A circle is one of the most common of shapes. Whether you look at planets' lines of orbits in the solar system, the simple yet effective functioning of wheels, or even molecules at the molecular level, the circle keeps showing up! But how do we calculate the area of a circle? The area of a circle is the measure of the amount of space enclosed by the circle.
A circle is a shape in which all points that comprise the boundary are equidistant from a single point located at the center.
Elements of a circle
Before we discuss the area of circles, let's review the unique characteristics that define the circle's shape. The figure below depicts a circle with a centre O. Recall from the definition that all points located on the circle's boundary are equidistant (of equal distance) from this centre point O. The distance from the centre of the circle to its boundary is referred to as the radius, R.
The diameter, D, is the distance from one endpoint on a circle to another, passing through the centre of the circle. The diameter is always twice the length of the radius, so if we know one of these measurements, then we know the other as well! A chord is a distance from one endpoint to another on a circle that, unlike the diameter, does not have to pass through the centre point.
Circle illustration showing the diameter, radius, and the centre of the circle.
Formula of the area of the circle
Now that we've reviewed the elements of a circle, let's begin with the discussion of the area of a circle. First, we will start with a definition.
The area of a circle is the space a circle occupies on a surface or plane. The measurements of area are written using square units, such as ft2 and m2.
To find the area of a circle, we use the formula:
\[Area = \pi \cdot r^2\]
where:
- \(A\) is the area of the circle.
- \(π\) (pi) is a mathematical constant approximately equal to 3.14159.
- \(r\) is the radius of the circle, which is the distance from the center of the circle to any point on its circumference
For this formula, it is important to know that \(\pi\) is pi. What is pi? It is a constant represented by the Greek letter \(\pi\) and its value is equal to approximately 3.14159.
Pi is a mathematical constant that is defined as the ratio of the circumference to the diameter of a circle.
You don't have to memorize the value of pi because most calculators have a key for quick entry, shown as \(\pi\).
Area of a circles examples
Let's use the area formula in an example to see how we can apply this calculation in practice.
The radius of a circle is 8 m. Calculate its area.
Solution:
First, we substitute the value of the radius into the circle's area formula.
\[Area = \pi \cdot r^2 \rightarrow Area = \pi \cdot 8^2\]
Then, we square the radius value and multiply it by pi to find the area in square units. Keep in mind that \(r^2\) does not equal \(2 \cdot r\), but rather \(r^2\) is equal to \(r \cdot r\).
\[Area = \pi \cdot 64 \rightarrow Area = 201.062 m^2\].
Where does the formula of the area of a circle come from?
The area of a circle can be derived by cutting the circle into small pieces as follows.
A circle broke up into pieces to form an approximate rectangle.
If we break the circle into little triangular pieces (like that of a pizza slice) and put them together in such a way that a rectangle is formed, it may not look like an exact rectangle but if we cut the circle into thin enough slices, then we can approximate it to a rectangle.
Observe that we have divided the slices into two equal parts and coloured them blue and yellow to differentiate them. Hence the length of the rectangle formed will be half of the circumference of the circle which will be \(\pi r\). And the breadth will be the size of the slice, which is equal to the radius of the circle, r.
The reason why we did this, is that we have the formula to calculate the area of a rectangle: the length times the breadth. Thus, we have
\[A = (\pi r)r\]
\[A = \pi r^2\]
Verbally, the area of a circle with radius r is equal to \(\pi\) x the radius2. Hence the units of area are cm2, m2 or (unit)2 for appropriate units.
Calculating the area of circles with a diameter
We have seen the formula for the area of a circle, which uses the radius. However, we can also find the area of a circle by using its diameter. To do this, we divide the diameter's length by 2, which gives us the value of the radius to input into our formula. (Recall that a circle's diameter is twice the length of its radius.) Let's work through an example that uses this method.
A circle has a diameter of 12 meters. Find the area of the circle.
Solution:
Let's begin with the formula for the area of a circle:
\[Area = \pi \cdot r^2\]
From the formula, we see that we need the value of the radius. To find the circle's radius, we divide the diameter by 2, like so:
\[r = \frac{12}{2} = 6 \space meters\]
Now, we can input the radius value of 6 meters into the formula to solve for the area:
\[\begin{align} Area = \pi \cdot 6^2 \\ Area = 113.1 \space m^2 \end{align}\]
Calculating the area of circles with circumference
Apart from the area of a circle, another common and useful measure is its circumference.
The circumference of a circle is the perimeter or enclosing boundary of the shape. It is measured in length, which means the units are meters, feet, inches, etc.
