Composite Area

Composite shapes are made of simpler ones, and there are two ways to handle them. Split the shape into pieces you recognise and add their areas. Or fill the shape out to a bigger rectangle, work out that rectangle's area, and subtract the piece you added. Both give the same answer, so pick whichever needs fewer missing lengths worked out — often subtracting is quicker for an L-shape. Whatever you choose, draw the dividing line on the diagram and label each piece before you calculate anything. Most errors here come from doing the working in your head.

Before you can find any area you usually need to deduce a side or two, using the same rule as for perimeter: the horizontals going one way must total the horizontals going the other, and the same for verticals. Here 9 minus 4 gives the 5, and 11 minus 5 gives the 6. Write each deduced length straight onto the diagram. Then split at the dashed line: the top piece is 11 by 4, which is 44, and the bottom piece is 5 by 5, which is 25, for 69 square centimetres. Notice that once you split, each piece uses its own dimensions — the bottom rectangle is 5 wide, not 11.

Composite shapes often mix straight and curved pieces. A running-track shape is a rectangle with a half circle on each end. Work out the rectangle — 230 by 110, giving 25300 — and then notice something useful: two half circles of the same radius make one whole circle, so instead of computing two halves you compute π times 55 squared, about 9503. The total is roughly 34803 square units. Spotting that two halves make a whole saves a step and avoids rounding twice, which is where small errors creep in.

'Find the shaded area' questions are subtraction questions in disguise, but the first move is to look carefully at what's actually shaded. Here a circle is cut from a square and the four leftover corners are shaded, so the answer is the square minus the circle: 324 minus about 254.5, giving roughly 69.5 square centimetres. If the shading had been the circle itself, the answer would simply be the circle. One technical point matters more than usual on these: keep full accuracy through the working and round only at the very end, because rounding 254.5 to 250 partway through would shift the final answer by 4.5.

Six steps that work on any composite shape. Decide whether to split or to subtract. Draw the dividing line on the diagram so you can see the pieces. Deduce every missing length and write it down. Find each piece separately, labelling them A, B and C. Add or subtract as the shape requires. And round once at the end, with squared units on your answer. There's a marking reason to work this way as well: in an exam, marks are awarded for the individual pieces, so a student who shows A = 44 and B = 25 and then makes an arithmetic slip on the total still scores most of them.