10.07 Introducing the circle
A circle is the set of all points that are the same distance from a central point. However, the central point is not part of 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.
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".
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.
This diameter has been 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.
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.
The pronumeral $\pi$π is an irrational number representing the ratio between the circumference and diameter of a circle.
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.
Worked Examples
Example 1
A circle has a radius of $4$4. What is the exact diameter and circumference of the circle?
Think: We know that the diameter is equal to twice the radius. We can also use the formula $C=2\pi r$C=2πr to find the circumference using the radius.
Do: Since the diameter is equal to twice the radius, we know that the diameter of the circle is:
$d=8$d=8
Substituting $r=4$r=4 into the formula $C=2\pi r$C=2πr gives us:
$C=2\pi\times4$C=2π×4
So we can calculate the circumference of the circle to be:
$C=8\pi$C=8π
Example 2
A circle has an exact circumference of $14\pi$14π. What is the radius and diameter of the circle?
Think: We can use the formula $C=\pi d$C=πd to calculate the diameter, then halve the diameter to find the radius.
Do: Substituting $C=14\pi$C=14π into the formula $C=\pi d$C=πd gives us:
$14\pi=\pi d$14π=πd
By dividing both sides of the equation by $\pi$π we get:
$d=14$d=14
We can then halve the diameter to find the radius:
$r=7$r=7
Reflect: In both cases, we used the relationships between our given and missing values to find the unknown distances. In the first example we calculated both the diameter and circumference using the radius. In the second example we used the circumference to find the diameter and then used the diameter to find the radius.
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.
Worked Example
The circumference of a circle is $50$50. What is the exact radius of the circle? What is the radius of the circle, rounded to two decimal places?
Think: The formula relating the circumference and radius is $C=2\pi r$C=2πr. To find the exact radius, we treat $\pi$π as a pronumeral. We can then evaluate the exact radius by treating $\pi$π as a number and rounding to two decimal places.
Do: Using the formula $C=2\pi r$C=2πr, we can calculate the exact radius with the working:
| $C$C | $=$= | $2\pi r$2πr |
|
| $50$50 | $=$= | $2\pi r$2πr |
Substitute in the given value of the circumference |
| $25$25 | $=$= | $\pi r$πr |
Reverse the multiplication of $2$2 |
| $\frac{25}{\pi}$25π | $=$= | $r$r |
Reverse the multiplication of $\pi$π |
As such, the exact radius of the circle is $\frac{25}{\pi}$25π.
If we evaluate this using our calculator, we get:
$\frac{25}{\pi}=7.95774715459$25π=7.95774715459
Rounding this to two decimal places gives us a value of $7.96$7.96 for the radius of the circle.
Practice questions
Question 1
Caitlin and David calculate the circumference of this circle using different formulas.
Caitlin uses the formula $C=\pi d$C=πd to calculate the circumference.
What was her result as an exact value?
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?
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.
AThe diameter is equal to half the radius, so $\pi r$πr and $2\pi d$2πd are always equal.
BThe diameter is always equal to twice the radius, so $\pi d$πd and $2\pi r$2πr are always equal.
CDavid's formula is the right one, and Caitlin only got the same result by chance.
D
Question 2
This table lists the radii, diameters, and circumferences of various circles.
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
Question 3