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Course: Digital SAT Math > Unit 13
Lesson 4: Circle theorems: advancedCircle theorems — Harder example
Watch Sal work through a harder Circle theorems problem.
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how did sal get the whole angle was 2pi? why isnt it 3pi?(42 votes)
There's two ways you can measure a circle, you can use Degree or Radians
In this problem they have used radians.
We are familiar with measuring angles in degrees as they teach us that first in schools and it's also easy to calculate...
Here are some important conversions from degrees to radians or vice versa
360 degrees (1 * 360) = 2π (1 * 2π)
270 degrees (3/4 * 360) = 3π/2 or 3/2 π (3/4 * 2π)
180 degrees (1/2 * 360) = π (1/2 * 2π)
90 degrees (1/4 * 360) = π/2 or 1/2 π (1/4 * 2π)
Hope this helped and cleared your doubt!(34 votes)
Why exactly do we use radians?(8 votes)
Calculus is done entirely in radians, elsewise you'd have to use multiplicative constants far too frequently.
Also, when calculating the length of an arc, it is very, very easy to do so in radians.(5 votes)
What is a radian, exactly?(5 votes)
A radian is a cool way of measuring a circle. Think of taking the length of a radius, which we constantly refer to when dealing with a circle, and making it into a tape measure. Instead of stretching straight from the center to the edge of the circle, now it is bendy. One radius would not make a tape measure long enough to fit all the way around a circle, however. If you make your tape measure twice the radius times π, then the tape measure will fit EXACTLY once around the circle.
When you use a radius to measure around a circle that way, we call the measurement aradian
and it takes 2π radians to fit around a circle.
If you take an angle that covers the length of that radius along the circumference, that angle measuresone radian. And, if you go all the way around the circle (360 degrees) the angle is equal to 2π radians. Exacly half a circle is equal to one π radian.(37 votes)
Why aren't we using the formula we were given :
Arc length/circumference × central angle/360?
When I used it it gave me something different :((6 votes)- It should give you the same thing:
Arc length/circumference = central angle/360
Arc length = circumference * central angle / 360
Arc length = 10pi * 260 / 360
Arc length = 65pi / 9 feet
Is this different than the process you used?(4 votes)
why is it multiplied by the circumference?(7 votes)- you are trying to get the fraction of the circumference that the sector length is(0 votes)
- why didnt he just find the length of the arc and subtract that from the circumference, its much faster(4 votes)
The question WAS what the length of the arc was(3 votes)
what is central angle?(2 votes)- A central angle is an angle between two radii of a circle, so that the angle comes from the center point of the circle. The main place you'll see central angles talked about on the SAT is questions about arc length. Arcs can be described by the central angle that they contain (if you take the two endpoints of the arc and draw lines towards the center of the circle, and have your central angle be the angle between those lines)(6 votes)
When would the first 7 digits of pi(3.14159) actually be used in an equation?(2 votes)- Typically, you would never see the actual digits of pi used in an equation, because the first however-many digits simply aren't equal to pi. It's more concise and precise to keep the actual pi symbol in the equation, and to only use a number approximation when using that equation to calculate a value. Personally, I don't remember ever seeing the digits of pi in a standard equation instead of pi itself.(4 votes)
at3:26, why did he divide 4pi/3 by 2pi and then multiply both by 3pi?(2 votes)- From the author:I hope this might help: 4π/3 out of 2π is the part/whole of the measure of the angles we're looking at. The big magenta angle = 4π/3 radians; the angle measure of the entire circle, by definition, is 2π radians. So, if we take that very same fraction of the length of the entire circumference, we'll have our answer, because those things are proportional. When Sal multiplies this fraction by 3π (the length of the total circumference), he is taking that same fraction of the measure of the entire circumference.
One way of looking at it is that the part/whole ratio of the angles is going to be the same as the part/whole ratio of the lengths of the arcs they chop the circle into.(4 votes)
- Pointing something out - if the whole circle's circumference is 2 pi, and the given central angle is 2 pi/3, the given angle is 1/3 of the circles circumference, or 120 degrees. You can then determine that the remaining arc's measure is 240 degrees (360-120), so why go through all the math to get there?(2 votes)
3:26Video transcript
- [Instructor] A circle has a
circumference of 10 pi feet. An arc, x, in this circle
has a central angle of 260 degrees. What is the length of x? So let's just visualize
what's going on here. So we have a circle. I can't draw a circle that well, but you get the point. So that's our circle. This is the center of our circle. It has a circumference of 10 pi feet, so if we were to go all
the way around the circle, this, it has a
circumference of 10 pi feet. Now you have an arc, x. You have an arc, x, in the circle that has a central angle of 260 degrees. So let's think about, so 260 degrees, so if you go straight up, that would, let's see, if you go this far it'd be 90 degrees, 180 degrees. If you were to go all the way here you'd get to 270, so it's
gonna be right around, and we just approximate,
right around there. So that would be be a
central angle of 260 degrees. And this is the arc. This is arc. Let me do this in a different color. Let me do it in purple. So this is arc x right over here. And we wanna figure out its length. Well think about it this way. Its central angle is 260 degrees. What's the central angle
of the circumference? Well the circumference is
going all the way around. So if you're going all the way around, that is 360 degrees. So the fraction that this, the arc x is, the length of x is of
the entire circumference, that's gonna be the same fraction that its central angle is of 360 degrees. So once again the entire
circumference is 10 pi feet. 10 pi feet. That's the circumference. Now the fraction of that circumference that's going to be arc x, that's going to be the fraction that 200. That's gonna be the same fraction of the central angle of that arc relative to the central angle if you were to go all the way around,
which would be 360 degrees. So it'll be 260/360 of the circumference. So what's that going to be? We'll let's see. We can simplify this a little bit. We could, if we divide the numerator and the denominator by 10, 10 divided by 10 is one. And let me make it. Actually, let me just write it this way. So we could write this as
260 times 10 pi is going to be 2600 pi over 360. And now we just need to simplify. If we divide the numerator
and the denominator by 10, this is gonna be 260. This is going to be 36. If we divide the numerator and the denominator by, let's see. It seems like they are
both divisible by four. 260 divided by four is
going to be, let's see. 200 divided by four is 50. 60 divided by four is 15. 15 plus 60 it's gonna be 65. So let me do this in a different color. This is gonna be 65 and 36
divided by four is nine. And just so you see I didn't, you know, I was tryna do it in my head, and it's good to get
some practice doing that, but all I did is divided
both the numerator and the denominator by four. Four goes into 260. Four goes into 26. Six times four is 24. Subtract, you get a two. Bring down that zero. Four goes into 20 five times. Five times four is 20. So that's all I did there. So it's 65 pi over nine and it looks like that's about as simplified
as I'm gonna get. And that's nice 'cause
that's one of the choices right over here.