Surface Areas and Volumes

This chapter extends your study of solid shapes to the right circular cone, the sphere and the hemisphere. A right circular cone is formed by rotating a right-angled triangle about one of its perpendicular sides, and its radius r, height h and slant height l are related by l^2 = r^2 + h^2. Unfolding the curved surface of a cone gives a sector of a circle, which leads to its curved surface area, pi r l, and total surface area, pi r (l + r). A sphere of radius r has surface area 4 pi r^2, and a hemisphere has curved surface area 2 pi r^2 and total surface area 3 pi r^2. You will learn that the volume of a cone is one-third the volume of a cylinder of the same base and height, giving (1/3) pi r^2 h, that the volume of a sphere is (4/3) pi r^3 and that of a hemisphere is (2/3) pi r^3. You will apply these formulas to tents, ice cream cones, domes, balls and bowls, usually taking pi as 22/7.

Detailed Worksheet: Surface Areas and Volumes Section A - Definitions (10 marks) 1. What is a right circular cone? Write the relation between its radius, height and slant height. (2 marks) 2. Write the formulas for the curved surface area and total surface area of a cone. (2 marks) 3. Write the formulas for the surface area of a sphere and the curved and total surface areas of a hemisphere. (2 marks) 4. Write the formulas for the volumes of a cone, a sphere and a hemisphere. (2 marks) 5. How is the volume of a cone related to the volume of a cylinder with the same base radius and height? (2 marks) Section B - Calculations and Applications (15 marks) 6. A cone has radius 7 cm and height 24 cm. Find its slant height, curved surface area and total surface area. (Take pi = 22/7.) (3 marks) 7. Find the volume of the cone in question 6. (3 marks) 8. Find the surface area and volume of a sphere of radius 21 cm. (Take pi = 22/7.) (3 marks) 9. A hemispherical bowl has an inner radius of 10.5 cm. Find the volume of milk it can hold, and its inner curved surface area. (Take pi = 22/7.) (3 marks) 10. The radius of a sphere is doubled. Find the ratio of the new surface area to the old one, and the ratio of the new volume to the old one. (3 marks) Section C - Diagrams (10 marks) 11. Draw a labelled diagram of a right circular cone showing its radius, height and slant height. Show how its curved surface, when cut and unfolded, forms a sector of a circle. (4 marks) 12. Draw a labelled diagram of a sphere and a hemisphere and mark the radius in each. Explain why the total surface area of a hemisphere is 3 pi r^2. (3 marks) 13. Draw a cone and a cylinder of the same base and height and explain, using an experiment of filling with sand or water, why the volume of the cone is one-third that of the cylinder. (3 marks) Section D - Analysis and Higher-order Thinking (15 marks) 14. A conical tent is 10 m high and the radius of its base is 24 m. Find its slant height and the cost of the canvas needed to make it at Rs 70 per square m. (Take pi = 22/7.) (5 marks) 15. A dome of a building is in the form of a hemisphere. From inside it was whitewashed at a cost of Rs 4989.60. If the cost of whitewashing is Rs 20 per square m, find the inside surface area of the dome and its radius. Then find the volume of air inside it. (Take pi = 22/7.) (5 marks) 16. A solid metal sphere of radius 6 cm is melted and recast into cones of radius 3 cm and height 4 cm. How many cones are formed? Also find the total curved surface area of all the cones. (Take pi = 3.14.) (5 marks) Instructions: Time allowed 2 hours. Attempt all sections. Write the formula, substitute values and give units with every answer.