Use ratios and scale drawings to interpret maps and plans, and use the trapezoidal rule to estimate the area of an irregular region

What this dot point is asking

NESA wants you to read scale notation, convert measurements between a drawing or map and reality, scale areas using the squared factor, and apply the trapezoidal rule to estimate the area of an irregular region from a set of equally-spaced offsets.

Two separate skills sit under one dot point. Scale drawings test whether you can move between paper and the real world without dropping a unit or forgetting that area grows by the square of the length factor. The trapezoidal rule tests whether you can apply one formula carefully: count the strips, double the right offsets, and keep the units. Neither idea is hard. Both reward neat, labelled work that carries its units, and that is exactly where the marks (and the lost marks) are.

The answer

Ratios and scale

A scale of $1:n$ means $1$ unit on the drawing represents $n$ units in reality, in the same unit.

• $1:100$. $1$ cm on the plan = $100$ cm = $1$ m in reality.
• $1:50000$. $1$ cm on the map = $50000$ cm = $500$ m in reality.

The scale itself is a pure ratio, so it has no units, but each side of the conversion must use the same unit. The safest habit has two steps. First scale the plan measurement, staying in the plan's unit (here centimetres). Then convert that result to a sensible real-world unit (metres or kilometres) at the very end. Mixing the two steps is where errors creep in.

Linear scaling

Real distance = drawing distance $\times$ scale factor.

A plan measurement of $7$ cm at scale $1:200$ represents $7 \times 200 = 1400$ cm = $14$ m in reality.

Area scaling

When you scale every linear dimension by a factor $k$, area scales by $k^2$.

A plan area of $50$ cm$^2$ at scale $1:100$ represents $50 \times 100^2 = 500000$ cm$^2$ = $50$ m$^2$ in reality.

The reason is that area is length times length. If each length grows $k$ times, the product grows $k \times k = k^2$ times. This is the most common scale-drawing trap. Doubling every side of a room does not double its floor area, it quadruples it; tripling them multiplies the area by nine. The figure above shows this directly. The plan rectangle and the real rectangle have the same shape, but the real one covers far more than $100$ times the area of the plan, because $100^2 = 10000$.

Volume scaling

When you scale linear dimensions by $k$, volume scales by $k^3$, since volume is length cubed.

A scale model with volume $100$ cm$^3$ at scale $1:50$ represents $100 \times 50^3 = 12500000$ cm$^3$ = $12.5$ m$^3$ in reality.

Reading maps

Australian topographic maps commonly use $1:25000$ or $1:50000$. A $4$ cm distance on a $1:25000$ map is $4 \times 25000 = 100000$ cm = $1$ km.

A clean route to kilometres straight from a map distance in centimetres: ground distance (km) = map distance (cm) $\times$ scale denominator $\div 100000$ (because $100000$ cm = $1$ km). On a $1:50000$ map, a $6.4$ cm separation is $6.4 \times 50000 \div 100000 = 3.2$ km.

The trapezoidal rule

The trapezoidal rule estimates the area of a region with one straight edge (the baseline) and one irregular curved edge, by slicing it into vertical strips and treating each strip as a trapezium. It is the standard HSC tool for the area of a paddock, a lake, a block of land or a cross-section, where there is no neat formula.

For a single strip (baseline length $h$, end offsets $a$ and $b$):

For multiple equal strips of width $h$ with offsets $y_0, y_1, \ldots, y_n$:

The end offsets ($y_0$ and $y_n$) are counted once; every interior offset is counted twice. That doubling is the rule's whole personality, and forgetting it (or doubling the ends as well) is the classic mistake.

Why the rule looks the way it does

It helps to see the formula assembled from individual trapezia rather than memorised. Each strip is a trapezium whose two parallel sides are the offsets at its left and right, $y_{i}$ and $y_{i+1}$, and whose width is $h$, so its area is $\frac{h}{2}(y_i + y_{i+1})$. Add the strips:

Every interior offset appears in two adjacent trapezia (it is the right side of one strip and the left side of the next), so it is added twice; the two end offsets appear in just one strip each. Collecting like terms gives exactly $\frac{h}{2}(y_0 + 2y_1 + 2y_2 + \cdots + y_n)$. The "double the middle ones" rule is not arbitrary; it is just bookkeeping for shared edges.

Watch the rule build up, strip by strip

Take a lake measured against a $120$ m baseline, with offsets $0, 25, 32, 20, 0$ m at $0, 30, 60, 90, 120$ m. Here is how the estimate is assembled.

Stage 1, identify the irregular region. The lake has one straight side (the baseline you measure along) and one curved side (the shoreline). There is no exact area formula for this shape, which is exactly when the trapezoidal rule earns its place.

Stage 2, measure equally spaced offsets. Divide the baseline into equal strips and measure the perpendicular distance (the offset) from the baseline up to the curve at each division. Here five offsets, $30$ m apart, give heights $0, 25, 32, 20, 0$ m. The strip width is $h = 30$ m, and there are four strips.

Stage 3, replace each curved top with a straight chord. Join the tops of adjacent offsets with a straight line. Each strip becomes a trapezium with parallel sides $y_i$ and $y_{i+1}$ and width $h$. The dashed accent line is the approximation; it follows the real curve (shown muted) closely, and more closely still as the strips get narrower.

Stage 4, add the trapezia. Sum the four trapezium areas. Using the collected formula, the interior offsets ($25$, $32$, $20$) are doubled and the ends ($0$, $0$) counted once:

When the trapezoidal rule applies, and how accurate it is

Use it when the region has one straight side (where you lay the baseline) and one irregular side (where you take the offsets at right angles to the baseline). The rule is exact when the boundary is itself a straight line, and only an estimate when the boundary curves. The estimate improves as the strip width $h$ shrinks, because shorter straight tops hug the curve more tightly. Whether you end up over or under the true area depends on the curve. Where the boundary bulges away from the baseline (like the top of the lake), the straight top cuts the corner, so the rule reads a little low. Where the boundary dips towards the baseline, the rule reads a little high.

How exam questions ask about this dot point

• "Find the real length / area / dimensions ..." from a plan or map at a stated scale. Multiply lengths by the scale factor; for an area, multiply by the factor squared (or convert both lengths first, then multiply).
• "A model / plan is drawn at $1:n$ ..." Set up the conversion in the plan's unit, then convert to metres or kilometres at the end.
• "Use the trapezoidal rule with ... strips / the offsets in the table ..." Read off $h$ (the gap between offsets, not the whole baseline), apply the formula, double the interior offsets only, and state units.
• "One application of the trapezoidal rule ..." means a single strip: $A \approx \frac{h}{2}(a + b)$, where $h$ is the whole baseline and $a$, $b$ are the two end offsets.
• "Estimate the area of the paddock / lake / block ..." with offsets given: the trapezoidal rule, almost always with several strips.