10.08 Area of a circle
The area of a circle is the 2D space within the circle's boundary. Knowing how this area relates to the other features of the circle can let us calculate the area of a circle from its other features or it can be used to find different measurements of the circle with a given area.
The area formula for a circle
We can calculate the area of a circle using the formula:
$A=\pi r^2$A=πr2
Where $A$A is the area and $r$r is the radius of the circle.
Using this formula, we can find the area of a circle using its radius and vice versa.
Worked Examples
Example 1
The radius of a circle is $6$6. What is the exact area of the circle?
Think: To find the area of the circle using the radius, we can substitute the value for the radius into the formula $A=\pi r^2$A=πr2 and solve for $A$A.
Do: If we substitute the radius $r=6$r=6 into the formula, we get:
| $A$A | $=$= | $\pi r^2$πr2 |
|
| $A$A | $=$= | $\pi\times6^2$π×62 |
Substitute in the value for the radius |
| $A$A | $=$= | $\pi\times36$π×36 |
Evaluate the square |
| $A$A | $=$= | $36\pi$36π |
Write $36$36 as the coefficient of $\pi$π |
As such, the area of this circle is $36\pi$36π.
Example 2
The area of a circle is $16$16. What is the exact radius of the circle?
Think: To find the radius of the circle using the area, we can substitute the value for the area into the formula $A=\pi r^2$A=πr2 and solve for $r$r.
Do: If we substitute the area $A=16$A=16 into the formula, we get:
| $A$A | $=$= | $\pi r^2$πr2 |
|
| $16$16 | $=$= | $\pi r^2$πr2 |
Substitute in the value for the area |
| $\frac{16}{\pi}$16π | $=$= | $r^2$r2 |
Reverse the multiplication of $\pi$π |
| $\sqrt{\frac{16}{\pi}}$√16π | $=$= | $r$r |
Square root both sides of the equation |
| $\frac{\sqrt{16}}{\sqrt{\pi}}$√16√π | $=$= | $r$r |
Apply the square root to both parts of the fraction |
| $\frac{4}{\sqrt{\pi}}$4√π | $=$= | $r$r |
Evaluate the square root in the numerator |
As such, the exact radius of this circle is $\frac{4}{\sqrt{\pi}}$4√π.
Reflect: In both case, we substituted the known value into the circle's area formula and solved to find the missing value. We also treated $\pi$π as a pronumeral since we wanted exact values. In the case where we want a rounded answer, we evaluate and round the answer as required.
Practice questions
Question 1
The formula for the area of a circle is
$A=\pi r^2$A=πr2
where $A$A is the area and $r$r is the radius.
Consider the circle below:
What is the exact area of the circle?
What is the area of the circle rounded to two decimal places?
Question 2
Christa is finding the radius of a circle, knowing only that its area is $64\pi$64π.
Fill in the blanks to complete Christa's working out.
$\pi r^2$πr2 $=$= $A$A (Formula for the area of a circle) $\pi r^2$πr2 $=$= $\editable{}$ (Substitute the area given in the question) $r^2$r2 $=$= $\editable{}$ (Divide both sides by $\pi$π) $r$r $=$= $\editable{}$ (Take the square root of both sides to find $r$r)
Connecting other features to the area
Since we now have a way to relate the area of a circle to its radius, we can use the radius to connect the area to the other distances in a circle.
Since the radius is equal to half the diameter, we can replace the radius $r$r in the area formula with $\frac{d}{2}$d2, this gives us:
$A=\pi\left(\frac{d}{2}\right)^2$A=π(d2)2
Which can be expanded to:
$A=\frac{1}{4}\pi d^2$A=14πd2
In the cases where the diameter is a nicer number to work with than the radius, this version of the area formula can be useful.
In a similar way, we can connect the area of a circle to its circumference. We know that the circumference of a circle is related to the radius by the formula $C=2\pi r$C=2πr while the area is related by the formula $A=\pi r^2$A=πr2.
Unlike with the diameter, there isn't a nice formula that emerges when combining these two relationships. Instead, we can find the area using the given area or circumference and then use that radius to calculate the missing value.
Worked Examples
Example 1
A circle has a diameter of $9$9. What is the area of the circle, rounded to two decimal places?
Think: We can substitute our value for the diameter into the formula $A=\frac{1}{4}\pi d^2$A=14πd2 and solve for $A$A to find the area of the circle.
Do: If we substitute the diameter $d=9$d=9 into the formula, we get:
| $A$A | $=$= | $\frac{1}{4}\pi d^2$14πd2 |
|
| $A$A | $=$= | $\frac{1}{4}\pi\times9^2$14π×92 |
Substitute in the value for the diameter |
| $A$A | $=$= | $\frac{1}{4}\pi\times81$14π×81 |
Evaluate the square |
| $A$A | $=$= | $\frac{81}{4}\pi$814π |
Multiply $81$81 with the coefficient $\frac{1}{4}$14 |
This means that the exact area of the circle is $\frac{81}{4}\pi$814π. Evaluating this gives us:
$\frac{81}{4}\pi=63.6172512352$814π=63.6172512352$\dots$…
Rounding this to two decimal places gives us a value of $63.62$63.62 for the area of the circle.
Reflect: We could have also calculated the radius from the diameter and used that value in the formula $A=\pi r^2$A=πr2. This would give us the same answer, but would also require us to find the square of $\frac{9}{2}$92 which a bit more effort than squaring just $9$9.
Example 2
A circle has a circumference of $22\pi$22π. What is the exact area of the circle?
Think: Using the formula $C=2\pi r$C=2πr we can find the radius from the circumference. We can then substitute this value for the radius into the formula $A=\pi r^2$A=πr2 to find the area of the circle.
Do: We can find the radius of the circle by substituting $C=22\pi$C=22π into the formula to get:
| $C$C | $=$= | $2\pi r$2πr |
|
| $22\pi$22π | $=$= | $2\pi r$2πr |
Substitute in the value for the circumference |
| $11\pi$11π | $=$= | $\pi r$πr |
Reverse the multiplication of $2$2 |
| $11$11 | $=$= | $r$r |
Reverse the multiplication of $\pi$π |
We have found that the radius of this circle is $11$11.
Substituting this value into the area formula of the circle gives us:
| $A$A | $=$= | $\pi r^2$πr2 |
|
| $A$A | $=$= | $\pi\times11^2$π×112 |
Substitute in the value for the radius |
| $A$A | $=$= | $\pi\times121$π×121 |
Evaluate the square |
| $A$A | $=$= | $121\pi$121π |
Write $121$121 as the coefficient of $\pi$π |
As such, the area of this circle is $121\pi$121π.
We can perform similar calculations when finding the diameter or circumference using some given area.
Practice questions
Question 3
The engineering team at Rocket Surgery are building a rocket for an upcoming Mars mission.
A critical piece is the circular connective disc that connects the booster rocket to the rest of the spacecraft. This disc must completely cover the top of the booster rocket.
The booster rocket has a diameter of precisely $713.5$713.5 centimetres.
Find the required area of the connective disc.
Give your answer to two decimal places.
Instead of using the exact value, an engineer uses the approximation $3.14$3.14 for $\pi$π.
What value does the engineer calculate for the area?
Give your answer to two decimal places.
If the connective disc is more than $100$100 cm2 too small, the disc will malfunction, resulting in catastrophic launch failure.
Is there a risk of malfunction if the disc is built according to the engineer's calculation?
Yes
ANo
B
Question 4
A circle has an area of $25\pi$25π cm2.
What is the radius of the circle?
What is its exact circumference?