scale.
Along with this three-step process, we saw that there’re different types of angles to measure. And those five angle types are acute angles, where the angle is between zero and 90 degrees; right angles, where 𝜃 is exactly 90 degrees; obtuse angles, where 𝜃 is between 90 and 180; straight angles, where 𝜃 is exactly 180 degrees; and finally reflex angles, where 𝜃 is between 180 and 360 degrees. And we saw that all five of these angle types can be measured using a protractor.