Surface Areas and Volumes

Many objects around us are combinations of basic solids: a capsule is a cylinder with two hemispherical ends, a tent is a cylinder topped by a cone, an ice-cream cone is a cone with a hemisphere on top, and a toy top is a cone fixed to a hemisphere. In this chapter you will recall the surface area and volume formulae of the cuboid, cube, right circular cylinder, cone, sphere and hemisphere, and learn to apply them to such combinations. For the surface area of a combined solid, you add only the areas of the surfaces that are actually exposed, leaving out the faces where two solids are joined. For example, the surface of a toy made of a cone on a hemisphere is the curved surface of the cone plus the curved surface of the hemisphere. For volume, the volumes of the parts are simply added, or subtracted when a part is scooped out. You will solve practical problems on capsules, tents, pen stands, pipes, gulab jamuns and storage tanks.

Detailed Worksheet: Surface Areas and Volumes Section A - Definitions (10 marks) 1. Write the formulae for the curved surface area, total surface area and volume of a right circular cylinder of radius r and height h. (2 marks) 2. Write the formulae for the slant height, curved surface area and volume of a right circular cone. (2 marks) 3. Write the formulae for the surface area and volume of a sphere and the curved surface area, total surface area and volume of a hemisphere. (2 marks) 4. Explain why the total surface area of a combination of solids is not the sum of the total surface areas of the individual solids. (2 marks) 5. Explain why the volume of a solid formed by joining two solids is the sum of their volumes. When is a volume subtracted? (2 marks) Section B - Calculations and Applications (15 marks) 6. 2 cubes each of volume 64 cm^3 are joined end to end. Find the surface area of the resulting cuboid. (3 marks) 7. A toy is in the form of a cone of radius 3.5 cm mounted on a hemisphere of the same radius. The total height of the toy is 15.5 cm. Find the total surface area of the toy. (Use pi = 22/7) (3 marks) 8. A medicine capsule is in the shape of a cylinder with two hemispheres stuck to each of its ends. The length of the entire capsule is 14 mm and the diameter of the capsule is 5 mm. Find its surface area. (Use pi = 22/7) (3 marks) 9. A solid toy is in the form of a hemisphere surmounted by a right circular cone. The height of the cone is 2 cm and the diameter of the base is 4 cm. Find the volume of the toy. (Use pi = 3.14) (3 marks) 10. A cubical block of side 7 cm is surmounted by a hemisphere. What is the greatest diameter the hemisphere can have? Find the surface area of the solid. (Use pi = 22/7) (3 marks) Section C - Diagrams (10 marks) 11. Draw a neat labelled sketch of a tent in the form of a cylinder of height 2.1 m and diameter 4 m surmounted by a cone of slant height 2.8 m. Find the area of canvas used and the cost at Rs 500 per m^2. (Use pi = 22/7) (4 marks) 12. Draw a sketch of a vessel in the form of a hollow hemisphere mounted by a hollow cylinder. The diameter of the hemisphere is 14 cm and the total height of the vessel is 13 cm. Find the inner surface area of the vessel. (Use pi = 22/7) (3 marks) 13. Draw a labelled sketch of a wooden article made by scooping out a hemisphere from each end of a solid cylinder of height 10 cm and base radius 3.5 cm. Find the total surface area of the article. (Use pi = 22/7) (3 marks) Section D - Analysis and Higher-order Thinking (15 marks) 14. From a solid cylinder whose height is 2.4 cm and diameter 1.4 cm, a conical cavity of the same height and same diameter is hollowed out. Find the total surface area of the remaining solid to the nearest cm^2. Explain which surfaces are included and why the base of the cone is not. (Use pi = 22/7) (5 marks) 15. A pen stand made of wood is in the shape of a cuboid 15 cm by 10 cm by 3.5 cm with four conical depressions to hold pens. The radius of each depression is 0.5 cm and the depth is 1.4 cm. Find the volume of wood in the entire stand. Why is the answer only slightly less than the volume of the cuboid? (Use pi = 22/7) (5 marks) 16. A vessel is in the form of an inverted cone of height 8 cm and radius of the top 5 cm. It is filled with water up to the brim. When lead shots, each a sphere of radius 0.5 cm, are dropped into the vessel, one-fourth of the water flows out. Find the number of lead shots dropped, and explain the principle used. (5 marks) Instructions: Time allowed 2 hours. Attempt all sections. Draw a neat sketch of every solid, mark all dimensions, and use the value of pi stated in each question.