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CHAPTER 12
SURFACE AREAS AND VOLUMES
(A) Main Concepts and Results
• The surface area of an object formed by combining any two of the basic solids,
namely, cuboid, cone, cylinder, sphere and hemisphere.
• The volume of an object formed by combining any two of the basic solids
namely, cuboid, cone, cylinder, sphere and hemisphere.
• The formulae involving the frustum of a cone are:
1
(i) Volume of the frustum of the cone = π h[ r12 + r22 + r1r2 ]
3
(ii) Curved surface area of the frustum of the cone = π(r1+r2)l,
(iii) Total surface area of the frustum of the solid cone
= πl(r1+r2)+ π r12 + π r22 , where l = h 2 + ( r1 – r2 ) 2 ,
h = vertical height of the frustum, l = slant height of the frustum and
r1 and r2 are radii of the two bases (ends) of the frustum.
• Solid hemisphere: If r is the radius of a hemisphere, then
curved surface area = 2πr2
2 3
total surface area = 3πr2, and volume = π r
3
4
• (
3 3
)
Volume of a spherical shell = π r1 – r2 , where r1 and r2 are respectively its
3
external and internal radii. 22
Throughout this chapter, take π = , if not stated otherwise.
7
03/05/18
,SURFACE AREAS AND VOLUMES 137
(B) Multiple Choice Questions :
Choose the correct answer from the given four options:
Sample Question 1 : A funnel (see Fig.12.1) is the
combination of
(A) a cone and a cylinder (B) frustum of a cone and a cylinder
(C) a hemisphere and a cylinder (D) a hemisphere and a cone
Solution : Answer (B)
Sample Question 2 : If a marble of radius 2.1 cm is put into a cylindrical cup full of water
of radius 5cm and height 6 cm, then how much water flows out of the cylindrical cup?
(A) 38.8 cm3 (B) 55.4 cm3 (C) 19.4 cm3 (D) 471.4 cm3
Solution : Answer (A)
Sample Question 3 : A cubical ice cream brick of edge 22 cm is to be distributed
among some children by filling ice cream cones of radius 2 cm and height 7 cm upto its
brim. How many children will get the ice cream cones?
(A) 163 (B) 263 (C) 363 (D) 463
Solution : Answer (C)
Sample Question 4 : The radii of the ends of a frustum of a cone of height h cm are
r1 cm and r2 cm. The volume in cm3 of the frustum of the cone is
1 1
(A) π h [ r12 + r22 + r1 r2 ] (B) π h [ r12 + r22 – r1r2 ]
3 3
1 1
(C) π h [ r12 – r22 + r1r2 ] (D) π h [ r12 – r22 – r1r2 ]
3 3
Solution : Answer (A)
Sample Question 5 : The volume of the largest right circular cone that can be cut out
from a cube of edge 4.2 cm is
(A) 9.7 cm3 (B) 77.6 cm3 (C) 58.2 cm3 (D) 19.4 cm3
Solution : Answer (D)
03/05/18
, 138 EXEMPLAR PROBLEMS
EXERCISE 12.1
Choose the correct answer from the given four options:
1. A cylindrical pencil sharpened at one edge is the combination of
(A) a cone and a cylinder (B) frustum of a cone and a cylinder
(C) a hemisphere and a cylinder (D) two cylinders.
2. A surahi is the combination of
(A) a sphere and a cylinder (B) a hemisphere and a cylinder
(C) two hemispheres (D) a cylinder and a cone.
3. A plumbline (sahul) is the combination of (see Fig. 12.2)
(A) a cone and a cylinder (B) a hemisphere and a cone
(C) frustum of a cone and a cylinder (D) sphere and cylinder
4. The shape of a glass (tumbler) (see Fig. 12.3) is usually in the form of
(A) a cone (B) frustum of a cone
(C) a cylinder (D) a sphere
03/05/18
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