To find the area of a compound shape, split it into simple pieces, find each area, then add (or subtract). The perimeter is different: it is only the outside edge, so any line you drew to make the cut is not counted.
This skill opens the module on area, perimeter and surface area and is used again when you work with prisms and sectors.
How do you decide where to cut?
Look for the simple shapes you already know: rectangles, triangles, trapeziums and parts of circles. Draw the cut lines lightly on the diagram so you can see them.
Then ask two questions. Which lengths are given, and which do I need to work out? A good cut uses lengths that are already on the diagram or easy to find by adding or subtracting.
Step by step
- Sketch the cuts and label each piece A, B, C.
- Write down the missing lengths, for example a side equal to 12 − 7.
- Find each area with its own formula, keeping full calculator values.
- Add the areas, or subtract a removed piece from the larger shape.
- For perimeter, list only outside edges, including any curved edge.
- Round at the end and include units: cm, m or cm² and m².
Worked example
A building plan is a rectangle 12 m long and 8 m wide, with a semicircle attached to one 8 m side. Find the total area and perimeter.
Area. The rectangle is 12 × 8 = 96 m². The semicircle has diameter 8 m, so radius 4 m. Its area is ½ × π × 4² = 8π = 25.13 m². Total area = 96 + 25.13 = 121.1 m² (1 d.p.).
Perimeter. The outside edge has two sides of 12 m, the opposite 8 m side, and the curved edge. The curve is half a circle: ½ × π × 8 = 4π = 12.57 m. The 8 m side where the semicircle is attached is now inside the shape, so it is not counted. Perimeter = 12 + 12 + 8 + 12.57 = 44.6 m (1 d.p.).
The mistake to watch for
A student adds the full rectangle perimeter and then adds the semicircle arc on top.
Mistaken answer: 12 + 12 + 8 + 8 + 12.57 = 52.6 m
The 8 m side under the semicircle is no longer on the outside, so it should not be there.
The correction is to trace the boundary with a pencil without lifting it. Every length you trace is in the perimeter, and a length you never touch is not. Here the pencil goes along 12, then 8, then 12, then around the curve.
Check yourself
1. An L-shape is made from a 10 cm by 7 cm rectangle with a 4 cm by 3 cm rectangle removed from one corner. Find the area and the perimeter.
Show answer
Area = 10 × 7 − 4 × 3 = 70 − 12 = 58 cm². Cutting a corner out of a rectangle keeps the perimeter the same as the original rectangle: 2 × (10 + 7) = 34 cm.
2. A house shape is a rectangle 6 cm wide and 5 cm tall with a triangle of base 6 cm and height 4 cm on top. Find the total area.
Show answer
Rectangle: 6 × 5 = 30. Triangle: ½ × 6 × 4 = 12. Total = 42 cm².
3. A square of side 10 cm has a quarter circle of radius 10 cm drawn inside it from one corner. Find the area of the square outside the quarter circle.
Show answer
Square: 100 cm². Quarter circle: ¼ × π × 10² = 25π = 78.54 cm². Difference = 100 − 78.54 = 21.46, so 21.5 cm² (1 d.p.).
Where this leads next
Compound shapes often hide unit changes, so continue with calculating area after converting units. Then test yourself on the mixed practice set. The non-calculator working trainer helps you keep each step of a multi-part calculation tidy.
Some students can cut a shape correctly but still lose marks on the perimeter. Our teachers look for exactly that kind of pattern in online one-to-one Mathematics tuition.