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8.03 The area of a circle

Lesson

Introduction

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 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.

Exploration

Take a look at the following applet by moving the sliders:

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The more triangles we use, the closer this area gets to the area of the circle, and the closer the base of the resulting parallelogram gets to being half of the circumference of the circle. So we can see that the area of a circle is given by \pi r^2.

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 circle with a radius of 8 centimetres.
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^2Use the formula
\displaystyle =\displaystyle \pi \times 8^2Substitute the radius
\displaystyle =\displaystyle 64\pi \text{ cm}^2Simplify
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\piUse the previous result
\displaystyle =\displaystyle 201.061\,929\ldotsEvaluate using calculator
\displaystyle =\displaystyle 201.06\, \text{cm}^2Round the answer

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^2Use the formula
\displaystyle 16\pi\displaystyle =\displaystyle \pi r^2Substitute 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}.

Idea summary

We can calculate the area of a circle using the formula:

\displaystyle A=\pi r^2
\bm{A}
is the area
\bm{r}
is the radius of the circle

This formula can also be used to find the radius if we know the area of the circle.

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 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

Example 3

A circle has an area of 47\text{ mm}^2. What is its circumference, rounded to two decimal places?

Worked Solution
Create a strategy

Use the formula A=\pi r^2 to find the radius so we can then find the circumference.

Apply the idea
\displaystyle A\displaystyle =\displaystyle \pi r^2Use the formula
\displaystyle 47\displaystyle =\displaystyle \pi r^2Substitute the area
\displaystyle r^2\displaystyle =\displaystyle \frac{47}{\pi} Divide by \pi
\displaystyle r\displaystyle =\displaystyle \sqrt{\frac{47}{\pi}}Square root both sides

Now we can substitute the radius into the circumference formula.

\displaystyle C\displaystyle =\displaystyle 2\pi rUse the formula
\displaystyle =\displaystyle 2 \pi \times \sqrt{\frac{47}{\pi}}Substitute the radius
\displaystyle \approx\displaystyle 24.30 Evaluate and round
Idea summary

We can calculate the area of a circle using the formula:

\displaystyle A=\frac{1}{4}\pi d^2
\bm{A}
is the area
\bm{d}
is the diameter of the circle

If we know the circumference of a circle, we can use the formula C=2\pi r to find the radius, and then we can find the area of the circle using the formula A=\pi r^2.

Similarly, if we know the area of a circle, we can use the formula A=\pi r^2 to find the radius, and then we can find the circumference using the formula C=2\pi r.

Outcomes

VCMNA275

Investigate the concept of irrational numbers, including π

VCMMG288

Investigate the relationship between features of circles such as circumference, area, radius and diameter. Use formulas to solve problems involving determining radius, diameter, circumference and area from each other

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