Paths & Borders
A path or border is a strip of uniform width w running around or inside a figure, or roads crossing a field. The outer-minus-inner principle handles all of it, but two formulas save real time in the exam. An outer path of width w around a rectangle l × b has area (l + 2w)(b + 2w) − lb; the dimensions grow by 2w on each axis because the path is added to both sides. An inner border (a margin inside the figure) shrinks the inner rectangle to (l − 2w)(b − 2w), so its area is lb minus that. For a circular ring or running track of inner radius r and width w, the area is π[(r + w)² − r²] = πw(2r + w). The crossing-roads case is the famous trap: two roads of width w, one along the length and one along the breadth, have combined area lw + bw − w² — you subtract w² because the square where they intersect would otherwise be counted twice. Always decide first whether the strip is added outside or carved inside; that sign error is the single biggest score-killer here.
✅ Solved examples
✏️ Practice — try these, take hints as needed
📝 Topic test — 8 questions
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Formula Reference Sheet
Base area formulas
| Rectangle | A = length × breadth |
|---|---|
| Triangle | A = ½ × base × height |
| Circle / ring | A = πr²; ring = π(R² − r²) |
| Trapezium | A = ½ × (sum of parallel sides) × height |
| Sector of a circle | A = (θ/360) × πr² |
Composite & path tools
| Shaded region | A(shaded) = A(outer) − A(inner) |
|---|---|
| Outer border, width w (rectangle l×b) | A = (l + 2w)(b + 2w) − lb |
| Inner border, width w | A = lb − (l − 2w)(b − 2w) |
| Two crossing roads, width w (field l×b) | A = lw + bw − w² |
| Circular ring / track, width w | A = π[(r + w)² − r²] = πw(2r + w) |