Area and circumference: rectangles, triangles and circles

One twelve-inch pizza, or two eight-inch pizzas for the same price: which gets you more pizza? Two feels like more, but it isn't. The single twelve-inch pizza has about an eighth more, and the reason is the rule for a circle's area, which grows much faster than its width.

This is video two of four in this chapter of maths skills. A triangle's area leans on the triangle facts, so if they feel shaky, go back to Angles, triangles and tessellation first.

Area is the flat space a material covers, like the acrylic in a sign or the tiles on a wall. It is counted in squares, so its units are square millimetres, square centimetres or square metres. A rectangle three centimetres long and two wide holds two rows of three one-centimetre squares, six square centimetres. That is why a rectangle's area is length times width. A triangle is half a rectangle. Draw a rectangle round it, as wide as the triangle's base and as tall as its height, and the triangle fills exactly half of it. The area of a triangle is a half, times the base, times the height. The half is there because the triangle is half of its rectangle. A house-number plaque is laser-cut from acrylic in the shape of a house. It is a rectangle two hundred millimetres wide and one hundred and fifty tall, with a triangular roof on top, also two hundred wide. The whole plaque stands two hundred and forty millimetres tall. A composite shape is one built from simpler shapes, and it is split into its pieces before any multiplying. How tall is the plaque's triangular roof, from its base to its tip? Ninety millimetres. Two hundred and forty take away one hundred and fifty is ninety. Now the pieces. The rectangle is two hundred times one hundred and fifty, which is thirty thousand square millimetres. Two students find the roof's area. One gets eighteen thousand square millimetres, and the other gets nine thousand. Which roof area is right, and what did the other student forget? Nine thousand square millimetres is right. A half, times two hundred, times ninety, is nine thousand. Eighteen thousand forgot the half, so it is the area of the rectangle round the roof. One examiner's report puts it plainly: Many candidates multiplied the length by the width but did not divide it by two. Those answers found the rectangle and stopped. The habit that fixes it is to draw the rectangle round every triangle, so the half is in front of you before you multiply. The plaque is the rectangle plus the roof. Thirty thousand plus nine thousand is thirty-nine thousand square millimetres of acrylic.

Thirty-nine thousand square millimetres is a clumsy number for a plaque, and the job asks for square centimetres. An area does not convert the way a length does. One centimetre is ten millimetres. A square one centimetre on each side is ten millimetres by ten millimetres, so it holds ten rows of ten tiny squares. One square centimetre is one hundred square millimetres. For an area, you square the length factor, and ten squared is one hundred. Metres work the same way. One metre is one hundred centimetres, so one square metre is one hundred times one hundred, ten thousand square centimetres. Back to the plaque, which has to be quoted in the bigger unit. What is thirty-nine thousand square millimetres in square centimetres? Three hundred and ninety square centimetres. Each square centimetre holds one hundred square millimetres, so divide by one hundred, not by ten. One examiner's report records this: The part of the question that was most challenging for candidates was the conversion of units within the context of a cross sectional area rather than conversion on a linear measure, hence the large proportion of almost correct answers six, sixty, six hundred, six thousand et cetera. A cross-sectional area is the area of a slice through a part. Those almost-correct answers had the right digits and the wrong power of ten, because an area was converted like a length. The fix is to convert the lengths before you multiply, or to square the factor. A café wants to cover thirteen point seven square metres of wall with square ceramic tiles, thirty centimetres along each side. Ignore gaps and cutting. Convert the length first: thirty centimetres is nought point three metres. One tile is nought point three times nought point three, which is nought point nought nine square metres. The tiles needed are the wall's area divided by one tile's area. Thirteen point seven divided by nought point nought nine is one hundred and fifty-two point two, and the twos keep going. Does the café buy one hundred and fifty-two tiles, or one hundred and fifty-three? One hundred and fifty-three. Tiles come whole, and one hundred and fifty-two would leave a bare patch, so you round up, however small the decimal. One examiner's report, on a question about covering a patio with paving stones, says: It was evident from the number of correct answers that students had a good level of mathematical knowledge to complete the required calculation. The same report adds: Where rounding up to the nearest whole paving stone was not indicated, a mark was lost. Both are true. The division was right, and the loss came at the last step. The fix is to write the rounding up as its own line: one hundred and fifty-two point two, round up, one hundred and fifty-three tiles. A second route works in square centimetres from the start, and it should land in the same place. Now try that route: thirteen point seven square metres in square centimetres is what? One hundred and thirty-seven thousand. A square metre is ten thousand square centimetres, so multiply by ten thousand. One tile is thirty times thirty, nine hundred square centimetres. One hundred and thirty-seven thousand divided by nine hundred is one hundred and fifty-two point two again. Multiply by one hundred instead, as if an area were a length, and the wall shrinks to about one and a half tiles.

