Area of triangles, parallelograms and trapezia

A triangular flag: twelve centimetres along the bottom, and both slanting edges ten centimetres. Three numbers, and the one the area needs is not among them. Nothing on that flag is wrong. It simply does not hand over the height, and the height is the whole job. Three easy steps get you there, and the first one is a tile pushed over sideways.

This is video two of four in Perimeter, Area and Circle Mensuration. If today feels shaky, the one before it, Perimeter of 2D and composite shapes, does the edges first.

Area is the space inside a shape, counted in squares. On a shape this size, square centimetres. All three shapes coming up are examined at Foundation and at Higher, so nobody gets to skip them. A parallelogram is a pushed-over rectangle, with both pairs of opposite sides parallel. Cut the triangle off one end, slide it round to the other, and you have a rectangle back. The first of the three formulae, and it is the one the other two grow out of: the area of a parallelogram is the base times the perpendicular height. Perpendicular height just means the height that meets the base at a right angle, and that is the one measurement the formula insists on. This tile has a base of nine centimetres, a height of five, and a slanting side of six. Which two does the area use: nine and six, nine and five, or six and five? Nine and five. Those are the two that meet each other at a right angle. Now that pair goes into the formula. Remember, area is the base times the perpendicular height, so here that is nine times five. Nine times five is forty-five. And because we multiplied one length by another length, that answer is forty-five square centimetres. The slanting six is there to be ignored. A slant never goes into an area calculation. A triangle is half of that. Two identical triangles push together into a parallelogram, so a triangle covers half of what that parallelogram covers. The second formula, and it is worth saying slowly. The area of a triangle is half the base times the perpendicular height. A trapezium has just one pair of parallel sides. Two copies of it make a parallelogram whose base is those two parallel sides added together. So a trapezium is half of that: add the two parallel sides, halve, then times the height. Take a trapezium tile with parallel sides of six and ten centimetres and a height of four. We start by adding the two parallel sides. Six and ten make sixteen, and that goes down as its own line. Next we halve that sixteen, which leaves eight. And last, eight times the height, four, gives thirty-two square centimetres. Three formulae, and every one of them wants the height. Not a side. The height.

Back to the flag: twelve centimetres along the bottom, and both slanting edges ten. Which number on that flag is the perpendicular height? None of them. One is the base and the other two are slants, and the perpendicular height was never written down. The height is the straight-up distance from the base to the top corner. It is never just any old side you are handed. One examiner's report on a Foundation paper, about a triangle whose height had to be found first, lists what students used instead. Many used seventeen centimetres, or seventeen take away eight equals nine, as the height. Both of those numbers came straight off the diagram. Neither one was the perpendicular distance to the base, so every area after it was wrong. The habit that earns the mark is a question. Before you use a number as the height, ask whether it meets the base at a right angle. On this flag the two slanting edges are equal, so the straight-up line from the top lands exactly in the middle of the base. That cuts twelve into six and six, and it leaves a right-angled triangle with a slant of ten and a short side of six. That needs Pythagoras. In a right-angled triangle, each of the two shorter sides multiplied by itself, added together, comes to the longest side multiplied by itself. Rearranged, that rule hands you a missing shorter side. Ten is the longest side and six is along the bottom. How tall is that right-angled triangle? Eight centimetres tall, and the walk to it goes one line at a time. The slant of ten is the longest side, so we square that one first. Ten squared means ten times ten, which is one hundred. Next we square the short side along the bottom. Six squared means six times six, which is thirty-six. Then we take that thirty-six off the one hundred, and it leaves sixty-four on the next line down. And that sixty-four is the height multiplied by itself, so the height is the square root of sixty-four, which is eight centimetres. Now the area, with a height that is finally the right one. Remember, a triangle is half the base times the perpendicular height. The base here is twelve centimetres, and the height is the eight we have just built. First we multiply the two of them. Twelve times eight is ninety-six. Then we halve it, and half of ninety-six is forty-eight. Forty-eight square centimetres, and a flag with no height written on it anywhere has handed over its area. Square centimetres, because area covers a surface. Lengths come in centimetres, coverings come in square centimetres.

Now a second flag, same idea with different numbers: a base of sixteen centimetres and both slants ten. The base splits into eight and eight, and Pythagoras turns that into a height of six. Those two steps are already done for you, so only the last one is left. Work out the area of that second flag yourself. Forty-eight square centimetres for that second flag's area, and the working goes line by line. Half the base times the height, so we multiply first. Sixteen times six is ninety-six. And then we halve it. Half of ninety-six is forty-eight square centimetres, the same area as the first flag. Two triangles that look nothing alike, covering exactly the same amount, because the base grew as the height shrank. One examiner's report on another Foundation paper names a second slip. Candidates very commonly calculated base times perpendicular height for the area of the triangle, without dividing by two. This resulted in zero marks being given here. The habit there is to write the halving down as its own line, so it cannot quietly go missing. All three formulae back, once, before the end. A parallelogram is the base times the perpendicular height. A triangle is half the base times the perpendicular height. And a trapezium adds the two parallel sides, halves them, then times the height.

The chapter's picture again: perimeter is the fence, area is the grass. And grass is measured straight back from the fence, never along a slanting path - which is how that flag, with no height written on it anywhere, still gave up forty-eight square centimetres.

Time to find out what stuck. Three of them, and you answer first. All three formulae need one measurement besides the base. Which one? The height, and always the perpendicular one - the straight-up distance from the base. Try this. A triangle with a base of ten centimetres and a height of seven. What is the area? Thirty-five square centimetres. Half the base times the height, so ten sevens are seventy, and half of seventy is thirty-five. Now a harder one. A parallelogram: base eight, slant seven, height five centimetres. Area? Forty square centimetres. Base times height, so eight fives, and the slanting seven stays out of it.

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Next in the chapter: Circle circumference and area.

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