Pariksha SaathiLesson 7 of 8 · Mathematics, Classes VI-VIII (Paper II)
Mensuration: Perimeter, Area, Surface Area and Volume (Cube, Cuboid, Cylinder)
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Perimeter
The perimeter is the length of the boundary of a closed figure: 2(l + b) for a rectangle, 4a for a square, the sum of the sides for a triangle and n × side for a regular polygon of n sides. Fencing a field 120 m by 80 m needs 2(120 + 80) = 400 m of wire, which at ₹15 a metre costs ₹6,000. The perimeter of a circle, its circumference, is 2πr or πd, with π taken as 22/7 or 3.14. A wheel of radius 35 cm covers 2 × 22/7 × 35 = 220 cm in one turn, so it makes 1,10,000 ÷ 220 = 500 turns in 1.1 km. A semicircle of radius 7 cm has perimeter πr + 2r = 22 + 14 = 36 cm; leaving out the diameter is a common error.
Area of plane figures
Area is the surface enclosed by a figure, measured in square units: 1 m² = 10,000 cm² and 1 hectare = 10,000 m². A rectangle has area l × b and a square a²; a parallelogram base × height, since cutting a triangle from one end and fixing it at the other makes a rectangle; a triangle 1/2 × base × height, half the parallelogram on the same base and height; a rhombus 1/2 × product of the diagonals; a trapezium 1/2 × sum of the parallel sides × distance between them; and a circle πr². So a parallelogram of base 12 cm and height 7.5 cm has area 90 cm²; a trapezium with parallel sides of 10 m and 16 m that are 8 m apart has 104 m²; a rhombus with diagonals of 12 cm and 18 cm has 108 cm²; and a circle of radius 14 cm has 22/7 × 196 = 616 cm².
Paths, rings and scaling
The area of a path or border is the outer area minus the inner area. A path 2.5 m wide round the outside of a garden 40 m by 30 m makes an outer rectangle 45 m by 35 m, so the path covers 1,575 − 1,200 = 375 m². A ring between circles of radius 14 cm and 7 cm has area 22/7 × (196 − 49) = 462 cm². Area changes with the square of the scale: doubling the side of a square multiplies its area by 4, and tripling the radius of a circle multiplies its area by 9. Area and perimeter are not tied together: a 6 cm by 4 cm rectangle and a 5 cm square both have perimeter 20 cm but areas of 24 cm² and 25 cm².
Surface area
The surface area of a solid is the total area of its faces. A cuboid has total surface area 2(lb + bh + hl) and lateral surface area, the four walls, 2h(l + b); a cube has 6a² and 4a²; a cylinder has curved surface area 2πrh and total surface area 2πr(r + h). A cuboid 10 cm × 8 cm × 5 cm has surface area 2(80 + 40 + 50) = 340 cm². The four walls of a room 6 m long, 5 m wide and 4 m high have area 2 × 4 × (6 + 5) = 88 m², so painting the walls and ceiling covers 88 + 30 = 118 m². A closed cylinder of radius 7 cm and height 10 cm has curved surface area 2 × 22/7 × 7 × 10 = 440 cm² and total surface area 2 × 22/7 × 7 × 17 = 748 cm².
Volume and capacity
Volume is the space a solid occupies, in cubic units: l × b × h for a cuboid, a³ for a cube and πr²h for a cylinder. Capacity is measured in litres: 1 cm³ = 1 mL, 1,000 cm³ = 1 L and 1 m³ = 1,000 L. A tank 2 m × 1.5 m × 1 m holds 3 m³, or 3,000 L; a cylinder of radius 7 cm and height 10 cm holds 22/7 × 49 × 10 = 1,540 cm³, or 1.54 L. A cuboid 8 cm × 6 cm × 4 cm holds 4 × 3 × 2 = 24 cubes of side 2 cm, and a cuboid of volume 440 cm³ on a base of 88 cm² is 5 cm high. Doubling every edge of a cube multiplies its surface area by 4 and its volume by 8.
In the classroom
The usual errors are mixing up area and perimeter, combining unlike units (cm with m, or cm with cm²), forgetting to square the radius, and confusing surface area with volume. Before the formulas, pupils count unit squares on grid paper and build cuboids from unit cubes; they cut a circle into many equal sectors and rearrange them into a near-parallelogram of length πr and height r to see why its area is πr²; and they unroll the label of a can to see that the curved surface of a cylinder is a rectangle of length 2πr and height h.
Key facts
- Fencing: a field 120 m × 80 m has perimeter 400 m; at ₹15 a metre the wire costs ₹6,000.
- Wheel of radius 35 cm: circumference 220 cm, so it turns 1,10,000 ÷ 220 = 500 times in 1.1 km.
- Semicircle of radius 7 cm: perimeter 22 + 14 = 36 cm and area 1/2 × 22/7 × 49 = 77 cm².
- Areas: parallelogram b × h; triangle 1/2 × b × h; rhombus 1/2 × d₁ × d₂; trapezium 1/2 × (a + b) × h; circle πr², so r = 14 cm gives 616 cm².
- A path 2.5 m wide outside a garden 40 m × 30 m has area 45 × 35 − 40 × 30 = 1,575 − 1,200 = 375 m².
- Ring between radii 14 cm and 7 cm: 22/7 × (196 − 49) = 22/7 × 147 = 462 cm².
- Room 6 m × 5 m × 4 m: the four walls are 2 × 4 × 11 = 88 m², walls and ceiling together 118 m².
- Cylinder of radius 7 cm and height 10 cm: curved surface 440 cm², total surface 748 cm², volume 1,540 cm³ = 1.54 L.
- Units: 1 m² = 10,000 cm²; 1 hectare = 10,000 m²; 1 L = 1,000 cm³; 1 m³ = 1,000 L.
- Same perimeter, different area: a 6 cm × 4 cm rectangle and a 5 cm square both have perimeter 20 cm, but areas 24 cm² and 25 cm².
Quick revision
Rectangle P = 2(l + b), A = lb; circle C = 2πr, A = πr²; parallelogram bh; triangle 1/2 bh; rhombus 1/2 d₁d₂; trapezium 1/2 (a + b)h; path = outer − inner area; cuboid TSA 2(lb + bh + hl), walls 2h(l + b); cube 6a²; cylinder CSA 2πrh, TSA 2πr(r + h); volumes lbh, a³, πr²h; 1 L = 1,000 cm³, 1 m³ = 1,000 L; doubling the edge of a cube: area ×4, volume ×8.