Using three point one four to approximate 𝜋, what is the diameter of a circle of circumference one hundred and twenty-five point six centimetres.
So here’s a little sketch of what we know. The circumference is a hundred and twenty-five point six centimetres, and we’re trying to find the diameter of this circle. Now we’ve got two formulae that tell us about the circumference of a circle. One is that the circumference is equal to 𝜋 times the diameter of the circle, and the other is that it’s equal to two times 𝜋 times the radius of the circle. Now remember, the diameter is the distance from one part of the circumference to the opposite part of the circumference in a straight line, via the centre of the circle. And the radius is the distance from the centre of the circle to the circumference of the circle, so that’s half of the diameter.
Now we know the circumference and we’re looking for the diameter. So it looks like that first formula is gonna be the most useful for us. Okay then. Let’s put in the information we’ve got, and let’s see if we can rearrange this equation to find the diameter. We’re told that the circumference is a hundred and twenty-five point six centimetres. We’re told to use three point one four as an approximation for 𝜋. And we want to know what the diameter is.
So let’s just call that d. So a hundred and twenty five point six is equal to three point one four times d. Now I just wanna know what one times d is. So if I divide the right-hand side of this equation by three point one four, then divide the top and divide the bottom by three point one four, those things are going to cancel. But now I’ve unbalanced my equation, so I need to do the same to the other side. Three point one four is to go on the denominator. So d is equal to a hundred and twenty-five point six over three point one four. And the units are centimetres. But let’s face it. We’re not gonna get away with writing that as a- our answer. There’s still a bit of calculation to do. And we’ve got a fraction that’s got decimals in it, so that’s not great.
So let’s see if we can simplify this fraction at all. Well what I’m gonna do, is multiply the bottom by a hundred and multiply the top by a hundred, so that I can get rid of all those decimal places. So I’m multiplying the top and the bottom by the same number so they would cancel out. So I’m not changing the size of this number, I’m just changing it to an equivalent fraction. So multiplying the numerator by a hundred gives me twelve thousand five hundred and sixty. And multiplying the denominator by a hundred gives me three hundred and fourteen. Now if you’re very good at your three hundred and fourteen times table, you’ll know that three hundred and fourteen times forty is equal to twelve thousand five hundred and sixty. And you’ll immediately be able to get to the right answer. But let’s assume you’re not quite that good. And let’s just do a bit of simpler cancelling with slightly smaller numbers, to start off with. Let’s divide the top and the bottom by two. And half of twelve thousand five hundred and sixty is six thousand two hundred and eighty. And half of three hundred and fourteen is one hundred and fifty-seven.
Now when we were doing that, well I spotted that three hundred and fourteen times two is six hundred and twenty-eight, so it looks like this is gonna cancel. Let’s rewrite six thousand two hundred and eighty as three hundred and fourteen times something. Well three hundred and fourteen times two is six hundred and twenty-eight. But I’m looking for six thousand two hundred and eighty, so I need to multiply by twenty to get six thousand two hundred and eighty. Now we know that we’ve just divided three hundred and fourteen by two to get one hundred and fifty-seven. So we can rewrite three hundred and fourteen as a hundred and fifty-seven times two, which makes our fraction a hundred and fifty-seven times two times twenty over a hundred and fifty-seven. Now if I divide the top by a hundred and fifty-seven and the bottom by a hundred and fifty-seven, they both cancel out. So my answer is two times twenty, which is forty.
And not forgetting that our units were in centimetres, the answer is that the diameter is forty centimetres.