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.

Cross-multiply:

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. Here’s a practice question.

## 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

(A)

(B)

(C)

(D)

(E)

## 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**.

Thanks MIke. The link clears the concept even more.

Dear Shawn,

You are quite welcome, my friend. I’m glad these articles cleared this up for you! Best of luck in the future!

Mike

Dear Mike,

I understand that the “angle of the arc, its arc measure, is just the same as the angle at the center of the circle”

I can also see that ∠AOB = 60°, so the measure of arc AB is 60°. (In the first figure above)

What I cannot make out is -> Where is the the angle of the arc? I can see the angle at the centre of the circle (∠AOB = 60°), but I cannot see the angle of the arc. Is there a picture or image in which angle of the arc is depicted? I cant visualize it.

Thanks

Shawn

Dear Shawn,

What you are asking is much easier to show in person, and hard to convey in words. One way to think about it is that a circle, a full circle is 360° — what does that mean? Well, it could mean that if I stand at the center of the circle, and follow it all the way around (turning my body, not just my head) then by the time I saw the whole circle, I would have turned 360°. Much in the same way: if you are standing at the center and looking or pointing at arc AB — if you start at A and rotate to B, or vice versa, you will have rotated 60°.

Does all this make sense?

Mike

Thanks Mike your explanation is really helpful.

Thanks Mike your explanation is really helpful

I have one more question, but that’s on the statement below:

“Since the angle 60° is one sixth of the full angle of a circle, the arclength is one sixth of the circumference”

Question : Is it always true that -> IF the angle X° (Angle at the centre of the circle) is 1/Y of the full angle of a circle, THEN the arclength is ALSO 1 / Y of the circumference

Dear Shawn,

I’m happy to respond. Yes, there is strict proportionality between central angle to 360° and arclength to circumference. In other words, if the central angle is 1/n of the full 360° of the circle, then the arc must be 1/n of the circumference. See:

http://magoosh.com/gmat/2013/slicing-up-gmat-circles-arclength-sectors-and-pi/

I hope you find that helpful.

Mike