Class 10 › Mathematics Surface Areas and Volumes Class 10 Mathematics · Chapter 12
Practice all → Summary Questions & answers Practise this chapter → View all Page 1 free about 8 min read , Full kit about 55 min read Page 1 free about 8 min read , Full kit about 55 min read View all 12 amazing Marks Booster kits → This chapter extends mensuration to solids and combinations of solids. It revisits the surface areas and volumes of the cuboid, cube, cylinder, cone, sphere and hemisphere, and teaches how to find these quantities for objects formed by joining such solids — for example a cone mounted on a cylinder. Problems also involve converting one solid into another (recasting), where the volume stays constant while the shape changes. These skills apply directly to real objects like containers, tents and toys.
Surface area
The total area of all the outer faces of a solid.
Volume
The amount of space a solid occupies.
Combination of solids
A shape formed by joining two or more basic solids.
Recasting
Melting and reforming a solid so its volume is conserved.
Q1 Write the volume of a sphere of radius \(r\). Answer Q2 When a solid is melted and recast into another shape, which quantity stays the same? Answer Q3 Write the total surface area and volume of a cylinder of radius r and height h. Answer Q4 Write the volume of a cone and of a sphere. Answer Q5 Write the curved surface area of a cone with slant height l. Answer Q6 What is meant by a combination of solids? Give an example. Answer Q7 When a solid is melted and recast into another shape, which quantity is conserved? Answer Q8 How do you find the surface area of a combined solid? Answer Q9 Find the volume of a hemisphere of radius 3 cm. (Take \(\pi = 22/7\).) Answer Q10 A cone and a cylinder have the same radius and height. What is the ratio of their volumes? Answer Q1 1. 2 cubes each of volume \(64\text{ cm}^3\) are joined end to end. Find the surface area of the resulting cuboid. Answer Q2 2. A vessel is in the form of a hollow hemisphere mounted by a hollow cylinder. The diameter of the hemisphere is \(14\text{ cm}\) and the total height of the vessel is \(13\text{ cm}\). Find the inner surface area of the vessel. Answer Q3 3. A toy is in the form of a cone of radius \(3.5\text{ cm}\) mounted on a hemisphere of same radius. The total height of the toy is \(15.5\text{ cm}\). Find the total surface area of the toy. Answer Q4 4. A cubical block of side \(7\text{ cm}\) is surmounted by a hemisphere. What is the greatest diameter the hemisphere can have? Find the surface area of the solid. Answer Q5 5. A hemispherical depression is cut out from one face of a cubical wooden block such that the diameter \(l\) of the hemisphere is equal to the edge of the cube. Determine the surface area of the remaining solid. Answer Q6 6. A medicine capsule is in the shape of a cylinder with two hemispheres stuck to each of its ends (see Fig. 12.10). The length of the entire capsule is \(14\text{ mm}\) and the diameter of the capsule is \(5\text{ mm}\). Find its surface area. Answer Q7 7. A tent is in the shape of a cylinder surmounted by a conical top. If the height and diameter of the cylindrical part are \(2.1\text{ m}\) and \(4\text{ m}\) respectively, and the slant height of the top is \(2.8\text{ m}\), find the area of the canvas used for making the tent. Also, find the cost of the canvas of the tent at the rate of ₹ \(500\) per \(\text{m}^2\). (Note that the base of the tent will not be covered with canvas.) Answer Q8 8. From a solid cylinder whose height is \(2.4\text{ cm}\) and diameter \(1.4\text{ cm}\), 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 \(\text{cm}^2\). Answer Q9 9. A wooden article was made by scooping out a hemisphere from each end of a solid cylinder, as shown in Fig. 12.11. If the height of the cylinder is \(10\text{ cm}\), and its base is of radius \(3.5\text{ cm}\), find the total surface area of the article. Answer Q1 1. A solid is in the shape of a cone standing on a hemisphere with both their radii being equal to \(1\text{ cm}\) and the height of the cone is equal to its radius. Find the volume of the solid in terms of \(\pi\). Answer Q2 2. Rachel, an engineering student, was asked to make a model shaped like a cylinder with two cones attached at its two ends by using a thin aluminium sheet. The diameter of the model is \(3\text{ cm}\) and its length is \(12\text{ cm}\). If each cone has a height of \(2\text{ cm}\), find the volume of air contained in the model that Rachel made. (Assume the outer and inner dimensions of the model to be nearly the same.) Answer Q3 3. A gulab jamun, contains sugar syrup up to about \(30\%\) of its volume. Find approximately how much syrup would be found in \(45\) gulab jamuns, each shaped like a cylinder with two hemispherical ends with length \(5\text{ cm}\) and diameter \(2.8\text{ cm}\) (see Fig. 12.15). Answer Q4 4. A pen stand made of wood is in the shape of a cuboid with four conical depressions to hold pens. The dimensions of the cuboid are \(15\text{ cm}\) by \(10\text{ cm}\) by \(3.5\text{ cm}\). The radius of each of the depressions is \(0.5\text{ cm}\) and the depth is \(1.4\text{ cm}\). Find the volume of wood in the entire stand (see Fig. 12.16). Answer Q5 5. A vessel is in the form of an inverted cone. Its height is \(8\text{ cm}\) and the radius of its top, which is open, is \(5\text{ cm}\). It is filled with water up to the brim. When lead shots, each of which is a sphere of radius \(0.5\text{ cm}\) are dropped into the vessel, one-fourth of the water flows out. Find the number of lead shots dropped in the vessel. Answer Q6 6. A solid iron pole consists of a cylinder of height \(220\text{ cm}\) and base diameter \(24\text{ cm}\), which is surmounted by another cylinder of height \(60\text{ cm}\) and radius \(8\text{ cm}\). Find the mass of the pole, given that \(1\text{ cm}^3\) of iron has approximately \(8\text{ g}\) mass. (Use \(\pi=3.14\)) Answer Q7 7. A solid consisting of a right circular cone of height \(120\text{ cm}\) and radius \(60\text{ cm}\) standing on a hemisphere of radius \(60\text{ cm}\) is placed upright in a right circular cylinder full of water such that it touches the bottom. Find the volume of water left in the cylinder, if the radius of the cylinder is \(60\text{ cm}\) and its height is \(180\text{ cm}\). Answer Q8 8. A spherical glass vessel has a cylindrical neck \(8\text{ cm}\) long, \(2\text{ cm}\) in diameter; the diameter of the spherical part is \(8.5\text{ cm}\). By measuring the amount of water it holds, a child finds its volume to be \(345\text{ cm}^3\). Check whether she is correct, taking the above as the inside measurements, and \(\pi=3.14\). Answer
The volume of a sphere of radius r is:
A 4πr² B (2/3)πr³ C (4/3)πr³ D πr²h
Flashcards & interactive practice 
