Circle circumference and area

One full turn of a bike wheel carries you forward by the distance round the outside of that wheel.

Unroll the tyre, lay it out flat along the ground, and that length is exactly how far one turn takes you.

That distance round the outside is called the circumference, and it comes out of one measurement: the radius.

Video three of four in Perimeter, Area and Circle Mensuration. If the shapes with straight edges are still shaky, Area of triangles, parallelograms and trapezia comes first.

The radius is the distance from the centre of a circle out to the edge. It is the measurement everything else is built on. The diameter goes all the way across, through the centre, so the diameter is simply twice the radius. A circle measures nine centimetres across. What is its radius? Four point five centimetres. Halving is allowed to leave you a decimal, and nothing has gone wrong when it does. Measure the way round any circle, then measure the way across it. The way round is always a bit over three times the way across. Always, for any size of circle, the same number: three point one four and on for ever. That number is pi. Which is exactly what the circumference formula says, and it is worth saying slowly. The way round a circle is two times pi times the radius. Written the other way round, it is pi times the diameter. The same thing said twice, because a diameter is two radiuses. Area is a different job. Cut a circle into thin wedges and lay them out head to tail, and they push into something very close to a rectangle. That rectangle is the radius tall, and half the way round wide. The second formula, and again worth taking slowly. The area of a circle is pi times the radius squared. These two formulae are on every tier, whichever pair of names your paper uses for them. Fencing round a pond: two pi r, pi r squared, or pi times the diameter squared? Two pi r, the fencing one. A fence is a length, and the other two multiply two lengths together, so they come out in squares. Two formulae, one radius between them, and most of the job is choosing which one a question wants.

A clock face has a radius of seven centimetres, and the question wants the distance round the rim. The diameter is twice the radius, so the way across this clock is fourteen centimetres. The way across is fourteen centimetres. Roughly how far is it all the way round? A bit over forty-two centimetres, because the way round is always a bit over three times the way across. Now the exact version. Remember, the way round is two times pi times the radius, and the radius on this clock is seven centimetres. We put that seven in: two times pi times seven. Two sevens are fourteen, with the pi riding along, so it is fourteen pi centimetres. Fourteen pi is the exact answer. Nothing has been rounded, and pi is still standing in it. For the decimal, press two, times, pi, times, seven, equals. The display reads forty-three point nine eight two two nine seven one five. Round once, at the very end, never in the middle. To one decimal place, forty-four point zero centimetres. The zero stays on the end, because one decimal place is what was asked for.

Now a rug, and this time the area. The rug measures one point eight metres across. Here is a first attempt. Pi times one point eight squared. One point eight squared is three point two four, and pi times that comes out at about ten point one eight square metres. That attempt is wrong before the arithmetic even starts. What has gone in wrong? That number is the whole way across, not the radius. The formula wants the radius, so it has to be halved first. So we halve. Half of one point eight is nought point nine, so the radius is nought point nine metres. Next we square that radius. Nought point nine times nought point nine is nought point eight one, so the area is nought point eight one pi square metres. Which is two point five four square metres, to two decimal places. The first attempt was four times too big, and that is what a diameter always does in a squared formula. One examiner's report on a Foundation paper names that error. A common error in the work with circle area was to multiply pi by the diameter squared rather than radius squared, suggesting confusion over radius and diameter, or use of the wrong circle formula, finding circumference instead of area. The habit is one extra line at the top of your working. Write down the radius before you write anything else. Last one. A circular flower bed sits in a square lawn, and the question asks for the bed alone. The bed is four metres across, so its radius is two metres. Nothing else about that lawn matters here. The bed is the whole question. Over to you. What is the area of the bed, in terms of pi? Four pi square metres for that bed, and the working is two short steps. Remember, area is pi times the radius squared. The radius is two metres, and two squared is four, so the area is four pi square metres. As a decimal, that comes to about twelve point five seven square metres. That is a whole circle's area, written on its own line with its units on it, and that matters for what comes next. Any question about a piece of a circle starts by finding the whole circle first. Get it down, units and all, before you touch the piece. Both formulae back, once, before the end. The way round is two times pi times the radius. The area is pi times the radius squared. One radius, feeding both.

Perimeter is the fence, area is the grass, and round a circle the grass only ever eats the radius. So halve the diameter before you square it.

Last stretch. Three questions, and the answers wait for you. Which of the two is a length: two pi r, or pi r squared? The first one, two pi r. That is the fence round the circle, and pi r squared is the grass inside it. Here's a different one. A circle of radius three. Circumference in terms of pi? Six pi. The circumference is two times three, which is six, with the pi riding along. Now try another. A circle ten centimetres across. Area in terms of pi? Twenty-five pi square centimetres. Remember, area is pi times the radius squared, so ten across means a radius of five, and five squared is twenty-five.

This chapter is four videos long. Thumbs up whichever ones you are comfortable with, and they never need watching again. The ones with no thumb are the ones to return to. Circles come with practice, so save this and let it settle for a day or two.

Next in the chapter: Arc length and sector area, including major sectors.

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