Area of Compound Shapes

Split, find, and add

A compound (or composite) shape is made of simpler shapes joined together. An L-shape, for instance, is just two rectangles stuck side by side.

The trick is to split the shape into pieces you already know how to measure — usually rectangles — find the area of each piece with its own formula, then add the areas together:

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

Sometimes it is easier to start with a big rectangle that contains the whole shape and subtract the missing corner piece instead:

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

Before you can multiply, you often have to find a missing side length — opposite sides of the outline must agree, so a missing length is the difference (or sum) of the ones you are given.

The method

Whether you add or subtract, the recipe is the same: break the shape down, measure each piece, and combine.

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.