C5.5; E5.5 - Compound areas and perimeters

Decomposing and subtracting areas

Cut a rectangular corner from a sheet. Its area decreases, but must its perimeter decrease too? Keep that question in mind as you first work out how much material remains.

A compound shape is formed from simpler shapes. You can find its area by adding non-overlapping pieces that exactly fill it, or by subtracting missing pieces from a larger shape. The cuts used to split the shape are imaginary: they do not add or remove surface.

For an L-shape, a surrounding rectangle often makes subtraction shorter. Alternatively, draw a horizontal or vertical line to split the L into two rectangles. Find each missing length by comparing aligned distances: an outside width of $12\,\mathrm{cm}$ with a $4\,\mathrm{cm}$ corner removed leaves $8\,\mathrm{cm}$.

[DIAGRAM: asset_name: Two L-shaped areas; asset_slug: compound_area_subtraction; caption: A surrounding rectangle makes the missing corner easy to subtract.; description: Two right-angled L-shapes each have outer width 12 and height 8 centimetres and top-right removed width 4 centimetres. The first missing corner has depth 3 centimetres and the second depth 5 centimetres. Dashed edges complete the surrounding rectangles and dimension arrows identify both removed corners. The drawn lengths agree exactly with the labels.]

Change only the removed corner's depth from $3\,\mathrm{cm}$ to $5\,\mathrm{cm}$. The larger cut should leave a smaller area.

The additive checks work because their two rectangles meet along a line without overlapping surface.

Describe the surrounding rectangle and the missing part explicitly in your working.

Tracking the outside perimeter

For perimeter, trace the outside boundary continuously until you return to your starting point. A line used to split an area into pieces lies inside the shape, so it is not included.

If you add two rectangle perimeters when the rectangles share an edge, that common edge has been counted twice, once for each rectangle. Subtract twice its length to leave the outer perimeter. Tracing the outside directly usually avoids this correction.

A rectangular corner cut gives a useful surprise. It removes one horizontal and one vertical boundary length, then exposes equal horizontal and vertical lengths inside the old rectangle. The total perimeter is unchanged. A notch in the middle of an edge behaves differently: travelling in and out adds two extra depths.

[DIAGRAM: asset_name: Corner cut compared with an edge notch; asset_slug: compound_outside_boundary; caption: Moving the same cut-out changes the outside journey without changing the removed area.; description: A 12 by 8 centimetre rectangle with a 4 by 3 top-right corner removed is compared with the same rectangle with a 4 centimetre deep and 3 centimetre high notch in the middle of its right edge. For the notch, the right-edge segments above and below are 2 and 3 centimetres. All outside edges are solid, construction lines are dashed, and missing side lengths are labelled with subtraction expressions.]

Move only the same cut-out from the corner to the middle of the right edge. The area removed stays the same; the boundary route changes.