In the first post in these series, I discussed the basic circle facts. This post concerns angles and circles, and dividing circles by angle.
Angles of a Circle
Suppose you stand at the center of a circle and turn around that you face each and every point on the circle. You would turn all the way around, which is an angle of 360º. In this sense, a whole circle has an angle of 360º. If you divided a circle equally, you could calculate the angle of each “slice”. Here are a few division results that could help you to know on test day (I’m just giving the ones that come out as nice round numbers, not the ones that result in ugly decimals):
360/2 = 180
360/3 = 120
360/4 = 90
360/5 = 72
360/6 = 60
360/8 = 45
360/9 = 40
360/10 = 36
360/12 = 30
Arcs and Arclength
Suppose we look at a “slice” of a circle, like a slice of pizza.
The curved line from A to B, a part of the circle itself, is called an arc. This corresponds to the crust of the pizza.
We can talk about the size of an arc in one of two ways: (a) its angle, sometimes called “arc angle” or “arc measure“, and (b), its length, called arclength. The angle of the arc, its arc measure, is just the same as the angle at the center of the circle. Here ∠AOB = 60°, so the measure of arc AB is 60°.
We find the arclength by setting up a proportion of part-to-whole. The angle is part of the whole angle of a circle, 360°. The arclength is part of the length all the way around, i.e. the circumference. Therefore:
Here, let’s say the radius is r = 12. Then, the circumference is . Since the angle is 60°, the ratio on the left side, angle/360, becomes 1/6. Call the arclength x.
In other words, since the angle 60° is one sixth of the full angle of a circle, the arclength is one sixth of the circumference.
In the next post, I will discuss straight lines and circles. This post is the second in a series. Here’s the whole series.
2) GMAT Geometry: Circles and Angles
Finally, here’s a practice question.
1) In the shaded region above, ∠KOL = 120°, and the area of the entire circle is . The perimeter of the shaded region is
Practice Question Explanation
1) The area , so r = 12. This means KO = 12 and OL = 12, so those two sides together are 24. The remaining side is arc KL. The whole circumference is . The angle of 120° is 1/3 of a circle, so the arclength is 1/3 of the circumference. This means, , and therefore the entire perimeter is . Answer = C.