Area
Interactive curriculum lessons, worked examples, and geometric problem-solving techniques designed to help Kenyan students master CBC, KPSEA, KCSE, and IGCSE mathematics.
First Principles
Objective: Understand and calculate the area of rectangles, squares, and simple composite shapes using grid counting and multiplication formulas.
Core Principles of Area
1. What is Area? Area measures the total 2-dimensional surface covered by a flat shape. It is always measured in square units (such as \(\text{cm}^2\), \(\text{m}^2\), or \(\text{km}^2\)).
2. From Counting Tiles to Multiplication:
- Imagine paving a classroom veranda in Nakuru measuring \(5\text{ m}\) long and \(3\text{ m}\) wide with \(1\text{ m} \times 1\text{ m}\) square tiles.
- Along the length, \(5\) tiles fit in one row.
- Along the width, there are \(3\) identical rows.
- Instead of counting \(1, 2, 3, \dots, 15\) tiles one by one, we simply multiply: \[5 \times 3 = 15\text{ square metres } (\text{m}^2)\]
Key Formulas
Where \(l\) is the length and \(w\) is the width.
Where \(s\) is the side length (since all sides of a square are equal).
Split the composite shape into non-overlapping standard rectangles, find each area, and add them together.
Calculate the total outer area and subtract the area of the unshaded cut-out or internal feature.
Worked Examples
A tailor in Gikomba market makes a rectangular table mat that measures \(80\text{ cm}\) in length and \(45\text{ cm}\) in width. Find its area in square centimetres (\(\text{cm}^2\)).
- Identify given values: Length \(l = 80\text{ cm}\), Width \(w = 45\text{ cm}\).
- Apply the formula: \(A = l \times w\).
- Compute: \[A = 80 \times 45 = 3{,}600\text{ cm}^2\]
- Final Answer: \(3{,}600\text{ cm}^2\).
A school garden in Machakos is L-shaped. It can be split into two rectangles: Region A measuring \(12\text{ m} \times 5\text{ m}\) and Region B measuring \(8\text{ m} \times 4\text{ m}\). Calculate the total area of the garden.
- Area of Region A: \[A_1 = 12 \times 5 = 60\text{ m}^2\]
- Area of Region B: \[A_2 = 8 \times 4 = 32\text{ m}^2\]
- Total Area: Add both regions: \[A_{\text{total}} = A_1 + A_2 = 60 + 32 = 92\text{ m}^2\]
- Final Answer: \(92\text{ m}^2\).
A rectangular courtyard in Mombasa measures \(18\text{ m}\) long and \(12\text{ m}\) wide. In the centre, there is a square tiled fountain with a side length of \(4\text{ m}\). The remaining ground is covered in grass. Find the area covered by grass.
- Area of entire courtyard: \[A_{\text{outer}} = 18 \times 12 = 216\text{ m}^2\]
- Area of square fountain: \[A_{\text{inner}} = 4 \times 4 = 16\text{ m}^2\]
- Remaining grass area: Subtract inner area from outer area: \[A_{\text{grass}} = 216 - 16 = 200\text{ m}^2\]
- Final Answer: \(200\text{ m}^2\).
Common Mistakes
Incorrect Action: When asked for the area of a \(6\text{ m} \times 4\text{ m}\) room, a learner calculates \(6 + 4 + 6 + 4 = 20\text{ m}^2\).
Why it feels right: The learner remembers that finding dimensions involves "all the sides" and uses addition.
The Correction: Adding all outer sides gives the Perimeter (the total distance around the boundary, \(20\text{ m}\)). Area measures the flat surface inside by multiplying the two perpendicular dimensions: \(6 \times 4 = 24\text{ m}^2\).
Incorrect Action: Stating the area of a \(5\text{ m} \times 3\text{ m}\) room as \(15\text{ m}\) instead of \(15\text{ m}^2\).
The Correction: Multiplying metres by metres gives \(\text{m} \times \text{m} = \text{m}^2\) (square metres). Area is 2-dimensional, so its units must always have the exponent \(^2\).
Incorrect Action: Splitting an L-shape into two rectangles but including the corner overlap twice.
The Correction: Make a single, clean cut across the shape so the sub-rectangles touch edge-to-edge without overlapping.
Real World
1. Tiling Floors & Verandas in Kenya
A builder in Eldoret is tiling a living room floor measuring \(6\text{ m} \times 4\text{ m}\). The total area is \(24\text{ m}^2\). If each box of tiles covers \(2\text{ m}^2\), the builder needs \(24 \div 2 = 12\) boxes of tiles.
2. Agricultural Planning & Shamba Farming
Farmers in Kitale calculate shamba areas to determine seed and fertiliser quantities. If \(1\text{ kg}\) of maize seed plants \(200\text{ m}^2\), knowing the rectangular dimensions of a plot allows the farmer to buy exact quantities and avoid wastage.
3. School Whiteboards and Noticeboards
School administrators calculate whiteboard surface areas to ensure adequate teaching display space per classroom according to Ministry of Education guidelines.
Practice