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# Area of a Circle

Use our extensive free resources below to learn about Area of a Circle and download SQA past paper questions that are directly relevant to this topic.

This material is an extract from our National 5 Mathematics: Curriculum Breakdown course led by instructor Andrew Eadie. Enrol in the full course now and gain access to over 100 detailed topic breakdowns, 48 video tutorials (20 hours) and 39 quizzes spanning the entire curriculum.

# Area of a Circle

Area is the amount of space inside the boundary of a flat (i.e. 2-dimensional) object.

By this stage you will be familiar with calculating the areas of most standard shapes, including circles. The key area formulas you should know are given below as a reminder:

Calculating the area of a circle is too basic a problem to be asked directly at the National 5 level. At this level, you’re much more likely to be aksed to calculate the area of a sector:

#### Area of a Sector

Remember that a sector is the “pizza slice” created when an angle is subtended by an arc:

The area of a sector is determined by the angle subtended by the arc (\boldsymbol{x^\circ}):

\begin{aligned}\textbf{Sector area}=\boldsymbol{\frac{x}{360}\times\pi r^{2}}\end{aligned}

In simple terms, this formula works out the “small section” or fraction (\frac{x}{360}) of the overall area (\pi r^{2}) of the circle that the sector represents. The overall circle has an angle of 360^\circ, so the angle subtended by the arc must be less than this. This fraction (\frac{x}{360}) is then multiplied by the \pi r^{2} term which we know is just the area of the full circle (since A=\pi r^{2}), resulting in an answer which is a fraction of the area. This should make sense since a sector is just a small section of the overall circle’s area!

Using the area of a sector formula should not be too challenging, as the following examples show:

Example 1

Calculate the area of the sector shown:

We have all we need to jump right into the sector area calculation:

\begin{aligned}\text{Sector area}&=\frac{x}{360}\times\pi r^{2} \\[12pt]\text{Sector area}&=\frac{120}{360}\times\pi (5.2)^{2} \\[12pt]\text{Sector area}&=28.3cm^{2}\end{aligned}

Example 2

A sector of a circle has an area of 30cm^{2} and an angle of 72^\circ. Calculate the radius of the circle.

Once again we must rearrange our new formula to make the term we want to calculate (the radius) the subject:

\begin{aligned}\text{Sector area}&=\frac{x}{360}\times\pi r^{2} \\[12pt]360\times\text{Sector area}&=x\times\pi r^{2} \\[12pt]\frac{360\times\text{Sector area}}{x}&=\pi r^{2} \\[12pt]\frac{360\times\text{Sector area}}{x\pi}&=r^{2}\end{aligned}

Substituting what we know into the equation:

\begin{aligned}r^{2}&=\frac{360\times (30)}{(72)\pi} \\[12pt]r^{2}&=47.75 \\[12pt]r&=\sqrt{47.75} \\[12pt]r&=6.9cm\end{aligned}

### Key Outcomes

The area of a circle is given by the equation:

\begin{aligned}\text{Area}&=\pi r^{2}\end{aligned}

The area of a sector is determined by the angle subtended by the arc (x^\circ):

\begin{aligned}\text{Sector area}&=\frac{x}{360}\times\pi r^{2} \\[18pt]\end{aligned}

## Area of a Circle

Click the links below to download SQA past paper questions that are directly relevant to Area of a Circle:

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