Mensuration (2D)

The master rule: areas scale as the SQUARE

If every length of a figure is multiplied by $k$ (sides doubled, "similar figures", map scales, photo enlargements):

• lengths, perimeters, diagonals → $\times k$
• areas → $\times k^2$

Going backwards (area ratio → length ratio) needs the square root. Perimeter ratio $p:q$ → area ratio $p^2:q^2$; area ratio $a:b$ → perimeter ratio $\sqrt a : \sqrt b$. This single rule answers most of the subtopic.

Percentage side-changes

Side $+x\%$ → area $+\left(2x + \frac{x^2}{100}\right)\%$. Memorise the common pair: $+20\% \to +44\%$, $-10\% \to -19\%$. A negative change is a DECREASE in area, still computed as $k = (1 - \frac{x}{100})^2$: $0.9^2 = 0.81$ → $-19\%$.

Cost, paint, tiles, grass

Any "cost of carpeting/painting/fencing" question scales with the QUANTITY it prices:

• paint/carpet/grass → area → $k^2$.
• fencing/border → perimeter → $k$. If the side triples, an area-based cost becomes $9 \times$; a fence cost becomes $3 \times$.

Maps, models, photos

Map scale $1:n$ means lengths on the map are $\frac{1}{n}$ of reality, so areas are $\frac{1}{n^2}$. Actual area $=$ map area $\times n^2$ (watch the unit conversion: $1\ \text{m}^2 = 10^4\ \text{cm}^2$, $1\ \text{ha} = 10^4\ \text{m}^2$). Photo enlarged "to 250% of original" means $k = 2.5$ → area $\times 6.25$.

Same perimeter, different areas

Of all shapes with a given perimeter, the circle encloses the most area; among rectangles, the square wins. Wire-bending questions: a wire of length $L$ bent into a square gives side $\frac{L}{4}$, into a circle gives $r = \frac{L}{2\pi}$ — the circle's area is bigger (for $L = 44$: square $121$, circle $154$).

A worked chain and the two traps

Typical chain: perimeters $3:5$ → sides $3:5$ → areas $9:25$ → if the larger area is 200, the smaller is $200 \times \frac{9}{25} = 72$. Trap one: applying the linear ratio to areas (giving $120$ here) — the most common wrong option offered. Trap two: forgetting that a DECREASE also squares: $-10\%$ on the side is $0.9^2 = 0.81$, i.e. $-19\%$ (the recovery term $\frac{x^2}{100}$ stops it from being $-20\%$). And when volumes appear (boxes, tanks), the law is $k^3$ — that belongs to mensuration-3d, but a 2-D question offering a $k^3$-style answer is bait, not a route.

Quick revision

• Lengths $\times k$; areas $\times k^2$; reverse needs $\sqrt{\ }$.
• Side $+x\%$ → area $+\left(2x + \frac{x^2}{100}\right)\%$; $+20\% \to 44\%$, $-10\% \to -19\%$.
• Area costs (paint/carpet) scale $k^2$; fence costs scale $k$.
• Map $1:n$: areas $\times \frac{1}{n^2}$. Photo to 250% → area $\times 6.25$.
• Same perimeter: circle area > square area.