ACARA v9 CONTENT DESCRIPTION “solve problems involving the area and perimeter of irregular and composite shapes using appropriate units”
What a composite shape is
A composite shape is built by joining simple shapes that you already know, usually rectangles and triangles. Many real outlines are composite rather than a single neat rectangle: the floor plan of a room, the boundary of a block of land, or the edge of a garden bed will often have steps and corners in it. The useful idea is that any such outline can be broken into a small number of simple pieces. Once you can see those pieces, both the area and the perimeter become a short sequence of familiar calculations rather than one hard problem.
A composite shape splits into rectangles
An L-shape is two rectangles joined; the dashed line shows the split.
a composite shape is made by joining simple shapes; an L-shape, for example, splits into two rectangles.
Area by splitting into simple shapes
To find the area of a composite shape, decompose it into rectangles and triangles, find the area of each piece, and add the results. The area of a rectangle is length x width, and the area of a triangle is (1/2) x base x height; both are tools carried over from earlier years. Take the L-shape shown above: split it into a lower rectangle of 8 x 2 = 16 and an upper rectangle of 3 x 4 = 12, then add to get 16 + 12 = 28 cm^2. Keep every length in the same unit so the pieces add cleanly, and write the area in square units.
Area by adding the pieces
Find each rectangle, then add the areas.
split the shape into rectangles, find each area with length x width, and add them: here 16 + 12 = 28 cm^2.
Area by subtracting a missing piece
Sometimes a shape is easier to see as a large shape with a piece removed rather than as several pieces joined together. In that case, find the area of the whole large shape and subtract the area of the missing piece. For a 10 x 6 rectangle with a 4 x 3 corner taken out, the whole is 60 and the cut-out is 12, so the area that remains is 60 - 12 = 48 cm^2. Adding pieces and subtracting a piece give the same answer, so you can choose whichever route uses the simpler shapes and the fewer steps.
Area by subtracting a piece
Find the whole, then subtract the cut-out piece.
for a shape with a piece removed, find the whole area and subtract the missing piece: 60 - 12 = 48 cm^2.
Perimeter is the outside edge only
The perimeter of a composite shape is the total length of its outside edges only. When you split a shape to work out its area, the internal line you draw is just a helper, not a real edge, so it is never included in the perimeter. Add only the lengths that run around the outside. A common step is finding a missing side from the others: because opposite sides of the overall span must match, the long bottom edge of the L equals the top edge plus the step, so 8 = 3 + 5. Tracing the outside of the L and adding gives 8 + 2 + 5 + 4 + 3 + 6 = 28 cm. Always state the unit, such as cm or m.
Perimeter is the outside edge only
Add the outside edges only; the split line does not count.
perimeter is the total length of the outside edges only; the internal split line is not part of it.
Estimating the area of an irregular shape
Not every shape has straight sides. To estimate the area of an irregular shape, place it on a grid of unit squares and count. First count the squares that are completely inside the shape, then look at the squares that the boundary passes through. A simple and reliable rule is to count each part square as roughly a half, or to pair up part squares so that two of them count as about one whole. The total is an estimate rather than an exact value, and a finer grid, with smaller squares, gives a closer estimate.
Estimating area on a grid
Count whole squares, then estimate the part squares.
estimate an irregular area by counting unit squares, taking full squares plus roughly half of the part squares.
Why this matters
Real surfaces are rarely simple rectangles, so the practical skills are decomposing a shape, adding or subtracting the pieces, and estimating when the edges are not straight. These are the calculations behind laying flooring, painting a wall, fencing a yard, and measuring land. Two pitfalls cause most of the errors: mixing up area, which is measured in square units, with perimeter, which is measured in units of length; and accidentally counting an internal split line as part of the perimeter. Keeping those two ideas clear is the main aim of this unit. It stays with straight-sided polygons and grid estimates; shapes with curved edges, such as circles, come in a later unit.
Quick self-check
1. An L-shape splits into a 8 x 2 rectangle and a 3 x 4 rectangle. What is its total area?
2. A 10 x 6 rectangle has a 4 x 3 rectangle cut out. What area remains?
3. When finding the perimeter of a composite shape, which lengths do you add?
4. A triangle has base 6 cm and height 4 cm. What is its area?
5. An irregular shape on a 1 cm grid covers 12 full squares and about 8 part squares. A good area estimate is...