In this video, we’re going to look at problems related to calculating area and circumference of a circle. First of all, just a quick reminder of some of the different formula that we’re going to need.
So in the diagram, I’ve drawn the diameter of the circle in purple and labelled it 𝑑. Remember that’s the line that goes from one side of the circumference all the way to the other passing through the centre of the circle. And in green, I’ve given drawn the radius and labelled it 𝑟. That’s a line that starts at the circumference of circle and joins it to the centre of the circle. So the different formulae for the circumference we can either use π𝑑 or we can use two π𝑟 depending on whether we prefer to work with the diameter or the radius.
And for the area of the circle, we have the formula π𝑟 squared So we use these formulae a lot, but we use them in the context of different problems to do with circles.
So here’s the first problem; it’s a worded problem. A spinner has six equal sectors, half of which are red and half of which are black. If the radius of the spin is three inches, what is the area of the red sectors? So if you haven’t been given a diagram, it’s always a good idea just to do a quick sketch first so you can visualise the problem better, so I draw a diagram of this spinner.
So we represent it as a circle and we’re told has six equal sectors. So there they are roughly equal in my diagram, and it says half are red and half are black. So I coloured them in and I imagine the spinner looks something like this.
Now we’re told the radius of the spinner is three inches, so I’ll add that piece of information on in green. And we’re asked to find the area of the red sectors. So these red sectors, well only if they’re all equal-sized and three out six of them are red, then we’re basically being asked to find half of the area of this total circle here.
So we need to recall our area formula, which is that the area is equal to π𝑟 squared, and we know the radius of the circle is three inches, so I can just substitute that directly into the area formula.
However I only want half the circle for the red sectors, so I’m gonna need to divide this area by two. So I have the area of the red sectors is π multiplied by three squared and then divided by two.
So if I were to give my answer as a fraction, it would be nine π over two or I can evaluate it using a calculator as a decimal. And that gives me the answer of fourteen point one square inches if I round it to one decimal place.
Okay, this is the next problem we’re going to look at. An athletics track is composed of a rectangle of length eighty-five metres with semicircles of radius thirty-six point five metres at either end. We’re asked to calculate the total length of the inside lane and also the area of space inside the track. So as before, a diagram would be very helpful here.
So the athletics track looks something like this. The straight section of length eighty-five metres and then semicircles on either end with a radius of thirty-six point five metres. So for the first part of this question, we’re asked to calculate the total length of the inside lane.
So this inside line is composed of the two straight sections at eighty-five metres each and then two semicircular arcs, which of course if you put together would form a complete circle. So we want to work out the circumference of this circle. So our circumference formulae, circumference is π𝑑 or two π𝑟. I’ll use two π𝑟 as it’s the radius that I’ve been given.
So I have the circumference of this circle is equal to two multiplied by π multiplied by thirty-six point five, which gives me a value of seventy-three π for the circumference of this circle, which, remember, is composed of two semicircular arcs.
Let’s answer the question then. The total length of the inside lane, so this length is the two semicircular arcs plus the two straight sections. So it’s seventy-three π plus eighty-five plus eighty-five again, which gives me an exact answer of a hundred and seventy plus seventy-three π.
Or if I use my calculator to evaluate this, it gives an answer of three hundred and ninety-nine metres to the nearest metre. Now that the length of the white line, on the very inside of the inside lane. For the second part of the question, we’re looking to find the area inside the track.
So we’re looking for all this area that I’ve marked in orange. Now that area is composed of three parts: there’s a rectangle in the middle, and then there are the two semicircles at the two ends. So we need to work those areas out.
Let’s look at the rectangle first of all. Well it has a length of eighty-five metres and then its width will be the full diameter of these semicircles, so it’ll be twice thirty-six point five, which is seventy-three. So the area of the rectangular part is eighty-five multiplied by seventy-three, which is six thousand two hundred and five metres squared.
Now let’s look at the area of the semicircles. Well there are two of them so together they form of full circle, and it’s a full circle of radius thirty-six point five. So we need our area formula again, which tells us that the area is equal to π𝑟 squared.
So the area of this circle is π multiplied by thirty-six point five squared. I could work out the semicircles separately, but I’d have to divide it by two and then double it because there are two of them, so I just treat it as a full circle. This gives an exact value of five thousand three hundred and twenty-nine over four π for the area of the circle. Final stage is I need to add these two areas together.
So six thousand two hundred and five for the rectangle plus five thousand three hundred and twenty-nine over four π for the circle, which gives me an answer of ten thousand three hundred and ninety square metres, and that’s rounded to the nearest metre.
Our final question then, using three point one four as an approximation for π, calculate the shaded area. Now you may want to pause the video at this point and just look at the diagram and think about what strategy you would use in order to work out the shaded area. See if you can break it down into different shapes that you’d be able to find the area of.
Now my strategy will be just to draw our part of the diagram in order to be able to visualise it a little bit better. So I’ve drawn it out again, but I’ve only put in one of the curved portions this time, because now I can see a little bit more easily that there’s a square and then there’s a quarter circle within that Square.
Now I’m looking for that shaded area. But in order to calculate that, I’m actually gonna have to do it by working out some different areas and subtracting. So I can see that this orange area here is obviously the same as this orange area here. But because of symmetry, it’s also the same as this orange area here. So my strategy is gonna be to work out that orange area and then double it and then subtract that from the area of the square in order to get the shaded area that I’m looking for.
So using my diagram, I can see that that orange area can be worked out generating the area of the square and then subtracting the area of the quarter circle, so that’s how I’m gonna do it. So the area of the square, well it’s seven times seven; it’s forty-nine centimetres squared.
Now for the quarter circle, I imagine the full circle which has a radius of seven centimetres and I’m gonna find that area, but then I’m gonna divide it by four as I only have a quarter circle. Remember also I’m not using π. I’m using three point one four as an approximation. So the area is three point one four multiplied by seven squared and then divided by four for quarter of that circle, which gives me a value of thirty-eight point four six five centimetres squared for that quarter circle.
Right the next step is to work out the orange area, which I can do by doing the square minus the quarter circle. So it’s forty-nine minus thirty-eight point four six five, which gives the answer of ten point five three five centimetres squared for that orange area.
Right the final step then is to work out the shaded area we were originally asked for. So in order to do that, I’m gonna do the total square, forty-nine centimetres squared, and then subtract this orange area twice. So it’s forty-nine minus two lots of ten point five three five, which gives me a final answer of twenty-seven point nine three centimetres squared for that shaded area.
Remember in the question we were asked to use three point one four as an approximation for π. So at this stage in our calculation, that’s what we did. We didn’t use the π button on the calculator; we used three point one four instead.
So if you ever faced with a question like that and a diagram where you can’t really picture how you’re going to work out an area. Sometimes drawing out just part the diagram gives you an idea of a strategy to use.
So there you have it. In summary, we have looked at a couple of different types of questions that you might be asked to do with calculating areas and circumferences of circles.