Lesson video

Hello, my name's Miss Parnham.

In this lesson, we're going to learn how to find the length of an arc on a semicircle and quarter circle, and the perimeter of a semicircle and quarter circle.

Let's look at an example, where we find the arc length of a semicircle and the perimeter of that semicircle.

Let's start by looking at a complete circle.

And remind ourselves of the formula for the circumference.

Here we have a circle with diameter of 18 centimetres, and the circumference can be found by multiplying pi by diameter.

18 times pi would give us 18 pi, or if we had a radius in this circle, that would be nine centimetres, then double the radius and multiply by pi.

Either way he answer is 18 pi.

And if we want that to three significant figures, this is 56.

5 centimetres.

Now, let's look at the semicircle, which is exactly half of this circle.

So again, the diameter is 18 centimetres or the radius is nine centimetres, whichever way you want to look at it.

The arc length of the semicircle is the formula for the circumference divided by two.

So pi d divided by two or two pi divided by two or just simply pi r.

So, this semicircle, the arc length is nine pi or to three significant figures, 28.

3 centimetres.

If we want the perimeter, we need to consider the arc length and the straight side, which is equal to the diameter or two radii.

And we can write this either way.

So all we need to do is take our answer for the arc length and add 18 onto it.

So, to three significant figures, this is 46.

3 centimetres, or in terms of pi it is nine pi plus 18.

Here's some questions for you to try.

pause the video to complete the task and restart the video when you're finished.

Here are the answers.

Question two is broken into two parts, so that the arc length could be found first and then added to the diameter to find the perimeter.

Sometimes a question might directly ask you for the perimeter.

So you need to know to break it down into these two steps, in order to get the correct solution.

Here's some questions for you to try, pause the video to complete the task and restart the video when you're finished.

Here are the answers.

In part c and d, you just needed to double the radius in order to find the diameter, and then you could solve it and exactly the same way as you did with parts a and b.

Let's look at an example where we find the arc length of a quarter circle and the perimeter of that quarter circle.

Let's start by considering a complete circle with the diameter of 20 centimetres.

We know that the circumference formula is pi times diameter or two pi r, if we have the radius, which in this circle would be 10 centimetres.

Using either formula, we would get the answer of 20 pi, or if we wanted this to three significant figures, 62.

8 centimetres.

So the arc length on a quarter circle with the same radius as this circle, would it be a quarter of this circumference.

So here's that quarter circle.

So we take two pi r and we divide by four, and that will simplify to pi r over two.

So this is five pi in terms of pi and 15.

7 centimetres to three significant figures.

What if we want to find the perimeter of this, quarter circle, we simply need to take the arc length, which we've already calculated and add on to radii, which is five pi plus 20 in terms of pi and to three significant figures, this is 35.

7 centimetres.

Here's some questions for you to try.

Pause the video to complete the task and restart the video when you're finished.

Here are the answers, when finding the perimeter of a quarter to circle, it's quite easy to fall into the trap of only adding on one radius to the arc length.

This is often because in diagrams, the radius is only labelled once.

So please be careful.

If you get any questions like this in the future.

Here are some further questions for you to try.

Pause the video to complete the task and restart the video when you're finished.

Here are the answers, if your answers are very close to these, but not quite the same, then check that you haven't rounded partway through the process.

We must always calculate with full numbers and only round at the end when we give our answer.

That's all for this lesson.

Thank you for watching.