YOUR STUDY SPACE · CBSE

Surface Areas and Volumes — Class 10 Mathematics

Mathematics · Chapter 12

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Chapter summary

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 areas of basic solidsVolumes of basic solidsCombination of solidsConversion of solids

Key ideas & terms

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.

Extra questions & answers

Write the volume of a sphere of radius .

When a solid is melted and recast into another shape, which quantity stays the same?

The volume remains unchanged.

Write the total surface area and volume of a cylinder of radius r and height h.

Total surface area and volume .

Write the volume of a cone and of a sphere.

Volume of a cone and volume of a sphere .

Write the curved surface area of a cone with slant height l.

Curved surface area , where .

What is meant by a combination of solids? Give an example.

A shape formed by joining two or more basic solids, e.g. a cone mounted on a cylinder (an ice-cream cone) or a hemisphere on a cuboid.

When a solid is melted and recast into another shape, which quantity is conserved?

The volume remains the same, even though the surface area and shape change.

How do you find the surface area of a combined solid?

By adding the areas of the visible (exposed) surfaces of the individual solids that form it.

Find the volume of a hemisphere of radius 3 cm. (Take .)

Volume cm.

A cone and a cylinder have the same radius and height. What is the ratio of their volumes?

Volume of cone : volume of cylinder .

EXERCISE 12.1

1. 2 cubes each of volume are joined end to end. Find the surface area of the resulting cuboid.

For each cube, , so . After joining them end to end, the cuboid has dimensions . Therefore,

2. A vessel is in the form of a hollow hemisphere mounted by a hollow cylinder. The diameter of the hemisphere is and the total height of the vessel is . Find the inner surface area of the vessel.

The common radius is , and the height of the cylindrical part is . The inner surface area is the sum of the curved surface areas of the cylinder and hemisphere:

3. A toy is in the form of a cone of radius mounted on a hemisphere of same radius. The total height of the toy is . Find the total surface area of the toy.

The cone has height . Its slant height is Hence, the exposed surface area is

4. A cubical block of side is surmounted by a hemisphere. What is the greatest diameter the hemisphere can have? Find the surface area of the solid.

The greatest possible diameter equals the side of the cube, so it is , and . The circular part of the cube covered by the hemisphere is not exposed. Thus,

5. A hemispherical depression is cut out from one face of a cubical wooden block such that the diameter of the hemisphere is equal to the edge of the cube. Determine the surface area of the remaining solid.

The radius of the hemispherical depression is . From the surface area of the cube, the circular opening is removed and the curved surface of the depression is added: Taking , the surface area is

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 and the diameter of the capsule is . Find its surface area.

The radius is , and the cylindrical length is . The two hemispheres form a sphere. Therefore,

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 and respectively, and the slant height of the top is , 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 ₹ per . (Note that the base of the tent will not be covered with canvas.)

Here, , the cylinder height is , and the cone slant height is . Since the base is uncovered, Therefore, the cost is

8. From a solid cylinder whose height is and diameter , 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 .

The radius is , and the slant height of the conical cavity is The remaining surface consists of the outer curved surface of the cylinder, one circular base, and the inner curved surface of the cone: To the nearest square centimetre, the surface area is .

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 , and its base is of radius , find the total surface area of the article.

The exposed surface consists of the curved surface of the cylinder and the curved surfaces of two hemispherical hollows: Thus,

EXERCISE 12.2

1. A solid is in the shape of a cone standing on a hemisphere with both their radii being equal to and the height of the cone is equal to its radius. Find the volume of the solid in terms of .

Here, and the cone height is . Therefore,

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 and its length is . If each cone has a height of , 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.)

The radius is . The cylindrical part has length . Hence, Taking , the volume is .

3. A gulab jamun, contains sugar syrup up to about of its volume. Find approximately how much syrup would be found in gulab jamuns, each shaped like a cylinder with two hemispherical ends with length and diameter (see Fig. 12.15).

The radius is , and the cylindrical length is . The volume of one gulab jamun is Therefore, the syrup in gulab jamuns is Thus, approximately of syrup would be present.

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 by by . The radius of each of the depressions is and the depth is . Find the volume of wood in the entire stand (see Fig. 12.16).

The volume of the cuboid is The total volume removed by the four conical depressions is Hence, the volume of wood is

5. A vessel is in the form of an inverted cone. Its height is and the radius of its top, which is open, is . It is filled with water up to the brim. When lead shots, each of which is a sphere of radius are dropped into the vessel, one-fourth of the water flows out. Find the number of lead shots dropped in the vessel.

The volume of the cone is One-fourth of this volume, equal to the volume displaced by the lead shots, is . The volume of one shot is Therefore, the number of shots is

6. A solid iron pole consists of a cylinder of height and base diameter , which is surmounted by another cylinder of height and radius . Find the mass of the pole, given that of iron has approximately mass. (Use )

For the lower cylinder, and . For the upper cylinder, and . Thus, The mass is Hence, the pole has a mass of approximately .

7. A solid consisting of a right circular cone of height and radius standing on a hemisphere of radius 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 and its height is .

The water left equals the cylinder volume minus the volumes of the cone and hemisphere: Therefore, Taking , this is approximately .

8. A spherical glass vessel has a cylindrical neck long, in diameter; the diameter of the spherical part is . By measuring the amount of water it holds, a child finds its volume to be . Check whether she is correct, taking the above as the inside measurements, and .

The spherical part has radius , and the cylindrical neck has radius and height . Its calculated capacity is Since , the child's measurement is not correct; it is about less than the calculated volume.