Area of Compound Shapes

Real rooms aren't rectangles

Stand in the doorway of a real house and look at the floor plan. Almost nothing is a plain rectangle. Rooms turn corners into an L; a running track is a rectangle with a half-circle stuck on each end; a garden has a rectangular pond cut out of the middle. Yet the carpet-fitter still has to know exactly how many square metres to order, and the surveyor still has to price the land.

The secret they all use is beautifully simple. A compound (or composite) shape is one built from simpler shapes joined together — and you already know how to find the area of the simple pieces. So you break the awkward shape into rectangles, triangles and half-circles, work out each area on its own, and then just add them up. An L-shaped room is nothing more fearsome than two rectangles holding hands.

Split, find, and add

The core move is to split the shape into pieces you can already measure — most often rectangles — find the area of each piece, then add the areas together:

A_{\text{total}} = A_1 + A_2 + \dots

Sometimes it is quicker to think the other way round: start with a big rectangle that swallows the whole shape, then subtract the missing corner or the hole:

A = A_{\text{big}} - A_{\text{cut-out}}

Adding and subtracting are two roads to the same town — for many shapes either will do, and a good habit is to solve it once each way and check the answers match.

To find the area of a compound shape:

split it into simple shapes — usually rectangles;
work out any missing side lengths first, using the lengths you are given (opposite sides of the outline must match up);
find the area of each piece with its own formula;
add the areas together — or, if you boxed the shape inside a big rectangle, subtract the cut-out piece.

See it

Step through the figure: meet the L-shape, cut it into two rectangles with a dashed line, then add their areas to get the total.

Worked example 1 — an L-shape, split two ways

Take the L-shape in the figure above: overall it is 6 wide and 4 tall, with a 4 \times 2 bite taken out of the top-right corner. Watch how two different cuts land on the same answer.

Way A — a horizontal cut. Slice along the inner step to get a wide bottom rectangle and a small top one:

\underbrace{6 \times 2}_{12} \;+\; \underbrace{2 \times 2}_{4} \;=\; 16 \text{ square units}

Way B — a vertical cut. Slice down the inner step instead: a tall left rectangle and a short right one:

\underbrace{2 \times 4}_{8} \;+\; \underbrace{4 \times 2}_{8} \;=\; 16 \text{ square units}

Same shape, same 16 — the cut you choose never changes the true area, only how tidy the arithmetic looks. Pick whichever split gives the friendliest numbers.

Worked example 2 — a shape with a semicircular end

A window is a rectangle 4 wide and 3 tall, topped by a semicircle whose flat edge is the top of the rectangle. The semicircle sits on the 4-wide side, so its diameter is 4 and its radius is r = 2.

A_{\text{rectangle}} = 4 \times 3 = 12

A_{\text{semicircle}} = \tfrac12 \pi r^2 = \tfrac12 \times \pi \times 2^2 = 2\pi \approx 6.28

A_{\text{total}} = 12 + 2\pi \approx 18.28 \text{ square units}

A half-circle is just \tfrac12 of the circle's \pi r^2 — the same "find each piece, then add" recipe, only now one of the pieces is round.

Worked example 3 — a rectangle with a hole, by subtraction

A metal plate is an 8 \times 5 rectangle with a 3 \times 2 rectangular window punched out of the middle. Splitting the frame into four thin strips is fiddly — subtraction is far cleaner:

A = \underbrace{8 \times 5}_{40} - \underbrace{3 \times 2}_{6} = 34 \text{ square units}

Step through it below: draw the big rectangle, punch the hole, take one from the other.

Three mistakes wreck more compound-area answers than anything else:

Find the missing lengths first. In an L-shape, an unlabelled side is not a free guess — it equals the difference of the two opposite sides. If the whole shape is 6 wide and the notch is 4 wide, the leftover top edge must be 6 - 4 = 2. Multiply by the wrong length and every area after it is wrong.
Decide add or subtract — and don't mix them up. Adding pieces that overlap, or forgetting to subtract the hole, silently double-counts area. Ask: "Is this region part of the shape, or a gap in it?"
Keep the units consistent. You cannot multiply 3\text{ m} by 40\text{ cm} and get a sensible area — convert to a single unit first, then answer in square units (\text{cm}^2, \text{m}^2).

Breaking a hard shape into easy pieces isn't just a schoolroom trick — it's how architects price a floor, how carpet-fitters cut to size, and how land surveyors measure a crooked field (they chop it into triangles and add them, a method as old as ancient Egypt).