Measurement of angles
circle with central angle "θ" and radius "r" marked, delineating the . A big part of the reason that I've explained the relationship between the circle formulas and. It is easy to convert between degree measurement and radian measurement. The length of an arc is is the radius r times the angle θ where the angle is If the gauge is 5 feet, find the difference in length of the two rails to the nearest. What is the connection between degrees and radians? Let's see A circle has a radius of 8 inches and central angle AOB makes an arc AB of length 2 feet.
So let me explain that. So if that's the center of the circle, and if we make this ray our starting point or one side of our angle, if you go all the way around the circle, that represents degrees. And the notation isand then this little superscript circle represents degrees. This could be read as degrees. Now, you might be saying, where did this number come from?
And no one knows for sure, but there's hints in history, and there's hints in just the way that the universe works, or at least the Earth's rotation around the sun.
You might recognize or you might already realize that there are days in a non-leap year, in a leap year.
And so you can imagine ancient astronomers might have said, well, you know, that's pretty close to And in fact, several ancient calendars, including the Persians and the Mayans, had days in their year. And is also a much neater number than It has many, many more factors.
It's another way of saying it's divisible by a bunch of things. But anyway, this has just been the convention, once again, what history has handed us, that a circle is viewed to have degrees. And so one way we could measure an angle is you could put one of the rays of an angle right over here at this part of the circle, and then the other ray of the angle will look something like this.
And then the fraction of the circle circumference that is intersected by these two rays, the measure of this angle would be that fraction of degrees. So in this case, this would be 60 degrees. I could do another example.
So let's say I had a circle like this, and I'll draw an angle. I'll put the vertex at the center of the angle. I'll put one of the rays right over here.
You could consider that to be 0 degrees. Or if the other ray was also here, it would be 0 degrees. And then I'll make the other ray of this angle, let's say it went straight up.
Let's say it went straight up like this. And the way that we oftentimes will denote that is by a symbol like this. But this literally means a degree angle. Let's do one more example. Let's do one more example of this, just to make sure that we understand what's going on. Actually, at least one more example. Maybe one more if we have time. So let's say that we have an angle that looks like this. Once more, I'm going to put its vertex at the center of the circle.
That's one ray of the angle.
And let's say that this is the other ray. This right over here is the other ray of the angle. If you want the arc length, the equation looks like this: The Central Angle Theorem Let's add a twist to your example where you're in the arena with your neighbor and your teacher.
Arc (geometry) - Wikipedia
Now there's a third person you know in the arena: And one more thing: You have to turn around to see them. Your neighbor is approximately across the arena from your friend and your teacher. From your neighbor's point of view, there's an angle formed by their line of sight to the friend and their line of sight to the teacher.
That's called an inscribed angle.
$$ S = r \theta $$ Formula and Equation
An inscribed angle is an angle formed by three points along a circle's circumference. The Central Angle Theorem explains the relationship between the size of the central angle, formed by you, and the inscribed angle, formed by your neighbor. The Central Angle Theorem states that the central angle is twice the inscribed angle. This assumes that you're using the same endpoints.
You're both looking at the teacher and the friend, not anybody else. Here's another way to write it. Let's call your friend's seat A, your teacher's seat B and your neighbor's seat C. You, at the center, can be O. This makes some sense. You're closer to the friend and the teacher, so to you they look further apart a larger angle.
To your neighbor on the other side of the stadium, they look much closer together a smaller angle. Exception to the Central Angle Theorem Now, let's shift things up. Your neighbor on the far side of the arena starts to move around! They still have a line of sight to the friend and the teacher, but the lines and angles keep shifting as the neighbor moves.
As long as the neighbor stays outside the arc between the friend and the neighbor, the Central Angle Theorem still holds true! But what happens when the neighbor moves between the friend and the teacher?
Now your neighbor is inside the minor arc, the relatively small distance between the friend and the teacher compared to the larger distance around the rest of the arena. Then you reach an exception to the Central Angle Theorem.