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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

## Want to join the conversation?

• How do we find the circumference when the radius is given? (<im a lil confuse)
• if the diameter is given we find the circumference by diameter x pi, so if the radius is half the value of the diameter then if you are only given the radius we find the circumference by radius x 2 x pi because radius x 2 = diameter
• can you tell me the drivation of this formula
• Here the Greek letter π represents a constant, approximately equal to 3.14159, which is equal to the ratio of the circumference of any circle to its diameter. One method of deriving this formula, which originated with Archimedes, involves viewing the circle as the limit of a sequence of regular polygons.
• what happens if the circle is not perfectly round?
• Then technically it's not a circle
• chicken sandwich
• why is this so hard :(
• it's not. you just have to understand
• What is zero divided by zero
• Interesting question! Think of 0 divided by 0 as the answer to the question “what number times 0 is 0?”. Because any number times 0 is 0, 0 divided by 0 can be anything! For this reason, 0 divided by 0 is called indeterminate.

If you take calculus later on, you will frequently encounter the indeterminate expression 0 divided by 0 in limit problems. What this means is that the result is inconclusive, so more work is required to calculate the limit or determine that the limit doesn’t exist.
• how do i find the circumference if the diameter is given
• Hi, to find the circumference and you have the diameter all you have to do is do the diameter times pi and the answer you get is the circumference. Another formula to find the circumference is if you have the diameter you divide the diameter by 2 and you get the radius. Once you have the radius you times the radius by 2 and times it by pie and then you get the circumference. Here are the two different formulas for finding the circumference:
C = πd
C = 2πr
d = diameter, C = circumference, and r = radius
Hope this helped :)