8.03 The area of a circle

We have defined the area of a unit square, then used this definition to find the areas of rectangles, triangles, and various other polygons. This works because polygons have straight edges, and we can divide them up and rearrange them into other shapes for which we already know the area.

Circles don't have straight edges, but we can still do something similar to find an expression for the area of a circle - by using a set of isosceles triangles which share a common vertex at the centre of the circle to approximate the area of the circle.

We can calculate the area of a circle using the formula \text{A}=\pi r^2 where A is the area and r is the radius of the circle.

Using this formula, we can find the area of a circle using its radius and vice versa.

Examples

Example 1

Consider the circle below:

a

What is the exact area of the circle?

Worked Solution

Create a strategy

Use the formula {A}=\pi r^2 and substitute the radius.

Apply the idea

\displaystyle \text{A}	\displaystyle =	\displaystyle \pi r^2	Use the formula
	\displaystyle =	\displaystyle \pi \times 8^2	Substitute the radius
	\displaystyle =	\displaystyle 64\pi \text{ cm}^2	Simplify

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b

What is the area of the circle rounded to two decimal places?

Worked Solution

Create a strategy

Evaluate the exact value found in the part (a) and round it to two decimal places.

Apply the idea

We know from the previous part that the exact value of the area is 64\pi.

\displaystyle A	\displaystyle =	\displaystyle 64\pi	Use the previous result
	\displaystyle =	\displaystyle 201.061\,929\ldots	Evaluate using calculator
	\displaystyle =	\displaystyle 201.06\, \text{cm}^2	Round the answer

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Example 2

A circle has an area of 16\pi \text{ cm}^2.

What is its exact radius?

Worked Solution

Create a strategy

Use the formula {A}=\pi r^2 and substitute the area.

Apply the idea

\displaystyle \text{A}	\displaystyle =	\displaystyle \pi r^2	Use the formula
\displaystyle 16\pi	\displaystyle =	\displaystyle \pi r^2	Substitute the area
\displaystyle r^2	\displaystyle =	\displaystyle 16	Divide by \pi
\displaystyle r	\displaystyle =	\displaystyle \sqrt{16}	Square root both sides
	\displaystyle =	\displaystyle 4	Evaluate

The exact radius of this circle is 4\text{ cm}.

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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 in the area formula with \dfrac{d}{2},

this gives us:A=\pi \left(\frac{d}{2}\right)^2

Which can be expanded to:A=\frac{1}{4}\pi d^2

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 while the area is related by the formula \text{A}=\pi r^2.

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.

Examples