Paths and composite figures

A garden has a path running all around it. A plot has a crosspath cutting through. A puzzle piece has corners snipped off. None of these are clean rectangles, but we can still find their areas , by breaking them into pieces we already know how to handle.

Concept

The three moves. Almost any composite figure can be tackled with one of these tricks:

1. Add up. Split the figure into rectangles and triangles. Compute each piece's area and add.
2. Subtract. If the figure equals "big rectangle minus a hole," find the big area, find the hole's area, and subtract.
3. Rearrange. Move pieces around (without overlapping) into a friendlier shape, then use a familiar formula.

These three together are powerful enough for almost every plane-area problem you will meet.

Example: path around a rectangle. A rectangular garden $EFGH$ sits inside a larger rectangle $ABCD$; the path is the strip between them. If the garden is $L \times W$ and the path has uniform width $w$ on each side, then the outer rectangle is $(L + 2w) \times (W + 2w)$.

$\text{Area of path} = (L + 2w)(W + 2w) - L \cdot W.$

Expanding gives $2w(L + W) + 4w^2$.

Example: crosspath through a plot. A $14 \mathrm{m} \times 12 \mathrm{m}$ plot has two perpendicular paths, each of width $1$ m, running through it. The total path area is (horizontal strip) $+$ (vertical strip) $-$ (overlap square at the centre):

$1 \times 14 + 1 \times 12 - 1 \times 1 = 25 \mathrm{m}^2.$

The subtraction of the overlap is crucial , otherwise you would count the central square twice.

Path-translation trick. If the inner rectangle (the park) slides around within the outer rectangle (but stays fully inside), the area of the path stays exactly the same. Why? The outer area is fixed; the inner park's area is fixed. Their difference doesn't depend on where the park sits.

Spiral or zig-zag tubes. A spiral "tube" of uniform width can be straightened out (mentally) into one long straight strip. Its area equals the area of that straight strip. This rearrangement trick is great for problems where the figure looks scary.

Sketching matters. Always draw a clean diagram and label every length. Mark the right-angles. The arithmetic is usually easy; it is the picture that makes or breaks the problem.

Worked examples

Example 1. A garden is $20 \mathrm{m} \times 12 \mathrm{m}$. A path $2$ m wide runs all around it on the outside. Find the path's area.

• Outer rectangle $= (20+4) \times (12+4) = 24 \times 16 = 384 \mathrm{m}^2$.
• Garden $= 240 \mathrm{m}^2$.
• Path $= 384 - 240 = 144 \mathrm{m}^2$.

Example 2. A $14 \mathrm{m} \times 12 \mathrm{m}$ plot has a horizontal path $1$ m wide and a vertical path $1$ m wide crossing it. Find total path area.

• Horizontal strip: $14 \times 1 = 14$. Vertical strip: $12 \times 1 = 12$. Overlap: $1 \times 1 = 1$.
• Total $= 14 + 12 - 1 = 25 \mathrm{m}^2$.

Example 3. A rectangular region has a smaller rectangle cut out from one corner. The outer is $10 \times 6$, the cut-out is $3 \times 2$. Find the remaining area.

• Outer $= 60$; cut-out $= 6$. Remaining $= 54 \mathrm{cm}^2$.

Example 4. A square of side $20$ cm contains an L-shaped path of width $1$ cm running along two adjacent sides. Find the path's area.

• The L-path is the union of a $20 \times 1$ horizontal strip and a $1 \times 20$ vertical strip minus their $1 \times 1$ overlap at the corner.
• Area $= 20 + 20 - 1 = 39 \mathrm{cm}^2$.

Try it yourself

1. A garden is $25 \mathrm{m} \times 15 \mathrm{m}$. A $3$-m-wide path runs outside it on all sides. Path area?
2. A $10 \mathrm{m} \times 8 \mathrm{m}$ plot has a single $1$-m path through the middle, parallel to the longer side. Path area?
3. Outer rectangle $12 \times 8$, inner $10 \times 6$ centred inside. Strip area?
4. A square of side $30$ cm has a square hole of side $10$ cm cut out at the centre. Area of the frame?
5. A $20 \mathrm{m} \times 16 \mathrm{m}$ plot has two perpendicular $2$-m-wide paths crossing through. Total path area.
6. If a park sits anywhere inside a larger rectangle, does sliding the park change the path's area? Why?
7. A rectangle has area $100$, with a triangle of base $4$ and height $5$ cut out. Remaining area?
8. The walls of a hallway $5 \mathrm{m} \times 8 \mathrm{m}$ are to be painted to height $3$ m. Total wall area (no windows)?

Activity

Build a path. On grid paper, draw a $10 \times 6$ rectangle for your "park". Now draw a path $2$ squares wide around it (on all four sides), going outside the rectangle. Count the squares in the path region directly, then verify with the formula $(L + 2w)(W + 2w) - LW$. Now move the park to the bottom-left corner of a bigger rectangle, so the path lies on only two sides , and check whether the total area covered by path is the same as before.