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Circles

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

So it’s very important to understand the distinction of 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 radius is abbreviated as r.

A segment with two endpoints on the circle is called a chord. Different chords can have different angles and difference 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’s no shortest chord. The short, 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 a 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 equals 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 equals pi times d. Circumference equals pi times diameter. And we could also write that as circumference equals 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, in fact, that decimal goes on forever.

There’s 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 three. 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 22/7.

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 the 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, and that way it makes it clear we didn’t go the other way through point D.

Finally, we can talk about the area of the circle 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 equals pi r squared. We notice that the area formula is in terms of the radius.

We can also express 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 of its other value. This indicates a primary strategy for circle problems. Whatever you are give, 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. 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 equal to pi r squared.

Cancel the pi’s, we get r squared equals 28. Or r equals the square root of 28. Now of course we can simply that, to 2 root 7. The circumference equals 2 pi r. So 2 pi times 2 root 7. That would be 4pi root 7, and that is the circumference of the circle.

In summary, all radia, 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 equal 2r. So that means we can express the circumference there as pi d or as circumference as 2pi r.

An arc is a piece of the curve of a circle denoted on the test by three points. Area equals pi r squared. And the number one circle strategy is find the radius first, and use the radius to find everything else.