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Learning Resources

Area

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

Grade 07 Pathway: N/A

First Principles

Objective: Understand and calculate the area of 2D plane figures including triangles, parallelograms, and composite figures by decomposing them into fundamental units.

(a) Concrete Scenario: Imagine laying out square grass sods (each \(1\text{ m} \times 1\text{ m} = 1\text{ m}^2\)) on a shamba in Nakuru. The total area is simply the count of unit squares needed to completely cover the ground without gaps or overlaps.

(b) Geometric Insight: Every 2-dimensional area is based on the product of two perpendicular dimensions (base and height). When a shape is slanted—like a parallelogram—we can slice off a triangle from one side and slide it to the other to form an equivalent rectangle of the same base and perpendicular height: \[\text{Area} = b \times h\] Similarly, any triangle is exactly half of a parallelogram with the same base and height: \[\text{Area} = \frac{1}{2} \times b \times h\]

(c) Composite Shapes Principle: Any irregular polygon can be partitioned (split) into non-overlapping fundamental shapes (rectangles, triangles, parallelograms), or calculated by enclosing it in a larger boundary and subtracting unshaded regions.

Interactive Composite Explorer: House Gable & Wall

Adjust the height of the triangular roof and the rectangular base wall to see how composite areas combine dynamically.




Base width (\(b\)): 8 m
Wall Area: 48.0 m²
Roof Area: 16.0 m²
Total Area: 64.0 m²

Shape Splitter

Drag the height slider to split the composite shape into a rectangle and a triangle. Watch each piece's area update live into the total.


Rectangle area: 0 units²
Triangle area: 0 units²
Total: 0 units²

Key Formulas

Rectangle:

\[ A = l \times w \]

where \(l\) is length and \(w\) is width.

Triangle:

\[ A = \frac{1}{2} \times b \times h \]

where \(b\) is the base and \(h\) is the perpendicular height (height at \(90^\circ\) to the base).

Parallelogram:

\[ A = b \times h \]

where \(b\) is the base and \(h\) is the vertical/perpendicular height (never the slant side).

Trapezium (Trapezoid):

\[ A = \frac{1}{2}(a + b)h \]

where \(a\) and \(b\) are the lengths of the two parallel sides, and \(h\) is the perpendicular distance between them.

Composite Shape (Additive Method):

\[ A_{\text{total}} = A_1 + A_2 + A_3 + \dots \]

Divide the complex shape into non-overlapping standard shapes and sum their areas.

Worked Examples

Example 1 (Easy): Area of a Triangular Plot

Problem: A roadside tea kiosk in Kericho sits on a triangular piece of land with a base of \(14\text{ m}\) and a perpendicular height of \(9\text{ m}\). Calculate its area.

Step-by-Step Solution:

  1. Identify the given dimensions: Base \(b = 14\text{ m}\), perpendicular height \(h = 9\text{ m}\).
  2. Apply the triangle area formula: \[ A = \frac{1}{2} \times b \times h \]
  3. Substitute the values: \[ A = \frac{1}{2} \times 14 \times 9 = 7 \times 9 = 63\text{ m}^2 \]

Final Answer: \( 63\text{ m}^2 \)

Example 2 (Medium): Area of a Parallelogram Garden

Problem: A farmer in Eldoret has a field shaped like a parallelogram. The base of the field is \(25\text{ m}\), its slanted side is \(15\text{ m}\), and the perpendicular distance between the parallel bases is \(12\text{ m}\). Find the area of the field.

Step-by-Step Solution:

  1. Identify the relevant parameters: Base \(b = 25\text{ m}\), perpendicular height \(h = 12\text{ m}\). (Note: The slant length \(15\text{ m}\) is extra information and is not the height).
  2. Apply the parallelogram formula: \[ A = b \times h \]
  3. Calculate: \[ A = 25 \times 12 = 300\text{ m}^2 \]

Final Answer: \( 300\text{ m}^2 \)

Example 3 (Hard): Area of a Composite Storefront Wall

Problem: The side wall of a grain store consists of a rectangular base of width \(10\text{ m}\) and height \(4\text{ m}\), topped by a triangular gable with the same base \(10\text{ m}\) and a perpendicular height of \(3\text{ m}\). A rectangular ventilation window of \(2\text{ m} \times 1.5\text{ m}\) is cut into the wall. Find the total paintable surface area.

