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Learn the relationship between the radius, diameter, and circumference of a circle.

## What is a circle?

We've all seen circles before. They have this perfectly round shape, which makes them perfect for hoola-hooping!
Every circle has a center, which is a point that lies exactly at the... well... center of the circle. A circle is a shape where distance from the center to the edge of the circle is always the same:
You might have suspected this before, but in fact, the distance from the center of a circle to any point on the circle itself is exactly the same.

This distance is called the radius of the circle.
Which of the segments in the circle below is a radius?

## Diameter of a circle

The diameter is the length of the line through the center that touches two points on the edge of the circle.
Which of the segments in the circle below is a diameter?

Notice that a diameter is really just made up of two radii (by the way, "radii" is just the plural form of radius):
So, the diameter d of a circle is twice the radius r:
d, equals, 2, r
Find the diameter of the circle shown below.
units

Find the radius of the circle shown below.
units

## Circumference of a circle

The circumference is the distance around a circle (its perimeter!):
Here are two circles with their circumference and diameter labeled:
Let's look at the ratio of the circumference to diameter of each circle:
Circle 1Circle 2
start fraction, start text, C, i, r, c, u, m, f, e, r, e, n, c, e, end text, divided by, start text, D, i, a, m, e, t, e, r, end text, end fraction:start fraction, 3, point, 14159, point, point, point, divided by, 1, end fraction, equals, start color #e84d39, 3, point, 14159, point, point, point, end color #e84d39start fraction, 6, point, 28318, point, point, point, divided by, 2, end fraction, equals, start color #e84d39, 3, point, 14159, point, point, point, end color #e84d39
Fascinating! The ratio of the circumference C to diameter d of both circles is start color #e84d39, 3, point, 14159, point, point, point, end color #e84d39
start fraction, C, divided by, d, end fraction, equals, start color #e84d39, 3, point, 14159, point, point, point, end color #e84d39
This turns out to be true for all circles, which makes the number start color #e84d39, 3, point, 14159, point, point, point, end color #e84d39 one of the most important numbers in all of math! We call the number pi (pronounced like the dessert!) and give it its own symbol start color #e84d39, pi, end color #e84d39.
start fraction, C, divided by, d, end fraction, equals, start color #e84d39, pi, end color #e84d39
Multiplying both sides of the formula by d gives us
C, equals, start color #e84d39, pi, end color #e84d39, d
which lets us find the circumference C of any circle as long as we know the diameter d.

## Using the formula $C = \pi d$C, equals, pi, d

Let's find the circumference of the following circle:
The diameter is 10, so we can plug d, equals, 10 into the formula C, equals, pi, d:
C, equals, pi, d
C, equals, pi, dot, 10
C, equals, 10, pi
That's it! We can just leave our answer like that in terms of pi. So, the circumference of the circle is 10, pi units.
Your turn to give it a try!
Find the circumference of the circle shown below.
Enter an exact answer in terms of pi.
units

## Challenge problem

Find the arc length of the semicircle.
Enter an exact answer in terms of pi.
units