12.09 Introducing the circle

Features of 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.

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

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

This diameter has been split into two radii.

The other distance we will look at is the circumference.

Relating features 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:

$d=2r$d=2r

Where $d$d is the diameter and $r$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: $\pi$π.

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

$C=\pi d$C=πd

Where $C$C is the circumference and $d$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:

$C=2\pi r$C=2πr

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

Calculating distances in a circle

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

Exact or rounded values?

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

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

$\pi=3.14159265359$π=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 $\pi$π. This is because $\pi$πis an irrational number, meaning that its decimal expansion continues forever.

This means that, in order to give an exact answer when working with $\pi$π, 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 $\pi$π as a number and then round to the number of decimal places required.

Practice questions

Question 1

Caitlin and David calculate the circumference of this circle using different formulas.

1. Caitlin uses the formula $C=\pi d$C=πd to calculate the circumference.

   What was her result as an exact value?

2. David determines that the radius is $5$5, and uses the formula $C=2\pi r$C=2πr to calculate the circumference, where $r$r is the radius.

   What was his result as an exact value?

3. Why did Caitlin and David get the same result even though they used different formulas?

   Caitlin's formula is the right one, and David only got the same result by chance.

   A

   The diameter is equal to half the radius, so $\pi r$πr and $2\pi d$2πd are always equal.

   B

   The diameter is always equal to twice the radius, so $\pi d$πd and $2\pi r$2πr are always equal.

   C

   David's formula is the right one, and Caitlin only got the same result by chance.

   D

Reveal SolutionWatch video

Question 2

This table lists the radii, diameters, and circumferences of various circles.

1. Complete the table. Give your answers as exact values, simplifying where possible.

   Radius (cm)	Diameter (cm)	Circumference (cm)
   $2$2	$4$4	$4\pi$4π
   $8$8	$\editable{}$	$\editable{}$
   $\editable{}$	$82$82	$\editable{}$
   $\editable{}$	$\editable{}$	$340$340

Reveal SolutionWatch video

Question 3