NSWMaths Standard 2Syllabus dot point

How are ratios and scale drawings used to read maps and plans, and how does the trapezoidal rule estimate irregular areas?

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

A focused answer to the HSC Maths Standard 2 dot point on ratios, scale drawings and the trapezoidal rule. Reading scale notation, converting distances, the trapezoidal rule formula with one or more applications, and worked examples for floor plans, maps and irregular Australian land areas.

Generated by Claude OpusReviewed by Better Tuition Academy9 min answerUpdated 2026-05-20

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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.

The answer

Ratios and scale

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

IMATH_8 . $1$ cm on the plan = $100$ cm = $1$ m in reality.
IMATH_12 . $1$ cm on the map = $50000$ cm = $500$ m in reality.

Always state the units. The scale itself is unitless (it is a ratio), but applications need consistent units.

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 linear dimensions 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.

Trap: doubling all dimensions quadruples the area.

Volume scaling

When you scale linear dimensions by $k$ , volume scales by $k^3$ .

A plan volume (e.g. a 3D scale model) of $100$ cm $^3$ at scale $1:50$ represents $100 \times 50^3 = 12500000$ cm $^3$ = $12.5$ m $^3$ .

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.

To find a distance on the ground in kilometres from a map distance in cm: ground distance (km) = map distance (cm) $\times$ scale denominator $\div 100000$ .

The trapezoidal rule

Estimates the area of a region with one straight edge (the baseline) and one irregular curve (the offsets).

For a baseline of length $h$ with end offsets $a$ and $b$ :

A \approx \frac{h}{2}(a + b).

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

A \approx \frac{h}{2} \left( y_0 + 2(y_1 + y_2 + \cdots + y_{n-1}) + y_n \right).

The end offsets are counted once; the interior offsets are counted twice.

When the trapezoidal rule applies

Use when the area is bounded by one straight side (where you measure the baseline) and one irregular curve (where the offsets are taken perpendicular to the baseline).

The estimate is exact for trapezoidal shapes and approximate otherwise; accuracy improves as the strip width $h$ decreases.

Worked example

Single-strip trapezoidal rule

A creek has two offsets to a property boundary: $8$ m at one end and $14$ m at the other, separated by $50$ m of straight baseline.

$A \approx \frac{50}{2}(8 + 14) = 25 \times 22 = 550$ m $^2$ .

Multiple-strip

A lake shoreline is measured against a $120$ m baseline. Offsets at $0, 30, 60, 90, 120$ m are $0, 25, 32, 20, 0$ m. Estimate the lake area.

Strip width: $h = 30$ m, $4$ strips, $5$ offsets.

$A \approx \frac{30}{2}(0 + 2(25 + 32 + 20) + 0) = 15 (2 \times 77) = 15 \times 154 = 2310$ m $^2$ .

Floor plan (Australian context)

A house floor plan at scale $1:100$ shows a rectangular kitchen $5.4$ cm by $4.2$ cm.

Real length: $5.4 \times 100 = 540$ cm = $5.4$ m. Real width: $4.2 \times 100 = 420$ cm = $4.2$ m. Real area: $5.4 \times 4.2 = 22.68$ m $^2$ .

Map distance to ground distance

On a $1:50000$ map, two summits are separated by $6.4$ cm. Ground distance:

$6.4 \times 50000 = 320000$ cm = $3200$ m = $3.2$ km.

Past exam questions, worked

Real questions from past NESA papers on this dot point, with our answer explainer.

2023 HSC Q214 marksA floor plan is drawn at a scale of

1:50

. On the plan, a rectangular living room measures

12

cm by

9

cm. Find the real area of the living room in square metres.

Show worked answer →

Real length: $12 \times 50 = 600$ cm $= 6$ m. Real width: $9 \times 50 = 450$ cm $= 4.5$ m.

Real area: $6 \times 4.5 = 27$ m $^2$ .

Alternatively, plan area is $12 \times 9 = 108$ cm $^2$ ; real area is $108 \times 50^2 = 108 \times 2500 = 270000$ cm $^2$ $= 27$ m $^2$ .

Markers reward either route, with the squared scale factor for area shown explicitly if the second route is used.

2022 HSC Q254 marksUse the trapezoidal rule with five equal strips to estimate the area of a paddock. Offsets at

0, 20, 40, 60, 80, 100

m are

0, 12, 18, 22, 15, 0

m respectively.

Show worked answer →

Strip width: $h = 20$ m. There are $6$ offsets and $5$ strips.

Trapezoidal rule for multiple strips: $A \approx \frac{h}{2} (y_0 + 2(y_1 + y_2 + y_3 + y_4) + y_5)$ .

$A \approx \frac{20}{2}(0 + 2(12 + 18 + 22 + 15) + 0)$ .

$= 10 (0 + 2 \times 67 + 0) = 10 \times 134 = 1340$ m $^2$ .

Markers reward the formula, the inner sum and the doubling, and an answer with units.