A circle has two key lengths. The diameter goes right across, through the centre. The radius goes from the centre to the edge, and it is half the diameter. The distance round the edge is the circumference. Unroll the edge of any circle into a straight line, and it measures a little over three diameters. The exact number of diameters is pi, three point one four one six and on. Circumference equals pi times d, the diameter. It is a length, so its units are millimetres or centimetres, never square units. A plant-pot stand is bent from steel rod: one circular ring two hundred and ten millimetres across, and four straight legs, each three hundred and fifty millimetres long. What length of rod bends the ring, in millimetres to the nearest one? Six hundred and sixty millimetres. Pi times two hundred and ten is six hundred and fifty-nine point seven three, which rounds to six hundred and sixty. The whole stand is asked for in centimetres. The legs are four times three hundred and fifty, one thousand four hundred millimetres. Add the unrounded ring to make two thousand and fifty-nine point seven three millimetres. Divide by ten for centimetres: two hundred and five point nine seven, so two hundred and six centimetres of rod. The ring stayed unrounded until this last step. The space inside a circle is its area. Cut a circle into thin slices and lay them top to tail, and they make a shape close to a rectangle, half the circumference long and one radius tall. Half the circumference is pi r, and pi r times r is pi r squared. Area of a circle equals pi r squared: halve the diameter first, then square the radius. A round cork mat, the same size as the ring, sits inside it. Work out the mat's area, to the nearest square centimetre. Three hundred and forty-six square centimetres. The radius is half of two hundred and ten, one hundred and five millimetres, which is ten point five centimetres. Converting that length first keeps the area in square centimetres. Ten point five times ten point five is one hundred and ten point two five, and times pi that is three hundred and forty-six point three six. Put the diameter in where the radius goes, and pi times twenty-one times twenty-one gives one thousand three hundred and eighty-five, four times too big. And a half circle's curved edge is half of pi times d, never the whole circumference. A different examiner's report says this: Many students who gained fewer marks confused the circumference and area often adding an area to a length. The fix is to ask what the material is doing. Running round the edge, it is a length, pi d. Filling the inside, it is an area, pi r squared, and a length and an area are never added together. Now the pizzas from the start. A twelve-inch pizza has a radius of six inches, and an eight-inch pizza has a radius of four. What is the area of one twelve-inch pizza, and of two eight-inch ones together? For the twelve-inch pizza, pi times six times six is about one hundred and thirteen square inches. Each eight-inch pizza is pi times four times four, about fifty point three, so two make about one hundred and one. The width went up by half, but the area more than doubled, because the radius is squared. Inches are an old imperial unit, and the maths works the same in millimetres.

That covers rectangles, triangles, units and circles. Here is one question on each of them.

A triangle has base one hundred and twenty and height eighty millimetres. Its area? Four thousand eight hundred square millimetres. A half, times one hundred and twenty, times eighty. Now convert: four thousand eight hundred square millimetres in square centimetres? Forty-eight square centimetres. Divide by one hundred, because one square centimetre is one hundred square millimetres. Here's a ring one hundred millimetres across. What length of rod bends it? About three hundred and fourteen millimetres. Pi times one hundred is three hundred and fourteen point one six. One more: a mat fills that ring. What is its area in square centimetres? About seventy-nine square centimetres. The radius is five centimetres, five squared is twenty-five, and twenty-five pi is seventy-eight point five four.

That is why the one twelve-inch pizza holds more than two eight-inch ones. Area grows with the radius squared, and two smaller circles cannot keep up.

If those came easily, you've got this down, and a thumbs up records that, so you can skip it next time round. If one slipped, write the café tile sum out in full on paper once, and the method tends to stay with you.

Next in the chapter: Surface area and volume: cuboids and cylinders, where flat areas add up into boxes and pots.

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