Geometry: Circles

Now we venture away from all the lines and angular shapes, and begin our discussion of circles. And remember that the transcript of the video is directly under it!

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Transcript of Video

Now, we can talk about circles. A circle is a set of all points equidistant from a fixed point, called the center. So here is an example of a circle. Point A is on the circle. Point B and the center inside the circle, or within the circle.

So it’s very important to understand the distinction between on the circle versus inside the circle.

A segment from the center to any point on the circle is called a radius and the plural of that word is radii. All radii of the same circle have the same length, by definition. The length of the radius is abbreviated as r.

A segment with two endpoints on the circle is called a chord. Different chords can have different angles and different lengths, so FG is a relatively short chord. EH is longer, and DJ that goes almost all the way through across the circle so that’s a much longer chord. There is no shortest chord, the chords can be anything down to zero.

But the longest chord is a chord that passes through the center. A chord that passes through the center of the circle is called the diameter. So a diameter is a chord, in fact, it is the maximum length chord–the longest possible chord. And notice that a diameter is made up of two radii. KO is a radius and MO is a radius, so we can say diameter = 2R.


The length around the whole circle is called the circumference, denoted by c. The ratio of c to d is one of the most special numbers in all of mathematics, pi. So c = pi times d, circumference = pi times d, and we could also write that as circumference = 2 pi times radius. That’s actually a more useful form as we’ll find in a few minutes. How big is pi?

Well, pi is an irrational number approximately equal to this decimal. And in fact, that decimal goes on forever, there is no repeating pattern, it just goes on forever. If we locate pi on the number line, notice it’s very close to 3. It’s between 3 and 4, relatively close to 3. In rough approximations, very rough approximations, we can simply approximate pi as 3.

And if we need a slightly better approximation, we can use 3.14 or 22 over 7, that’s actually a very useful approximation for pi. We can also talk about pieces of a circle. The highlighted curve from A to C is called an arc, so that is a piece of all the way around a circle. Technically, if we set arc AC, that would mean the short route around the circle.

But, the test is usually very careful to say arc ABC to specify a three letter name for the arc. In that way, it makes it clear that we didn’t go the other way through point D.


Finally, we can talk about the area of circles. This famous formula was discovered by the brilliant mathematician Archimedes. Archimedes was really one of the greatest mathematicians of all time.

And his formula is area = pi r squared. We notice that the area formula is in terms of the radius. We can also expressed the circumference in terms of the radius. So what that means is, if we know the radius of a circle, we can find out all its other values.

This indicates a primary strategy for circle problems.

Whatever you’re given, find the radius first, and then use the radius to find whatever else you need. That is a really important mindset to have when you’re dealing with circles.

Practice Problem

Here’s a very simple practice problem. Pause the video, and then we’ll talk about this.

Okay, the area of a circle is 28 pi.

What is the circumference? Well, set that area 28 pi = to pi r squared, cancel the pis, we get r squared = 28, or r = the square root of 28. Now of course, we can simplify that to 2 root 7. The circumference = 2pir. So 2 pi times 2 root 7 that would be 4 pi root 7.

And that is the circumference of the circle.


In summary, all radii of a circle are the same length. A chord is a segment that has both endpoints on the circle. A diameter is a chord through the center, this is the longest possible chord in a circle. Of course, the diameter = 2r.

So that means, that we can express the circumference either as pi d or as circumference = 2pir. An arc is a piece of the curve of a circle, denoted on the test by three points. Area = pi r squared. And the number 1 circle strategy is find the radius first, and use the radius to find everything else.

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  • Mike MᶜGarry

    Mike served as a GMAT Expert at Magoosh, helping create hundreds of lesson videos and practice questions to help guide GMAT students to success. He was also featured as "member of the month" for over two years at GMAT Club. Mike holds an A.B. in Physics (graduating magna cum laude) and an M.T.S. in Religions of the World, both from Harvard. Beyond standardized testing, Mike has over 20 years of both private and public high school teaching experience specializing in math and physics. In his free time, Mike likes smashing foosballs into orbit, and despite having no obvious cranial deficiency, he insists on rooting for the NY Mets. Learn more about the GMAT through Mike's Youtube video explanations and resources like What is a Good GMAT Score? and the GMAT Diagnostic Test.

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