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Area

Interactive curriculum lessons, worked examples, and geometric problem-solving techniques designed to help Kenyan students master CBC, KPSEA, KCSE, and IGCSE mathematics.

Grade 09 Pathway: N/A

First Principles

Objective: Find the area of regular and irregular polygons by decomposition.

Concrete Scenario

Imagine a family shamba drawn on paper — its boundary zigzags, so no single textbook formula covers it. However, if you draw straight lines (diagonals) from one corner to the other vertices, the entire plot splits neatly into triangles. Because you already know how to find the area of each triangle, summing them yields the total area of the shamba. This powerful mathematical strategy is called decomposition.

Geometric Insight

Every polygon with \(n\) sides can be partitioned into exactly \(n - 2\) non-overlapping triangles by drawing diagonals from a single chosen vertex. For example, a pentagon (\(5\) sides) splits into \(3\) triangles, while a hexagon (\(6\) sides) splits into \(4\) triangles. The fundamental rule of decomposition is that the sub-shapes must tile the polygon completely without gaps and without overlaps.

Algebraic Rule

If a polygon is decomposed into non-overlapping regions \(T_1, T_2, \dots, T_k\), then:

\[ A_{\text{polygon}} = A_{T_1} + A_{T_2} + \cdots + A_{T_k} \]

For each triangle, \(A = \frac{1}{2} b h\), where \(b\) is the base and \(h\) is the perpendicular height. Experiment with the interactive shape splitter below to see live decomposition in action!

🔷 Shape Splitter — Triangulate & Sum Live

Drag any blue vertex to reshape. Red vertex = fan point. Dashed red lines = diagonals.

Key Formulas

\[A_{\text{triangle}} = \frac{1}{2} b h\] — Triangle area: half of the base multiplied by the perpendicular height.
\[A_{\text{rectangle}} = \ell \times w\] — Rectangle area: length multiplied by width.
\[A_{\text{square}} = s^2\] — Square area: side length squared.
\[A_{\text{parallelogram}} = b \times h\] — Parallelogram area: base multiplied by perpendicular height.
\[A_{\text{trapezium}} = \frac{(a + b)\,h}{2}\] — Trapezium area: half the sum of parallel sides times perpendicular distance.
\[A_{\text{polygon}} = A_1 + A_2 + \dots + A_k\] — Area by Decomposition: sum of all non-overlapping constituent shapes.
\[\text{Number of Triangles} = n - 2\] — An \(n\)-sided polygon can always be triangulated into \(n - 2\) triangles from one vertex.
\[1\text{ ha} = 10\,000\text{ m}^2 \quad \vert \quad 1\text{ km}^2 = 1\,000\,000\text{ m}^2\] — Common land measurement units.

Worked Examples

Problem 1 (Easy): A rectangular garden in Nakuru measures \(14\text{ m}\) long by \(8\text{ m}\) wide. A diagonal path divides the garden into two identical right-angled triangles. Use decomposition to verify the area of one triangle and the total area.
  1. Identify Dimensions: The rectangle has length \(b = 14\text{ m}\) and width \(h = 8\text{ m}\).
  2. Calculate Single Triangle Area: \[ A_{T} = \frac{1}{2} \times b \times h = \frac{1}{2} \times 14 \times 8 = 56\text{ m}^2 \]
  3. Sum the Two Triangles: \[ A_{\text{total}} = 56\text{ m}^2 + 56\text{ m}^2 = 112\text{ m}^2 \]
  4. Check with Standard Formula: \(A = \ell \times w = 14 \times 8 = 112\text{ m}^2\).

Answer: Total area is \(112\text{ m}^2\) (each triangle is \(56\text{ m}^2\)).

Problem 2 (Medium): An L-shaped community plot in Eldoret is decomposed into two non-overlapping rectangles. Rectangle A is \(18\text{ m}\) by \(10\text{ m}\), and Rectangle B is \(12\text{ m}\) by \(6\text{ m}\). Find the total area of the plot.
  1. Identify Sub-regions: The shape consists of \(\text{Rectangle } A\) and \(\text{Rectangle } B\).
  2. Area of Rectangle A: \[ A_A = 18\text{ m} \times 10\text{ m} = 180\text{ m}^2 \]
  3. Area of Rectangle B: \[ A_B = 12\text{ m} \times 6\text{ m} = 72\text{ m}^2 \]
  4. Sum Decomposed Areas: \[ A_{\text{total}} = A_A + A_B = 180 + 72 = 252\text{ m}^2 \]

Answer: \(252\text{ m}^2\)

