Perimeter and Area of Composite Shapes Using the Trapezoidal Rule

Good to see you back. In the previous lesson, you separated a finance problem into parts before calculating: ordinary pay, overtime, commission, and so on. Measurement problems reward the same habit. A complicated-looking diagram becomes manageable when you identify its boundaries, split it into familiar shapes, and keep units precise.

This lesson covers three related tasks: finding a perimeter by tracing an outside boundary, finding an exact area of a composite shape by adding or subtracting simpler areas, and estimating an irregular area with the trapezoidal rule. These are common practical-exam contexts: fencing, flooring, land surveying, and material quantities.

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Perimeter: trace the outside only

The perimeter is the total distance around the outside edge of a shape. Its units are linear, such as $\text{cm}$, $\text{m}$, or $\text{km}$.

For an irregular block of land, a curved boundary may already have a stated length. You add that length just as you would a straight side; you do not need a special formula unless the curve’s length is unknown.

For the paddock perimeter diagram, the interior $220\text{ m}$ labels divide the base for later area work; they are not extra outside edges. Trace the boundary once:

$$\begin{array}{rl}P& =210+440+320+530\\ & =1500\text{ m}\end{array}$$

So $1500\text{ m}$ of fencing would be required.

Perimeter errors to avoid

• Counting an internal line: A line used to divide a shape into sections is not part of its perimeter.
• Ignoring a labelled curve: If a curved edge is given as $530\text{ m}$, include $530\text{ m}$.
• Using square units: Perimeter is distance, so write $\text{m}$, not ${\text{m}}^2$.
• Missing a short edge at an indentation: Trace systematically around the shape rather than adding only the “main” lengths.

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Composite areas: turn one shape into several familiar ones

A composite shape is made from two or more simpler shapes. Its area is exact when all straight dimensions are known.

The central decision is whether to:

1. Split and add areas of pieces that make up the shape, or
2. Enclose and subtract a missing section from a larger, simpler shape.

Use whichever method produces the least working and the fewest missing lengths.

The main formulas are:

$$A_{\text{rectangle}}=lw$$ $$A_{\text{triangle}}=\frac{1}{2}bh$$ $$A_{\text{trapezium}}=\frac{h}{2}(a+b)$$

In the trapezium formula, $a$ and $b$ are the parallel sides, and $h$ is their perpendicular separation. A slanted side is not automatically the height.

Worked example: subtracting a cut-out

A patio is formed from a $12\text{ m}$ by $9\text{ m}$ rectangle, with a $5\text{ m}$ by $4\text{ m}$ rectangular garden bed removed from one corner.

The whole rectangle has area:

$$A_{\text{whole}}=12\times 9=108{\text{ m}}^2$$

The garden bed has area:

$$A_{\text{cut out}}=5\times 4=20{\text{ m}}^2$$

Therefore, the patio area is:

$$\begin{array}{rl}{A}_{\text{patio}}& =108-20\\ & =88{\text{ m}}^2\end{array}$$

The word removed, cut out, hole, pond, or garden bed is a strong clue that subtraction is likely the efficient method.

Find missing lengths before areas

Composite diagrams often provide total lengths alongside partial lengths. Calculate the unknown segment first.

For example, if a $15\text{ m}$ total width is split into a known $6\text{ m}$ section and an unknown section, then:

$$\text{unknown width}=15-6=9\text{ m}$$

Write that deduction on the diagram before calculating areas. This reduces the chance of accidentally using a total dimension for a smaller component.

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The trapezoidal rule: estimating an irregular area

A composite shape has straight boundaries that allow an exact calculation. Land beside a creek, river, or curved coastline often does not. In that situation, we replace a curved boundary with straight segments, creating trapeziums whose total area is an estimate.

For one trapezium:

$$A\approx \frac{h}{2}(d_1+d_2)$$

where:

• $d_1$ is the first measured width;
• $d_2$ is the next measured width;
• $h$ is the distance between those measurements.

The approximate sign matters. The top boundary is curved in reality, but the calculation uses a straight line between two measured points.

