Perimeter of Composite Shapes

Perimeter is the distance all the way round the outside of a shape. For a composite shape — an L, a T, a staircase — there's no formula, and you don't need one: just walk the boundary and add every edge you cross. The shape here has six sides, 12 + 4 + 4 + 5 + 8 + 9, which comes to 42 cm. Two habits make this reliable. Count how many sides the shape has before you start adding, then check your sum used that many numbers. And tick each side as you use it, so nothing gets counted twice or skipped.

Exam questions usually leave one or two sides unlabelled, and there's a rule that always finds them. In a shape made of right angles, the horizontal sides going one way must add up to the horizontal sides going the other way, and the same holds for the verticals. So in this shape, the two verticals on the right must add to the 9 on the left: 9 minus 4 gives 5. And the two horizontals along the bottom must add to the 11 across the top: 11 minus 5 gives 6. Now every side is known, and the perimeter is 11 + 4 + 5 + 5 + 6 + 9 = 40 cm.

When part of the boundary is curved, that piece contributes an arc length instead of a straight side. For a rectangle capped with a semicircle: the three straight sides give 16 + 16 + 12, and the curved end gives half a circumference, which is half of π times the diameter 12, about 18.85 cm. The total is roughly 62.8 cm. The trap is the dashed line where the semicircle joins the rectangle — that's a construction line you drew to split the shape up, not an outside edge, so it isn't part of the perimeter. Only add the diameter as a straight side when the flat edge really is on the outside.

Questions often run the other way: you're given the perimeter and asked for a missing length. The method is to write the perimeter as an expression, set it equal to the number you're given, and solve. For a rectangle with sides x and 5 and a perimeter of 34: 2x + 2 times 5 equals 34, so 2x + 10 = 34, 2x = 24, and x = 12 cm. That's an ordinary linear equation, so nothing new is needed. Substitute back to check — 2 times 12 plus 10 is 34 — and sanity-check the answer, because a length has to be positive and smaller than the whole perimeter.

Five checks before you write down a perimeter. Are all the measurements in the same unit — adding 40 cm to 1.2 m gives nonsense until one of them changes. Are your units a length, like cm or m, and never cm squared, because only area gets squared units. Does the question want a rounded decimal or an exact answer left in terms of π. Did you use every side — count the sides, count your numbers, and make them match. And is it a fencing question, because a gate or an open side against a wall may mean you deliberately don't go all the way round. Remember 1 m = 100 cm, 1 km = 1000 m, 1 cm = 10 mm.