Perimeter of 2D and composite shapes

A running track: two long straights, with a curved end at each end. Somebody has to buy the edging that runs all the way round it. How many metres do they order? You will have that number by the end of this video, and it is only a few easy steps. The first one is a rectangle. Fence panels are sold by the metre. Grass seed is sold by the square metre. The distance all the way round the outside is the perimeter, the space inside is the area, and this video is the fence.

Start with a rectangle, seven metres along the bottom and three metres up the side, like a small patio. To find a perimeter you walk the boundary, the outside edge where a fence would stand, and you add every edge you cross. Perimeter sits on both tiers, Foundation and Higher, or Core and Extended if that is your board's wording. Three ways of using those two numbers, and only one of them is a perimeter. Is the perimeter seven times three, seven plus three, or all four sides added? All four sides added. Seven plus three plus seven plus three, which comes to twenty metres. Seven times three fills the inside instead. Seven plus three carries you along the bottom and up one side, then stops. The common slip here is adding one length and one width, then stopping. That is half the way round a rectangle, not the whole boundary. One examiner's report, on a Foundation paper, records that exact sum. Some students, it says, just added together the length and width, scoring zero. The fix is the walk. Pen on a corner, one way round, and tick each edge as it goes into the sum.

Now a garden path shaped like the letter L, the kind that runs round the corner of a lawn. Before any numbers at all: how many edges does an L shape have? Six. A rectangle has four corners and four edges, and the L has two more of each. Round that boundary the six edges measure eight metres, two, three, three, five and five. Start at the top left and go clockwise, keeping a running total. Eight, ten, thirteen, sixteen, twenty-one. The last edge takes it to twenty-six. The perimeter of the path is twenty-six metres. Now the same path, added up by somebody else, and their total comes out at twenty-four. Every number in that sum is a real edge. So why is twenty-four wrong? There are only five edges in that sum. The two-metre edge never got added, and a short edge is the easy one to miss. Count the edges before you add, then count the numbers in your sum. Six edges means six numbers, every time.

Now back to that running track, two long straights with a curved end at each end, and the curve is the new part. Each end is a semicircle, which is half a circle, and the radius is the distance from the centre out to the edge. Each straight here is forty metres, and each curved end has a radius of ten metres. The straights are the part you can already do. Forty and forty is eighty metres of boundary before the curve starts. Push the two curved ends together. What single shape's edge do they make? One whole circle. Two halves make a whole, so the two ends are one circle's worth of curve. The distance all the way round a full circle is two times pi times the radius. That is the one formula this bit needs, and it is worth saying slowly. Where that formula comes from is the job of the video on circle circumference and area. So we put the radius in. Two times pi times ten, and two tens are twenty, so the curve is twenty pi metres. As a decimal, twenty pi metres is about sixty-two point eight metres of curve. Now we add the straights back on. Eighty metres of straight, plus twenty pi metres of curve. Which comes to one hundred and forty-two point eight metres of edging, to one decimal place. If the question wants an exact answer, leave it as eighty plus twenty pi. If it wants a decimal, round once, right at the end.

Keep the picture. Perimeter is the fence, area is the grass. You buy fencing by the metre, a curved panel is still fencing, and the edging round that running track comes to one hundred and forty-two point eight metres of it.

Three quick ones before the end, and you go first on each of them. Fence or grass: which one of those two is the perimeter? The fence. Perimeter is the distance all the way round the outside. Next one. A rectangle, six by four metres. Someone writes ten metres. What is that? Half of it. Ten is one length plus one width, and the answer they want is twenty metres. Now a hexagon with six equal sides of five centimetres. What is its perimeter? Thirty centimetres. Six edges, five centimetres each, counted once each and added.

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Next in the chapter: Area of triangles, parallelograms and trapezia.

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