example:

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?

Choose all answers that apply:

Choose all answers that apply:

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 ddd of a circle is twice the radius rrr:

d = 2rd=2rd, equals, 2, r

Find the diameter of the circle shown below.

units

Explain

Find the radius of the circle shown below.

units

Explain

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 1 Circle 2

\dfrac{\text{Circumference}}{\text{Diameter}}

Diameter

Circumference

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: \dfrac{3.14159...}{1} = \redD{3.14159...}

1

3.14159...

=3.14159...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 #e84d39 \dfrac{6.28318...}{2} = \redD{3.14159...}

2

6.28318...

=3.14159...start 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 CCC to diameter ddd of both circles is \redD{3.14159...}3.14159...start color #e84d39, 3, point, 14159, point, point, point, end color #e84d39

\dfrac{C}{d} = \redD{3.14159...}

d

C

=3.14159...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 \redD{3.14159...}3.14159...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 \redD\piπstart color #e84d39, pi, end color #e84d39.

\dfrac{C}{d} = \redD{\pi}

d

C

=πstart fraction, C, divided by, d, end fraction, equals, start color #e84d39, pi, end color #e84d39

Multiplying both sides of the formula by ddd gives us

C = \redD\pi dC=πdC, equals, start color #e84d39, pi, end color #e84d39, d

which lets us find the circumference CCC of any circle as long as we know the diameter ddd.

Using the formula C = \pi dC=πdC, equals, pi, d

Let's find the circumference of the following circle:

The diameter is 101010, so we can plug d = 10d=10d, equals, 10 into the formula C = \pi dC=πdC, equals, pi, d:

C = \pi dC=πdC, equals, pi, d

C = \pi \cdot 10C=π⋅10C, equals, pi, dot, 10

C = 10\piC=10πC, equals, 10, pi

That's it! We can just leave our answer like that in terms of \piπpi. So, the circumference of the circle is 10 \pi10π10, pi units.

Your turn to give it a try!

Find the circumference of the circle shown below.

Enter an exact answer in terms of \piπpi.

units

Explain

Challenge problem

Find the arc length of the semicircle.

Enter an exact answer in terms of \piπpi.

units

Explain

Area and circumference of circles

Radius, diameter, circumference & π

Labeling parts of a circle

Radius, diameter, & circumference

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Practice: Radius and diameter

Radius & diameter from circumference

Relating circumference and area

Practice: Circumference of a circle

Area of a circle

Practice: Area of a circle

Partial circle area and arc length

Practice: Circumference of parts of circles