Step-by-Step Solution:

  1. Area of rectangular wall (\(A_1\)): \[ A_1 = l \times w = 10 \times 4 = 40\text{ m}^2 \]
  2. Area of triangular gable (\(A_2\)): \[ A_2 = \frac{1}{2} \times b \times h = \frac{1}{2} \times 10 \times 3 = 15\text{ m}^2 \]
  3. Gross wall area: \[ A_{\text{gross}} = A_1 + A_2 = 40 + 15 = 55\text{ m}^2 \]
  4. Area of window to subtract (\(A_{\text{window}}\)): \[ A_{\text{window}} = 2 \times 1.5 = 3\text{ m}^2 \]
  5. Net paintable area: \[ A_{\text{net}} = 55 - 3 = 52\text{ m}^2 \]

Final Answer: \( 52\text{ m}^2 \)

Common Mistakes

Mistake Using the slant side length as the height \(h\) in triangles or parallelograms.

Why it feels right The slanted edge is an easily visible outer boundary of the shape, making it tempting to measure along the edge.

Correction Height \(h\) must always be perpendicular (forming a \(90^\circ\) right angle) to the chosen base. The slant side is longer than the perpendicular height (hypotenuse in a right triangle).

Mistake Forgetting the factor of \(\frac{1}{2}\) when finding the area of a triangle.

Why it feels right Multiplying base by height directly (\(b \times h\)) gives the area of a rectangle or parallelogram, so learners forget that a triangle occupies only half of that space.

Correction Always write out the formula \[ A = \frac{1}{2} \times b \times h \] before substituting values.

Mistake Confusing perimeter with area when working with composite shapes.

Why it feels right When combining two shapes, learners sometimes add all the outer and inner boundary lengths together, confusing linear distance with 2D region coverage.

Correction Area measures internal 2D surface coverage in square units (\(\text{m}^2\)), calculated by summing independent component areas, whereas perimeter is the continuous distance around the outer boundary.

Real World

Land Demarcation (Shamba Surveying): Most agricultural plots in rural Kenya are not simple rectangles. Surveyors divide irregular plots into a series of right-angled triangles and trapeziums using a baseline, calculate each piece's area, and sum them to obtain the total title deed acreage.
Corrugated Iron Sheets (Mabati) for Roofing: Builders calculating roofing requirements for pitched roofs determine the exact triangular area of the gables and the rectangular area of the roof slopes to order the correct number of iron sheets, preventing costly material waste.
Paving & Tiling: Landscapers laying paving blocks in city plazas in Nairobi calculate the composite area of walkways and subtract circular flowerbeds to estimate paving stones and sand accurately.

Practice

A rectangular nursery seedbed in Kiambu is 15 metres long and 6 metres wide. What is its area in square metres? (Type only the number, e.g., 641)
Review the concepts above.
A triangular road sign along the Thika Superhighway has a base of 80 cm and a perpendicular height of 50 cm. Calculate its area in square centimetres. (Type only the number, e.g., 209)
Review the concepts above.
A flower garden is designed in the shape of a parallelogram with a base of 18 metres and a perpendicular height of 7 metres. What is the area of the garden in square metres? (Type only the number, e.g., 124)
Review the concepts above.
A grazing paddock is shaped as a trapezium with parallel sides of 14 metres and 26 metres, and a perpendicular distance between them of 10 metres. Find the area of the paddock in square metres. (Type only the number, e.g., 571)
Review the concepts above.
The gable end of a classroom consists of a rectangle 12 metres wide and 4 metres high, topped by a triangle of base 12 metres and perpendicular height 3 metres. What is the total composite area of the wall in square metres? (Type only the number, e.g., 217)
Review the concepts above.
An L-shaped compound is made from a large rectangle measuring 20 metres by 15 metres from which a rectangular corner section measuring 8 metres by 6 metres has been removed. What is the remaining area of the compound in square metres? (Type only the number, e.g., 580)
Review the concepts above.