15.01 Introducing the circle

We know from the definition of a circle that every point on the circle is the same distance from the central point. We call this distance the radius.

A line segment joining the central point to some point on the circle is called "a radius".

Any two radii of a given circle have the same length which is called "the radius" of the circle.

Another distance we should pay attention to in a circle is the diameter.

A line segment joining two points on the circle, passing through the central point, is called "a diameter".

A line segment joining two points on the circle, passing through the central point, is called "a diameter". Any two diameters of a given circle have the same length which is called "the diameter" of the circle.

We can also see that the diameter is equal to twice the radius, since any diameter can be split into two radii.

The other distance we will look at is the circumference.

The circumference of a circle is the distance around its boundary. We can think of the circumference as the perimeter of the circle.

We have already seen that a diameter can be split up into two radii. This tells us that the diameter is equal to twice the radius. We can express this using the formula:d=2r

Where d is the diameter and r is the radius of a circle. This formula also tells us that the radius of a circle is equal to half the diameter.

In order to relate the circumference of a circle to the radius or diameter, we will need to introduce another value: \pi . \pi is an irrational number representing the ratio between the circumference and diameter of a circle.

Knowing that \pi is the ratio between the circumference and the diameter, we can relate the two distances using the formula:C=\pi d

Where C is the circumference and d is the diameter of a circle.

We can then use the relationship between the radius and diameter to find the formula relating the circumference and radius:C=2\pi r

Where C is the circumference and r is the radius of a circle.

Now that we have some formulas to relate the different distances in a circle, we can use any one of them to find the other two.

Examples

Notice that, in the examples above, we treated \pi as a pronumeral rather than multiplying it as a number. This is because we wanted to find exact values.

If we type \pi into our calculator and try to evaluate it, the result will look something like this:\pi =3.14159265359

Some calculators will show more or less decimal places than what is shown here, but none of them will show the exact value of \pi. This is because \pi is an irrational number, meaning that its decimal expansion continues forever.

This means that, in order to give an exact answer when working with \pi, we need to leave it as a pronumeral.

However, if we are asked to give a rounded answer, we can then evaluate our answer using \pi as a number and then round to the number of decimal places required.

Examples