If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

# 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 aren't we using the formula we were given :
Arc length/circumference × central angle/360?
When I used it it gave me something different :(
• 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?
• 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
• what is central angle?
• 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)
• Are u provided a sheet for all the formulas in digital sat
• When would the first 7 digits of pi(3.14159) actually be used in an equation?