Arc length and sector area, including major sectors

A wedge is marked on a round table top, one hundred degrees wide. The area you want is everything else, and the angle you need for it is nowhere on the drawing. You will have that area before the end, and it is a couple of easy steps. The first is a quarter of a circle. A slice of a circle is a fraction of that circle, and the angle tells you which fraction.

Video four of four in Perimeter, Area and Circle Mensuration. Everything here leans on Circle circumference and area, so start there if the two circle formulae are not solid.

A sector is a slice of a circle, cut from the centre out, like a slice of cake. Its curved edge has a name of its own: an arc. A whole circle is three hundred and sixty degrees, and a slice with an angle of ninety takes ninety of them. So a ninety degree slice is what fraction of the circle: a half, a quarter, or a third? A quarter, because ninety is a quarter of the way round. Every measurement of that slice is a quarter as well. That is the method, all of it. The angle over three hundred and sixty is your fraction, and you take that fraction of whatever the whole circle has. Here is the first formula, and it is worth saying slowly. Arc length is the angle over three hundred and sixty, times two pi r - the fraction, times the whole way round. Here is a sector with an angle of forty degrees. Its radius, remember, is the distance from the centre out to the rim, and here that is nine centimetres. The question wants the arc. We build the whole circle first. Remember, the way round a whole circle is two times pi times the radius, and the radius here is nine. Nine doubled is eighteen, with the pi riding along, so the whole way round would be eighteen pi. The whole way round is eighteen pi. What is forty degrees' worth of it? Two pi centimetres' worth, and it comes in two steps. First the fraction. Forty over three hundred and sixty cancels down: divide top and bottom by forty, and it comes to one over nine. Next we take one ninth of the whole way round. Eighteen pi shared into nine is two pi, so the arc is two pi centimetres. About six point two eight centimetres, if the question asks for a decimal rather than an exact answer. Arcs and sectors turn up at both tiers on most papers, so check your own paper before skipping anything.

Sector area runs on the same fraction. A quarter of a circle holds a quarter of its area, and nothing new has to be learnt. The second formula, just as slowly. Sector area is the angle over three hundred and sixty, times pi r squared - the same fraction, times the whole circle's area. Now back to that table top, fifteen centimetres from the centre to the rim, with the wedge marked out on it. The marked wedge has an angle of one hundred degrees, and the question asks for the area of the section left over. So the wedge that is marked is the smaller one, and the piece you want is the bigger one. The bigger piece is the one you want. What angle does it have? Two hundred and sixty degrees for the bigger piece. Three hundred and sixty take away one hundred, and that subtraction gets a line of its own. On its own line, written down, every single time. The moment it hides inside a bigger calculation, it goes missing. Two hundred and sixty out of three hundred and sixty. What does that cancel down to? Thirteen eighteenths, and it comes down in a single move. Both of those numbers divide by twenty. Two hundred and sixty over twenty is thirteen, and three hundred and sixty over twenty is eighteen, so the fraction is thirteen over eighteen. Now the whole circle's area, before we take any fraction of it. Remember, the area of a circle is pi times the radius squared, and the radius here is fifteen centimetres. Fifteen squared means fifteen times fifteen, which is two hundred and twenty-five. The whole table top is two hundred and twenty-five pi square centimetres. Now we take thirteen eighteenths of that. Two hundred and twenty-five shared into eighteen is twelve point five. And twelve point five times thirteen is one hundred and sixty-two point five, so the piece you want is one hundred and sixty-two point five pi square centimetres. As a decimal, about five hundred and ten point five square centimetres, if the question wants one. Somebody uses the marked one hundred degrees instead. Is their answer too big or too small? Far too small. That answer is the little wedge, not the big piece, and most of the table goes missing. One examiner's report on a Higher paper sets out what happened on a major arc question of its own, with different numbers. For the major arc length, a number did not identify two hundred and forty degrees as the angle for the sector. Errors included finding the full circumference, using the minor sector angle one hundred and twenty degrees, or using the area formula for the sector. Three different wrong turns, and all of them happen before the arithmetic starts. Every one is a choice about which angle, or which formula, to use. The habit that earns the mark is two lines before any substituting. Which piece am I being asked for, and what is its angle.

One last kind. Show that the area of a sector with radius eight centimetres and angle ninety degrees is exactly sixteen pi square centimetres. The answer is printed inside the question, so the marks are not in the answer. They are in the working that arrives at it. Write the first line of that working yourself, before I do. The line is ninety over three hundred and sixty, times pi, times eight squared. The fraction and the formula, joined up. Now we take that line apart, a piece at a time. Ninety over three hundred and sixty cancels down to one quarter. Then the circle. Eight squared means eight times eight, which is sixty-four, so the whole circle's area is sixty-four pi. And a quarter of sixty-four pi is sixteen pi square centimetres, because sixty-four shared into four is sixteen. That is just what the question claimed, so the showing is done. One examiner's report on a Higher paper describes the students who scored on a question like that. The higher ability candidates set up a correct equation involving the formula for the area of the sector. An equation, then. Even when the answer is already printed, the fraction and the formula go down as one line of working. Both formulae back, once, before the end. Arc length is the angle over three hundred and sixty, times two pi r. Sector area is the angle over three hundred and sixty, times pi r squared.

Perimeter is the fence, area is the grass, and a sector is a slice of each. Slice the fence for the arc, slice the grass for the area, and the slice is the angle over three hundred and sixty. That is all the rest of that table top was: thirteen eighteenths of the grass, five hundred and ten point five square centimetres.

That is the chapter done, but not before three last questions. Every sector question starts on the same fraction. What is that fraction? The angle over three hundred and sixty. It scales the arc and the area in exactly the same way. Now for a different one. Radius six, angle one hundred and twenty. Arc length? Four pi centimetres. One hundred and twenty over three hundred and sixty is a third, the whole way round is two times pi times six, which is twelve pi, and a third of twelve pi is four pi. Now another. A minor sector is marked as eighty degrees. What angle has the major one? Two hundred and eighty degrees. Three hundred and sixty take away eighty, on its own line as always.

Four videos, and this is the last of them. Thumbs up every one you have fully understood, and your own gaps show up on their own. Whatever has no thumb on it is worth another go. Don't worry if it hasn't clicked yet - save the chapter and give it a second pass later in the week.

That completes our chapter on Perimeter, Area and Circle Mensuration. Next chapter: Constructions, Loci, Scale Drawings and Bearings.

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