Area of Parallelograms

A parallelogram is a four-sided shape with two pairs of parallel sides, and it is easiest to picture as a rectangle that has been pushed over so it leans. Opposite sides stay parallel and equal, so this one has two sides of 10 centimetres and two of 5. To find its area you need two measurements. The base is one of the sides, normally the one drawn flat along the bottom. The height is the straight-up gap between that base and the side opposite it, measured at a right angle — which is why it is drawn as a dashed line inside the shape rather than along an edge. Here the base is 10 and the height is 4, while that 5 is the slanted side.

Why does the formula work? Cut the parallelogram straight down along its height, taking the triangle off the left-hand end, then slide that triangle across to the right. Because the two slanted sides are parallel and equal, it fits perfectly, and what you are left with is an ordinary rectangle 10 centimetres by 4. You did not add anything or throw anything away, so the area never changed — which means the parallelogram has exactly the same area as that rectangle. So the formula is simply base times perpendicular height: 10 times 4 gives 40 square centimetres. Notice there is no half anywhere. The half belongs to triangles; a parallelogram is a whole rectangle's worth of space.

Exam diagrams usually give you three numbers when the formula only needs two, and the extra one is the slanted side. Here the base is 10, the perpendicular height drawn dashed inside is 4, and the slanted side is 5. The right answer is base times height, 10 times 4, which is 40 square centimetres. Using the slant instead gives 50, which is too big, and that is worth knowing: the slope is always longer than the straight-up gap, so this mistake always inflates your area. The other slip is halving, which gives 20 — that is the triangle formula wandering in where it does not belong. Pick the dashed number, multiply, and do not halve.

You do not have to use the side that happens to be drawn along the bottom. Any side can be the base, as long as you use the height that goes with it. This is one parallelogram shown twice. Standing on its 15 centimetre side, the perpendicular height is 8, so the area is 15 times 8, which is 120 square centimetres. Turn the very same shape so it stands on its 10 centimetre side, and the perpendicular height is now 12, giving 10 times 12 — still 120. Both are right, because a shape only has one area. What you must never do is pair a base with the wrong height: 15 times 12 would give 180, which no version of this shape has.

Last of all, the reverse question: you are told the area and asked for a missing measurement. Say a parallelogram has an area of 45 square centimetres and a base of 9. Write the formula, A equals b times h. Substitute what you know, so 45 equals 9 times h. Then undo the multiplication by dividing: h is 45 divided by 9, which is 5 centimetres. Always check by going forwards — 9 times 5 is 45, which matches. The same four steps handle a missing base. And these are real questions: a garden bed shaped like a parallelogram with a base of 6 metres and a height of 3 needs 18 square metres of soil, and that unit must be squared.