Problem 3 (Hard): A surveyor maps an irregular pentagonal plot by measuring along a central baseline from vertex \(A\) to \(D\). The polygon is decomposed into three sections: a right triangle \(T_1\) with base \(10\text{ m}\) and height \(8\text{ m}\), a trapezium \(Tr\) with parallel offsets of \(8\text{ m}\) and \(14\text{ m}\) across a baseline distance of \(15\text{ m}\), and a right triangle \(T_2\) with base \(12\text{ m}\) and height \(14\text{ m}\). Calculate the total plot area.
  1. Area of Triangle 1: \[ A_{T1} = \frac{1}{2} \times 10 \times 8 = 40\text{ m}^2 \]
  2. Area of Trapezium: \[ A_{Tr} = \frac{8 + 14}{2} \times 15 = 11 \times 15 = 165\text{ m}^2 \]
  3. Area of Triangle 2: \[ A_{T2} = \frac{1}{2} \times 12 \times 14 = 84\text{ m}^2 \]
  4. Sum all Decomposed Parts: \[ A_{\text{total}} = 40 + 165 + 84 = 289\text{ m}^2 \]

Answer: \(289\text{ m}^2\)

Common Mistakes

Mistake Splitting a shape into overlapping triangles and adding all individual areas together.
Correction Decomposition requires that shapes tile the polygon with zero overlap and no gaps. If regions overlap, the shared area is counted twice, leading to an artificially inflated result.
Why it feels right Each drawn triangle looks like a valid geometric shape, tempting the student to compute and add every triangle visible on the diagram.
Mistake Using a slant side length instead of the perpendicular height when applying \(A = \frac{1}{2}bh\).
Correction Height must always be measured at a \(90^\circ\) right angle to the chosen base line.
Why it feels right Slanted boundary sides are easy to measure directly on a perimeter fence, but they are longer than the true perpendicular altitude.
Mistake Multiplying side lengths of an irregular polygon directly like a rectangle.
Correction Only rectangles have \(A = \ell \times w\). Irregular shapes must be partitioned into known standard shapes (triangles, trapeziums, rectangles) before calculating area.

Real World

Shamba Boundary Surveying: Land boundaries in Kenyan counties such as Kiambu or Kakamega often follow natural features like rivers and paths, creating irregular polygons. Surveyors set a straight baseline and decompose the field into offset triangles and trapeziums (cross-staff surveying) to compute the true acreage.
Roof Construction & Iron Sheets: Builders constructing hip or multi-pitched roofs decompose irregular roof planes into triangles and trapeziums to calculate exactly how many corrugated iron sheets (mabati) to purchase from the hardware store.
Tile and Paving Estimation: Paving an irregular compound or courtyard requires decomposing the space into rectangles and triangles so contractors can accurately estimate paving blocks and sand without costly material waste.
Agricultural Seed & Fertiliser Application: Agricultural extension officers advise farmers to calculate their exact shamba area in square metres (and convert to hectares: \(1\text{ ha} = 10\,000\text{ m}^2\)) so that fertilizer (DAP/CAN) and maize seeds are applied at optimal density.

Practice

A rectangular maize plot measures 16 m in length and 10 m in width. A farmer divides it along a diagonal into two equal right-angled triangles. What is the area of one of these triangles in square metres? (Type only the number, e.g., 42)
Review the concepts above.
A pentagonal field is decomposed into 3 non-overlapping triangles from a single corner. The areas of the three triangles are 45 m², 65 m², and 30 m². What is the total area of the field in square metres? (Type only the number, e.g., 42)
Review the concepts above.
An L-shaped clinic compound is split into two non-overlapping rectangles. Rectangle 1 is 12 m long and 5 m wide. Rectangle 2 is 8 m long and 4 m wide. What is the total area of the compound in square metres? (Type only the number, e.g., 42)
Review the concepts above.
A plot shaped like a trapezium is decomposed into a rectangle of length 10 m and width 6 m, and an attached right-angled triangle with base 4 m and height 6 m. What is the total area of the plot in square metres? (Type only the number, e.g., 42)
Review the concepts above.
An irregular hexagonal plot of total area 250 m² is triangulated from one vertex into 4 non-overlapping triangles. Three of the triangles have areas of 55 m², 68 m², and 72 m². What is the area of the fourth triangle in square metres? (Type only the number, e.g., 42)
Review the concepts above.
A surveyor divides an irregular plot into two triangles and one trapezium along a baseline. Triangle A has base 8 m and height 6 m. Triangle B has base 10 m and height 8 m. The trapezium between them has parallel sides of 6 m and 8 m, with a height of 12 m. What is the total area of the plot in square metres? (Type only the number, e.g., 42)
Review the concepts above.