10.02c, 10.03e - Composite perimeters and areas

The exposed outer boundary

Start with two rectangles measuring $12\,\mathrm{m}$ by $6\,\mathrm{m}$. Attach a semicircle along a complete $6\,\mathrm{m}$ end of one, and cut the same semicircle out of an end of the other. Which needs more fencing?

The perimeter is the total length round the shape's boundary. Trace that boundary with your finger. A joining line inside a shape is not exposed, so it does not need fencing. A cut-out can create a new exposed curved boundary.

[DIAGRAM: asset_name: Semicircle attached or removed; asset_slug: composite-exposed-boundaries; caption: Each outline contains the same three straight sides and one semicircular arc. The dashed diameter is not exposed.; description: Two twelve-metre by six-metre rectangles. On the left a semicircle is attached externally along the complete right-hand six-metre end. On the right an equal semicircle is cut inward from the same end. Dark outlines show the exposed boundary. A dashed line marks the six-metre diameter in each, and the twelve-metre length and six-metre height are labelled.]

For a semicircle of radius $r$, the curved part has length $\frac12\times2\pi r=\pi r$. The perimeter of a separate semicircle would also include its straight diameter. When it is joined to another region, decide whether that diameter remains exposed.

We can now calculate the boundary of each garden.

Change only whether the semicircle bulges out or cuts in. The exposed curved length stays the same.

Surprisingly, cutting material away has increased the perimeter from $36\,\mathrm{m}$, just as attaching the semicircle did. Area and perimeter need separate calculations. If a region contains an enclosed hole and a question asks for its total boundary length, include the hole's boundary too.

The next description supplies every dimension needed; you can trace the boundary mentally.

Splitting composite regions

To find area, cover the whole region with simpler shapes which do not overlap. Their areas add, even when the lines used to split them are not exposed boundaries.

For an L-shaped region, a line from the inward corner can split it into two rectangles. A sloping boundary may suggest a triangle or trapezium. A curved addition may be a semicircle or a sector. Choose pieces whose dimensions you know or can find by subtraction.

[DIAGRAM: asset_name: Two ways to split an L-shaped region; asset_slug: composite-l-shape-splits; caption: Both splits cover the same L-shape without gaps or overlap. Missing lengths come from subtraction.; description: Two copies of an L-shaped region formed from a ten-by-seven centimetre rectangle with a four-by-three centimetre upper-right corner removed. One copy is split vertically into a six-by-seven rectangle and a four-by-four rectangle. The other is split horizontally into a ten-by-four rectangle and a six-by-three rectangle. Overall and cut-out dimensions are labelled, with derived six and four lengths shown in dark text.]

In this L-shape, the missing lengths are $10-4=6\,\mathrm{cm}$ and $7-3=4\,\mathrm{cm}$. The vertical split gives $6\times7+4\times4=58\,\mathrm{cm}^2$. The horizontal split gives $10\times4+6\times3=58\,\mathrm{cm}^2$. Different valid splits agree because they cover exactly the same region.

The garden's joining line is useful for area: it separates the rectangle from the semicircle without leaving gaps or creating overlap.

[DIAGRAM: asset_name: Adding non-overlapping component areas; asset_slug: composite-added-areas; caption: The dashed joins split each region into a rectangle and one or two semicircles. Add every component area once.; description: Two shapes use a twelve-by-six rectangle shaded pale blue. The first has one externally attached semicircle of diameter six on the right. The second has equal semicircles at both six-unit ends. The semicircles are shaded pale sage. Shared diameters are dashed, and rectangle dimensions are labelled in metres.]