Lesson Notes By Weeks and Term v3 - Junior Secondary 3
Area of plane figures
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Area of plane figures
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Subject: General Mathematics
Class: Junior Secondary 3
Term: 3rd Term
Week: 7
Theme: Mensuration And Geometry
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Find are as of triangles; Find area of a parallelograms Find are as of trapezium; Find area of circles and sectors Solve word problems in volving are as Solve quantitative aptitude problems on are as.
A triangle is a polygon with three edges and three vertices. The area of a triangle is half the product of its base and its perpendicular height.
Formula: Area (A) = 1⁄2 × base (b) × height (h) Or, A = (b × h) / 2 Explanation: The 'base' can be any side of the triangle. The 'height' (or altitude) is the perpendicular distance from the opposite vertex to that base. It's crucial that the height is perpendicular to the chosen base.
Worked Example 1 (Triangle): A triangular cassava farm plot in Enugu has a base length of 20 metres and a perpendicular height of 15 metres. Calculate the area of the farm plot.
Solution: Identify the given values: Base (b) = 20 m Height (h) = 15 m State the formula: A = 1⁄2 × b × h Substitute the values into the formula: A = 1⁄2 × 20 m × 15 m Calculate the area: A = 10 m × 15 m A = 150 m2 Therefore, the area of the cassava farm plot is 150 square metres. A parallelogram is a quadrilateral with two pairs of parallel sides. Opposite sides are equal in length, and opposite angles are equal. The area of a parallelogram is the product of its base and its perpendicular height.
Formula: Area (A) = base (b) × height (h)
Explanation: Similar to a triangle, the 'base' can be any side of the parallelogram. The 'height' is the perpendicular distance between the chosen base and its opposite parallel side. Visually, a parallelogram can be transformed into a rectangle by cutting off a right-angled triangle from one end and attaching it to the other end.
Worked Example 2 (Parallelogram): A tailor wants to cut a piece of Ankara fabric shaped like a parallelogram for a design. The fabric piece has a base of 30 cm and a perpendicular height of 18 cm. Find the area of the fabric piece.
Solution: Identify the given values: Base (b) = 30 cm Height (h) = 18 cm State the formula: A = b × h Substitute the values into the formula: A = 30 cm × 18 cm Calculate the area: A = 540 cm2 The area of the Ankara fabric piece is 540 square centimetres. A trapezium (or trapezoid) is a quadrilateral with at least one pair of parallel sides.
Formula: Area (A) = 1⁄2 × (sum of parallel sides) × height Or, A = 1⁄2 × (a + b) × h Where 'a' and 'b' are the lengths of the two parallel sides, and 'h' is the perpendicular distance (height) between these parallel sides.
Explanation: The height 'h' must be perpendicular to both parallel sides. The formula can be conceptualized by imagining two identical trapeziums forming a parallelogram, or by dividing the trapezium into a rectangle and two triangles.
Worked Example 3 (Trapezium): A proposed school garden in Abuja has the shape of a trapezium. Its parallel sides measure 25 metres and 35 metres, respectively. The perpendicular distance between these parallel sides is 12 metres. Calculate the area of the garden.
Solution: Identify the given values: Parallel side a = 25 m Parallel side b = 35 m Height (h) = 12 m State the formula: A = 1⁄2 × (a + b) × h Substitute the values into the formula: A = 1⁄2 × (25 m + 35 m) × 12 m A = 1⁄2 × (60 m) × 12 m Calculate the area: A = 30 m × 12 m A = 360 m2 The area of the school garden is 360 square metres. A circle is a set of all points in a plane that are at a fixed distance (radius) from a fixed point (centre).
Formula: Area (A) = π × radius2 (r2) Or, A = πr2 Where 'π' (pi) is a mathematical constant approximately equal to 3.142 or 22/7, and 'r' is the radius of the circle.
Explanation: The radius (r) is the distance from the centre of the circle to any point on its circumference. The diameter (d) is twice the radius (d = 2r). If the diameter is given, divide it by 2 to find the radius before applying the formula.
Worked Example 4 (Circle): A circular fish pond is to be dug in a compound. The pond has a radius of 3.5 metres. Using π = 22/7, calculate the area of the land it will occupy.
Solution: Identify the given values: Radius (r) = 3.5 m π = 22/7 State the formula: A = πr2 Substitute the values into the formula: A = (22/7) × (3.5 m)2 A = (22/7) × (3.5 × 3.5) m2 A = (22/7) × 12.25 m2 Calculate the area: A = 22 × (12.25 / 7) m2 A = 22 × 1.75 m2 A = 38.5 m2 The area of the circular fish pond is 38.5 square metres.
Understanding area has significant practical relevance in various aspects of Nigerian life and vocations.
Construction and Real Estate Development: Application: Architects and builders frequently calculate the area of floor plans for various rooms (bedrooms, living rooms, kitchens) to determine the amount of flooring materials (tiles, carpet, wood), paint for walls, and roofing sheets needed. Land surveyors calculate the area of plots for sale or allocation, which directly impacts the value and documentation of properties.
Nigerian Context: When building houses in rapidly growing cities like Lagos, Abuja, or Port Harcourt, accurate area calculations are vital for budgeting construction costs, obtaining permits, and ensuring fairness in land transactions. A farmer planning to build a new barn or poultry house will also need these skills.
Agriculture and Land Management: Application: Farmers need to calculate the area of their farm plots to determine the appropriate quantities of seeds, fertilizers, and pesticides required. This helps in optimizing yields and managing costs effectively.
Nigerian Context: A rice farmer in Kebbi State needs to know the exact area of his paddy field to calculate how much fertilizer to buy. Similarly, a cocoa farmer in Ondo State might calculate the area covered by his trees to estimate potential yield or pest control needs. Tailoring, Crafts, and Design: Application: Tailors and fashion designers calculate the area of fabric needed for different garment sections to minimize waste and estimate material costs. Craftspeople designing patterns for Ankara fabrics, making traditional mats, or creating local artworks also use area concepts for proportion and material planning.
Nigerian Context: A seamstress in Onitsha Market cutting out pieces for a traditional agbada or a contemporary dress must accurately calculate the area of fabric needed for each part to avoid shortages or excessive waste, which directly impacts profitability. Adire and tie-dye artists consider the surface area of fabric when planning their designs and dye quantities.