Area of Triangles and Parallelograms by Rearranging

Draw a rectangle six by four and cut it along a diagonal: two triangles, exactly equal, each half of the twenty-four. So a right-angled triangle with base six and height four has area twelve, and the rule for any triangle is base times height divided by two. On a grid you can see it by counting whole squares and pairing the half-squares along the slanted edge. Halve at the end, or halve one of the numbers first if it is even — ten times five, halved, is five fives.

A triangle that is not right-angled is still half a rectangle. Drop the height line from the top corner straight down to the base: it splits the triangle into two right-angled triangles, each half of its own rectangle, so the whole triangle is half of the eight by five rectangle around it — twenty. The height is that straight-up distance, at right angles to the base, never a slanted side. Any side can be the base, as long as the height is measured to that side. Two triangles with the same base and height have the same area however different they look.

A parallelogram is a rectangle in disguise. The slanted end is a triangle; cut it off and slide it to the other end, and the parallelogram becomes a rectangle with the same base and the same height. Nothing was added or lost, so the area is base times height — seven times four, twenty-eight. The height is the straight-up distance between the two parallel sides, not the slanted side. Slant changes the perimeter but never the area, and there is no halving, because a parallelogram is a whole rectangle rearranged.

When a shape is labelled with three numbers, two of them are for area and one is a trap. A parallelogram with base seven, height four and a slanted side of four and a half has area seven times four, twenty-eight; seven times four and a half is the trap. The height meets the base at a right angle — look for the square-corner mark or the dashed line — and it is always shorter than the slanted side. Perimeter uses the real sides; area uses base and height.

Any shape made of straight sides can be split into rectangles and right-angled triangles, and then it is pieces you know. A rectangle six by four with a triangle of base four and height four on the end is twenty-four plus eight, thirty-two. A trapezium is a rectangle with a triangle at each end. A square cut on the diagonal is two equal triangles. The skill the syllabus names is rearranging: everything here is rectangles and half-rectangles, put together or taken apart. Quiz and exercises now.