Class 8 Maths Chapter 11 Area of Polygon - Mensuration

Mensuration: Areas of Trapezium, Rhombus, and General Polygons

Mensuration is the branch of mathematics that deals with the measurement of geometric figures — their areas, perimeters, surface areas, and volumes. This chapter in CBSE Class 8 Mathematics extends the students' knowledge beyond rectangles and squares to cover the areas of trapeziums, rhombuses, and general polygons using coordinate methods. These skills are applied in practical contexts such as land measurement, architectural design, and construction.

The area of a trapezium (a quadrilateral with one pair of parallel sides) is given by: Area = ½ × (sum of parallel sides) × perpendicular distance between them = ½(a + b) × h, where a and b are the lengths of the parallel sides and h is the height (the perpendicular distance between them). This formula works because a trapezium can be divided into two triangles or rearranged into a rectangle of width (a+b)/2 and height h. For example, a trapezium with parallel sides 8 cm and 12 cm, and height 5 cm, has an area of ½(8+12) × 5 = 50 cm². The area of a rhombus can be calculated in two ways. Since a rhombus is a parallelogram, its area equals base × height. However, a more convenient formula uses its diagonals: Area = ½ × d₁ × d₂, where d₁ and d₂ are the lengths of the two diagonals. This works because the diagonals of a rhombus bisect each other at right angles, dividing the rhombus into four congruent right-angled triangles, and the sum of their areas gives ½ × d₁ × d₂. For example, a rhombus with diagonals 10 cm and 8 cm has area = ½ × 10 × 8 = 40 cm².

General polygons (polygons with any number of sides) can be measured by dividing them into simpler shapes (triangles and rectangles) and summing their areas. The method of coordinates provides a powerful general technique. For any polygon with vertices (x₁, y₁), (x₂, y₂), ..., (xn, yn) listed in order (either clockwise or anticlockwise), the area is: Area = ½|x₁(y₂−yₙ) + x₂(y₃−y₁) + ... + xn(y₁−yₙ₋₁)|. This is called the Shoelace Formula. For example, a quadrilateral with vertices A(1,1), B(5,1), C(4,4), D(0,4) has area = ½|1(1−4) + 5(4−1) + 4(4−1) + 0(1−1)| = ½|−3 + 15 + 12 + 0| = ½ × 24 = 12 square units. A simpler approach for common polygons is to divide them into rectangles and triangles, calculate the area of each using basic formulas, and add them up. This composite area method is practical for irregular plots of land and architectural floor plans. For regular polygons (all sides and angles equal), if the side length is s and there are n sides, the area can be calculated using the formula: Area = (n × s²)/(4 × tan(π/n)). The chapter also covers the concept of perimeter (the total boundary length) for irregular shapes and the difference between area and perimeter — a shape can have the same area but different perimeters (e.g., a long thin rectangle and a square). In practical applications, surveyors use these methods to measure land areas for property records, legal disputes, and urban planning.

• Trapezium area = ½(a+b) × h, where a and b are parallel sides and h is the perpendicular distance between them.
• Rhombus area = ½ × d₁ × d₂, using its diagonals; also equal to base × height (as a parallelogram).
• General polygon area: divide into triangles/rectangles and sum, or use the Shoelace Formula with vertex coordinates.
• For irregular land measurement, the coordinate method is the most practical approach for surveyors.
• Area and perimeter are independent — the same area can have very different perimeters; irregular shapes require decomposition.