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# Circle theorems — Harder example

Watch Sal work through a harder Circle theorems problem.

## Want to join the conversation?

• how did sal get the whole angle was 2pi? why isnt it 3pi?
• 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!
• Why exactly do we use radians?
• 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.
• What is a radian, exactly?
• 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 a `radian`
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 measures `one 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.
• why is it multiplied by the circumference?
• you are trying to get the fraction of the circumference that the sector length is
• why didnt he just find the length of the arc and subtract that from the circumference, its much faster
• The question WAS what the length of the arc was
• 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?
• at , why did he divide 4pi/3 by 2pi and then multiply both by 3pi?
• 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.
• At , Why did he multiply by the circumference (3pi)?