Video: AQA GCSE Mathematics Foundation Tier Pack 1 • Paper 2 • Question 26

A semicircle with diameter 13.2 cm is shown in the diagram. Calculate the area of the semicircle.

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Video Transcript

A semicircle with diameter 13.2 centimetres is shown in the diagram. Calculate the area of the semicircle.

The area of a two-dimensional shape is the amount of space inside that shape. There are standard formulae which we can apply, but we need to remember them. The formula for calculating the area of a full circle is 𝜋𝑟 squared, where 𝑟 represents the radius of the circle.

Notice that this means 𝜋 multiplied by 𝑟 squared. And by applying BIDMAS or the order of operations, we see that we square the radius first and then multiply by 𝜋. We don’t multiply the radius by 𝜋 and then square the result.

But this question isn’t about a circle. It’s about a semicircle. A semicircle is half of a circle. So the area of a semicircle is half the area of the full circle. We can therefore find the area by dividing our formula 𝜋𝑟 squared by two.

Now in this question, we haven’t been given the length of the semicircle’s radius. Instead, we’ve been given the length of the diameter. But the two are closely related. The diameter of a circle is twice the length of the radius. So we can use the length we’ve been given for the diameter to calculate the radius.

The radius will be equal to half of 13.2 or 13.2 over two. That’s 6.6. And the units for the radius are the same as the units for the diameter. They’re centimetres. So to calculate the area of this semicircle, we substitute the radius of 6.6 into our formula for the area of the semicircle. And we get 𝜋 multiplied by 6.6 squared over two. 6.6 squared is equal to 43.56. So we have 43.56𝜋 over two. And as 43.56 divided by two is equal to 21.78, the answer simplifies to 21.78𝜋.

Now if we didn’t have access to a calculator or if we needed an exact value for our answer, we could leave our answer in terms of 𝜋. So we just leave our answer as 21.78𝜋. This is sometimes called leaving your answer as a multiple of 𝜋. However, we do have access to a calculator in this question. So we can evaluate 21.78𝜋. And it gives 68.4238 continuing.

We haven’t been asked to round our answer to a particular degree of accuracy. So we’ll use three significant figures as our default. Our deciding number is the fourth significant figure, which is a two. It tells us that we’re rounding down. So the four in the third significant figure or first decimal place will remain a four. The units for the diameter were centimetres. So the units for the area will be centimetres squared. And we can give our answer as either 21.78𝜋 centimetres squared or 68.4 centimetres squared to three significant figures.

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