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Hey guys! Welcome to this video on the radius, diameter, and circumference of a circle.

Circles have been around for as long as the Earth has been around. People were able to see natural circles by observing the moon, the sun, and other various naturally circular shapes.

The first technological invention using a circular shape, however, wasn’t until 3500 BC, and it was the invention of the potter’s wheel. Then, 300 years later, they were used for the wheels of chariots. As people began to see the value and use for circular-shaped objects, they begin to study circles.

Things like radius, diameter, and circumference are terms that helps us to keep track of various measurements of a circle.

So, now, let’s take a look what each of these measurements represent.

## Definitions

### Midpoint of a Circle

First, let’s define **midpoint**, so you’ll understand what I’m talking about as I reference it. Here’s a circle:

The midpoint is the exact center of the circle, where the dot is.

Now, let’s look at these other terms.

### Radius of a Circle

Radius is the length from the midpoint of the circle to the outer edge of the circle. The radius is represented by the lowercase letter (r).

### Diameter of a Circle

Diameter is the full length of the circle running from the edge, through the midpoint, all the way to the other side. That is this whole length right here. The diameter of a circle is represented by the letter (d).

### Circumference of a Circle

Now, circumference is the distance around the outside edge of this circle. Circumference is represented by the uppercase letter (C).

Circumference is comparable to the perimeter of a shape, like a **parallelogram**. If you were to cut the line of a circle, as if it were a string, and lay it out to measure. This length would be equivalent to the circumference. However, since a circle has a continuous curve, we use the word circumference rather than perimeter to distinguish it.

Now that we’ve looked at what the radius, diameter, and circumference are, let’s look at how to calculate each one.

## Calculations

If someone were to just kinda hand you a piece of paper with a circle on it…. Well, actually, that would be pretty weird.

But let’s say we wanted to find the radius, diameter, and circumference of that circle, and all we have is a ruler.

The easiest thing to start with would be to take the ruler and measure, from the very center of the circle, the length between the outer edge. That would be the diameter.

Let’s say, that when we measured, we got a length of 9 cm for the diameter.

Well, we know that if our radius runs from the midpoint to the outer edge, then all we have to do to find the length of our radius would be to divide the length of the diameter by 2.

So, when we take 9 and divide it by 2 we get a radius length of 4.5 cm.

### Radius Formula

The formula for the radius can be written as (r=frac{d}{2})

### Diameter Formula

The formula for diameter can be written as (d=2r)

### Circumference Formula

The formula for the circumference of a circle is (C=pi times d), or it can be written as (C=2times pi times r). Either one works!

Now, you may be asking, “Well where did pi come from, and why do we all the sudden get the circumference if we multiply said pi by our diameter? Who decided that?” If you are not asking that question… You should, and I’m going to answer it anyways.

**Pi** is a symbol we use in mathematics to represent the number 3.14. And actually that is just pi rounded to the nearest hundredth. Pi actually has no end, and no predictable pattern. It just keeps going.

However, when you see the symbol (pi), generally (and in our case), 3.14 will suffice.

Pi is not a random number that mathematicians made up, and declared “we will multiply the diameter by the number every time, and call it a circumference.” On the contrary, pi was discovered to be the constant **ratio** between the circumference and the diameter.

That is why and how we got the formula for the circumference of a circle.

Now, let’s take the circle with the diameter of 9 cm, and radius of 4.5 cm, and calculate the circumference.

I’m going to use the formula with the diameter for this one.

So, circumference equals (I’m just gonna rewrite the formula to help us follow our work), (C=pi times d), equals pi times diameter. So now all we need to do is to plug in our number for diameter. This is equal to, and also we said pi is equal to 3.14, (C=(3.14)(9text{ cm})=28.26text{ cm}).

And here’s our answer! Now to practice, try drawing a circle on a piece of paper, and measure your diameter with a ruler. Then, find your radius, and circumference.

I hope that this video has been helpful for you.

See you guys next time!

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