Let's look at some formulas that relate the circumference to the circle's radius and diameter:
\[\frac{\text{Circumference}}{\text{Diameter}} = \pi \rightarrow \text{Circumference} = \pi \cdot \text{Diameter} \rightarrow \text{Circumference} = \pi \cdot 2 \cdot r\]
The formulas above show that we can multiply \(\pi\) by the diameter of a circle to calculate its circumference. Since the diameter is twice the length of the radius, we can replace it with \(2r\) if we need to modify the circumference equation.
You may be asked to find the area of a circle using its circumference. Let's work through an example.
The circumference of a circle is 10 m. Calculate the area of the circle.
Solution:
First, let's use the circumference formula to determine the radius of the circle:
\(\text{Circumference} = \pi \cdot 2 \cdot rr = \frac{\text{Circumference}}{\pi \cdot 2} r = \frac{10}{\pi \cdot 2} r = \frac{5}{\pi} m = 1.591 m\)
Now that we know the radius, we can use it to find the area of the circle:
\(\begin{align} \text{Area} = \pi \cdot r^2 \\ \text{Area} = \pi \cdot 1.591^2 \\ \text{Area} = 7.95 \space m^2 \end{align}\)
So, the area of the circle with a circumference of 10 m is 7.95 m2.
Area of semi-circles and quarter-circles with examples
We may also analyze the circle's shape in terms of halves or quarters. In this section, we will discuss the area of semi-circles (circles cut in half) and quarter-circles (circles cut in quarters).
Area and circumference of a semi-circle
A semi-circle is a half circle. It is formed by dividing a circle into two equal halves, cut along its diameter. The area of a semi-circle can be written as:
\(\text{Area of a semicircle} = \frac{\pi \cdot r^2}{2}\)
Where r is the radius of the semi-circle
To find the circumference of a semi-circle, we first halve the circumference of the whole circle, then add an additional length which is equal to the diameter d. This is because the perimeter or boundary of a semi-circle must include the diameter to close the arc. The formula for the circumference of a semi-circle is:
\[\text{Circumference of a semicircle} = \frac{\pi \cdot d}{2} + d\]
Calculate the area and circumference of a semi-circle that has a diameter of 8 cm.
Solution:
Since the diameter is 8 cm, the radius is 4 cm. We know this because the diameter of any circle is twice the length of its radius. Using the formula for the area of a semi-circle, we get:
\(\text{Area} = \frac{\pi \cdot r^2}{2} \rightarrow \text{Area} = \frac{\pi \cdot 4^2}{2} \rightarrow \text{Area} = 25.133 cm^2\)
For the circumference, we input the value of the diameter into the formula:
\(\text{Circumference} = \frac{\pi \cdot d}{2} + d \rightarrow \text{Circumference} = \frac{\pi \cdot 8}{2} + 8 \rightarrow \text{Circumference} = 20.566 cm\)
Area and circumference of a quarter-circle
A circle can be divided into four equal quarters, which produces four quarter-circles. To calculate the area of a quarter-circle, the equation is as follows:
\[\text{Area of a quartercircle} = \frac{\pi \cdot r^2}{4}\]
To get the circumference of a quarter-circle, we start by dividing the circumference of the full circle by four, but that only gives us the quarter-circle's arc length. We then have to add the length of the radius twice to complete the quarter-circle's boundary. This calculation can be performed using the following equation:
\(\text{Circumference of a quarter circle} = \frac{\pi \cdot d}{4} + 2r \rightarrow \text{Circumference of a quarter circle} = \frac{\pi \cdot d}{4} + d\)
Calculate the area and circumference of a quarter-circle with a radius of 5 cm.
Solution:
For the area, we get:
\(\text{Area} = \frac{\pi \cdot r^2}{4} \rightarrow \text{Area} = \frac{\pi \cdot 5^2}{4} \rightarrow \text{Area} = 19.6 cm^2\)
The circumference can be calculated as:
\(\text{Circumference} = \frac{\pi \cdot d}{4} + d \rightarrow \text{Circumference} = \frac{\pi \cdot 10}{4} + 10 \rightarrow \text{Circumference} = 17.9 cm\)
Area of circles - Key takeaways
- In a circle, all points which comprise the shape's boundary are equidistant from a point located at its centre.
- The line segment that spans from the centre of the circle to a point on its boundary is the radius.
- The diameter of a circle is the distance from one endpoint on a circle to another that passes through the centre of the circle.
- The circumference of a circle is the arc length of the circle.
- The area of a circle is \(\pi \cdot r^2\).
- The circumference of a circle is \(2 \cdot \pi \cdot r\).
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