The Area of a Circle
There are a few things we need to know when calculating the area of a circle, and it’s always good to start with the basics. First, a circle is a 2D shape — like any other 2D shape, its area is the amount of space it covers.
Unlike polygons (such as squares, triangles, or parallelograms), we can’t multiply together the length of the sides to find the area. So instead, we use the distance from the circle’s edge to the center.
There are two measurements you need to be aware of here:
- The radius is the distance from the circle’s edge to the center.
- The diameter is the distance from one side of the circle to the other through the center of the circle.
The diameter is twice the radius; the radius is half the diameter. Keep reading to find out how to work out the area of a circle using the diameter.
How to Calculate the Area
To find the area, we need to know the radius. If you were finding the area of any other shape, you would multiply two measurements together, for example, height × width in a rectangle. Because a circle has the same radius all the way around, we multiply the radius by the radius. We then take that answer and multiply it by π:
How to work out the area of a circle using the radius
- Area of a circle = π × radius × radius
Pi is a Greek letter, spelled pi, and pronounced like pie. Pi is a constant, which means it never changes. It is also irrational, which means it never ends and never repeats. The first 20 digits are:
- 3.14159265358979323846…
If you’re using π to find a circle’s area (or circumference), you can use the π button on your calculator or round it to 3.14.
Example 1 — Find the area of a circle using a radius of 5 cm. Give your answer correct to 1 decimal place.
Area = π × radius × radius
We know the radius, so put it into the formula above:
- Area = π × radius × radius
- Area = π × 5 × 5
- Area = 78.5 cm2 (to 1d.p.)
Make sure you give the correct units. Like any other area, the area of a circle is given in square units. That may be cm2, m2, mm2 or km2, among others.
Let’s go through another example.
Example 2 — Find the area of a circle using the radius of 17 cm. Give your answer correct to 1 decimal place.
Area = π × radius × radius
We know the radius, so put it into the formula above:
- Area = π × radius × radius
- Area = π × 17 × 17
- Area = 907.9 cm2 (to 1d.p.)
How to work out the area of a circle using the diameter
In maths, you will not always be given the radius of a circle. However, it is still possible to find the area. Here are the steps you must take to work out the area of a circle using the diameter:
Step 1: Find the radius
The radius is equal to half of the diameter. So, to find the radius, we must divide the diameter by 2:
radius = diameter ÷ 2
Step 2: Use the formula using the radius
Now that we know the radius, we can use the same formula for the area.
Area = π × radius × radius
We are multiplying r by itself in this equation, which means we are squaring it. So, the formula can be simplified into this:
- A= π ×r2
Finally, because this is an algebraic formula, we can remove the × sign:
- A= πr2
Let’s put this into practice with an example.
Example 1 — Find the area of a circle using a diameter of 14 m. Give your answer correct to 1 decimal place.
Be careful! We have been given the diameter in this question, but we need the radius to find the area.
The radius is half the diameter:
- radius = diameter ÷ 2
- radius = 14 ÷ 2
- radius = 7 m
We can now find the area as before:
- Area = π × radius × radius
- Area = π × 7 × 7
- Area = 153.9 m2
Here’s another example of how to find the area of a circle using the diameter:
Example 2 — Find the area of a circle using a diameter of 32 m. Give your answer correct to 1 decimal place.
The radius is half the diameter, so:
- radius = diameter ÷ 2
- radius = 32 ÷ 2
- radius = 16 m
Now, we can find the area using the original formula:
- Area = π × radius × radius
- Area = π × 16 × 16
- Area = 804.2 m2
How to work out the area of a circle using the circumference
In some cases, you will find that a circle’s radius and diameter are unknown. Fear not; you can still calculate the area of a circle using the circumference. The circumference is the total distance around the circle. The formula for finding the area of a circle using the circumference is:
C = 2πr
Key:
- C represents the circumference
- r represents the radius
- A represents the area
Step 1: Solve for ‘r’
We can take this formula for the circumference and use it to find the value of the radius. To do this, the formula will be:
r = C/2π
Step 2: Replace ‘r’ in the formula
Now, we can replace ‘r’ in the original formula for the circumference with this new expression:
A = π(C/2π)2
Step 3: Simplify the formula
This formula is quite confusing to look at, so let’s simplify it. The simplified version of this formula for the area of a circle is:
A = C2/4π
Now you have a fool-proof formula for calculating the area of a circle.
Let’s put all of this into practice with an example!
Example 1 — Find the area of a circle using the circumference of 30 m. Give your answer correct to 1 decimal place.
A = C2/4π
Now, we have to put the measurements of the circle into this formula.
A = 302/4π
A = 900/4π
Area = 71.6 m2
