Perimeter and Area of 2D Shapes

Perimeter and area answer two different questions about the same shape. Perimeter is the distance all the way around the edge — the length of fence you would need. Area is the space covered inside — the amount of turf. The units keep them apart: a plain unit like centimetres is a length, so it is a perimeter, while a squared unit like square centimetres is an area. And there is a reason for that little raised 2: area comes from multiplying two lengths, so centimetres times centimetres gives square centimetres. If you ever forget which answer you have found, look at the units.

Start with the rectangle, since everything else builds on it. For one measuring 12 cm by 7 cm, the perimeter is 2 times the length plus the width: 2 times 19, which is 38 centimetres. The area is length times width: 12 times 7, which is 84 square centimetres. A square is just a rectangle with both sides equal, so its perimeter is 4 times the side and its area is the side times itself — for a side of 7 that gives 28 centimetres and 49 square centimetres. One warning on the perimeter formula: it is 2 times the bracket, because both lengths and both widths get doubled. Dropping the bracket gives 2l + w, which is wrong.

Triangles and parallelograms both come back to the rectangle. A triangle sits inside a rectangle of the same base and height and fills exactly half of it, so its area is a half times base times height — with a base of 10 and a height of 6 that is half of 60, or 30 square centimetres. A parallelogram is a rectangle with a triangle slid from one end to the other, so its area is simply base times height — base 9, height 4 gives 36 square centimetres. Now the big trap. The height means the perpendicular height, the straight-up distance from the base measured at a right angle to it, not the length of the slanted side. Diagrams give you the slant length on purpose as a distractor, so look for the little right-angle marker or the dashed line drawn inside the shape.

Two more tools. A trapezium has one pair of parallel sides, and its area is a half times the sum of those two parallel sides, times the height — you average them, then multiply by the height. With parallel sides of 5 and 9 and a height of 4, that is half of 14 times 4, which is 28 square centimetres. For a composite shape, cut it into rectangles, find each area, and add them. But watch what happens to perimeter: area is built from the pieces, so the cut line is fine, while the perimeter goes around the outside only and the cut line is not part of it. Including the cut line in the perimeter is the most common mistake here. Trace the outline with your finger — if your finger never crosses the middle, you have it right.

Two things exams love. First, working backwards: given an area of 84 square centimetres and a width of 7, find the length. Write the formula, substitute what you know — 84 equals l times 7 — and solve, giving l = 12 centimetres. Formula, substitute, solve, every time. Second, area units do not scale the way length units do. One metre is 100 centimetres, but one square metre is 100 times 100, which is 10000 square centimetres. The reason is that you scaled both directions: a square metre is 100 centimetres across and 100 centimetres down, so it holds ten thousand little centimetre squares. Same idea for one square centimetre being 100 square millimetres. Square the conversion factor rather than using it once.