Mensuration

Mensuration is the branch of mathematics that deals with the measurement of lengths, areas and volumes. In this chapter you will find the area of a trapezium using the formula (1/2) x height x (sum of the parallel sides), the area of a rhombus as half the product of its diagonals, and the area of a general quadrilateral or an irregular polygon by splitting it into triangles and trapeziums, as surveyors do when measuring fields. You will then move from plane figures to solids. You will find the lateral and total surface areas of cubes and cuboids, and the curved and total surface areas of cylinders using 2 x pi x r x h and 2 x pi x r x (r + h), with pi = 22/7. You will calculate the volumes of cubes, cuboids and cylinders, and relate volume to capacity using 1 L = 1000 cm^3 and 1 m^3 = 1000 L, applying these ideas to tanks, rooms, boxes and pillars.

Detailed Worksheet: Mensuration Section A - Definitions (10 marks) 1. What is a trapezium? Write the formula for its area and explain each term. (2 marks) 2. Write the formula for the area of a rhombus in terms of its diagonals. Why does this formula work? (2 marks) 3. What is the difference between the lateral surface area and the total surface area of a cuboid? Write the formula for each. (2 marks) 4. Write the formulas for the curved surface area and the total surface area of a closed cylinder of radius r and height h. (2 marks) 5. Distinguish between volume and capacity. How many litres make 1 m^3, and how many cm^3 make 1 litre? (2 marks) Section B - Calculations and Applications (15 marks) 6. (i) The parallel sides of a trapezium-shaped table top are 1 m and 1.2 m, and the perpendicular distance between them is 0.8 m. Find its area. (ii) The area of a trapezium is 34 cm^2, one parallel side is 10 cm and the height is 4 cm. Find the other parallel side. (3 marks) 7. (i) Find the area of a rhombus whose diagonals are 7.5 cm and 12 cm. (ii) A rhombus has side 5 cm and altitude 4.8 cm. If one diagonal is 8 cm, find the other diagonal. (3 marks) 8. A cuboidal box measures 60 cm by 40 cm by 50 cm. Find its total surface area and its volume. How many litres can it hold? (3 marks) 9. A closed cylinder has radius 14 cm and height 8 cm. Find its curved surface area, total surface area and volume. (Take pi = 22/7.) (3 marks) 10. The total surface area of a cube is 600 cm^2. Find its edge and volume. What happens to the surface area if the edge is doubled? (3 marks) Section C - Diagrams (10 marks) 11. Draw a trapezium with parallel sides 10 cm and 6 cm and height 4 cm. Draw one diagonal to divide it into two triangles, and use the diagram to derive the formula for the area of a trapezium. Find its area. (4 marks) 12. Draw a net of a cuboid of length 4 cm, breadth 3 cm and height 2 cm. Label the dimensions of each face and find the total surface area. (3 marks) 13. Draw a cylinder of radius 7 cm and height 10 cm, and draw its net. State the length and breadth of the rectangular part and find the curved surface area. (3 marks) Section D - Analysis and Higher-order Thinking (15 marks) 14. The floor of a building has 3000 tiles, each in the shape of a rhombus with diagonals 45 cm and 30 cm. Find the total area of the floor in m^2 and the cost of polishing it at Rs 4 per m^2. (5 marks) 15. Cylinder P has diameter 7 cm and height 14 cm. Cylinder Q has diameter 14 cm and height 7 cm. Without calculating, predict which has the greater volume. Then find the volume and the total surface area of each, and check your prediction. Does the cylinder with the greater volume also have the greater surface area? (5 marks) 16. A rectangular water tank is 6 m long, 5 m wide and 4.5 m deep. (i) Find its capacity in litres. (ii) If 50 families each use 270 litres of water a day, for how many days will a full tank last? (iii) Explain why a cylindrical tank of the same capacity might be preferred. (5 marks) Instructions: Time allowed 2 hours. Attempt all sections. Take pi = 22/7 unless stated otherwise, write the formula before substituting, and give units in every answer.