Worked example: two trapeziums

In the paddock divided into trapeziums, each section has width:

$$h=220\text{ m}$$

For trapezium 1, the two measured widths are $210\text{ m}$ and $258\text{ m}$:

$$\begin{array}{rl}{A}_1& \approx \frac{220}{2}(210+258)\\ & =110(468)\\ & =51480{\text{ m}}^2\end{array}$$

For trapezium 2, the measured widths are $258\text{ m}$ and $320\text{ m}$:

$$\begin{array}{rl}{A}_2& \approx \frac{220}{2}(258+320)\\ & =110(578)\\ & =63580{\text{ m}}^2\end{array}$$

Add the two estimated areas:

$$\begin{array}{rl}{A}_{\text{total}}& \approx 51480+63580\\ & \approx 115060{\text{ m}}^2\end{array}$$

The estimated paddock area is:

$$\boxed{115060{\text{ m}}^2}$$

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Several trapeziums: why the middle measurements are doubled

Suppose measurements are equally spaced and recorded in order as:

$$d_1,d_2,d_3,\dots ,d_n$$

Each interior measurement belongs to the trapezium on its left and the trapezium on its right. That is why every middle measurement is counted twice.

The extended trapezoidal rule is:

$$A\approx \frac{h}{2}\left[d_1+2(d_2+d_3+\cdots +d_{n-1})+d_n\right]$$

Use this compact formula only when the spacings between measurements are equal.

Exam-safe setup

Before pressing calculator buttons:

1. Write the measurements in boundary order, from one end to the other.
2. Identify the common spacing $h$.
3. Circle the first and last measurements once.
4. Mark each interior measurement twice.
5. Keep brackets around the complete expression.
6. Give area in square units and use $\approx$, not $=$, for an estimate.

For instance, with $h=20\text{ m}$ and readings $12\text{ m}$, $15\text{ m}$, $19\text{ m}$, $18\text{ m}$, the setup is:

$$A\approx \frac{20}{2}\left[12+2(15+19)+18\right]$$

The key is not the arithmetic. It is ensuring that $12$ and $18$ appear once, while $15$ and $19$ appear twice.

Accuracy and reasonableness

More trapeziums generally give a better estimate because shorter straight segments usually follow a curved boundary more closely. However, the answer remains an approximation.

Do not assume the trapezoidal-rule estimate is always too large or always too small. Its relationship to the true area depends on the curve:

• if the straight approximation lies mostly above the boundary, the estimate is too large;
• if it lies mostly below the boundary, the estimate is too small;
• if the curve changes direction, some errors may offset each other.

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A quick decision guide

What the question gives you	Best approach
All outside boundary lengths	Add the outside lengths for perimeter
Straight-sided shape made from rectangles, triangles, or trapeziums	Split into components and add areas
Large outer shape with a missing inner section	Outer area minus inner area
Curved or irregular boundary, with widths measured at equal intervals	Trapezoidal rule
Material packs or whole containers	Calculate the required amount, then round up if only whole packs can be bought

Keep perimeter and area separate:

$$\text{perimeter uses linear units}$$ $$\text{area uses square units}$$

A fencing question is likely asking for perimeter. A turf, tile, paint, land, or floor question is likely asking for area.

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Key takeaways

For perimeter, trace and add every external edge exactly once. Ignore interior partition lines.

For composite area, choose the simplest valid decomposition: add separate pieces or subtract a cut-out from a larger shape. Find any missing side lengths before beginning area calculations.

For an irregular area, use the trapezoidal rule:

$$A\approx \frac{h}{2}(d_1+d_2)$$

For multiple, equally spaced intervals:

$$A\approx \frac{h}{2}\left[d_1+2(d_2+d_3+\cdots +d_{n-1})+d_n\right]$$

The first and last measurements occur once; every interior measurement occurs twice. Always state square units and show that the result is approximate.

Next, you will extend measurement into three dimensions: surface area, volume, mass, capacity, and the unit conversions that connect them.