CONE AND SPHERE
ASSIGNMENT - 1
1. A sector containing an angle of 90° is cut from a circle of radius 42 cm and folded into a cone. Find the radius and the curved
surface area of the cone.
2. The slant height and base radius of a cone are 17 cm and 8 cm respectively. Find the volume of the cone.
3. Two cones have their heights in the ratio 1 : 3 and the radii of their bases in the ratio 3 : 1. Find the ratio of their volumes.
4. The radius and height of a right circular cone are in the ratio 5 : 12 and its volume is 2512 cm3. Find the radius and slant
height of the cone. (Use = 3.14)
5. A sector of radius 35 cm is cut out of a thin cardboard with angle 180°. It is folded into a cone that of maximum size. Find
the curved surface area and the volume of the cone.
6. A wooden cone has an outer radius of 60 cm and an inner radius of 50 cm. The outer and inner heights are 40 cm and 36
cm respectively. Find the volume of wood in the cone. (Take = 3.14)
7. How many metres of canvas 1.25 m wide will be needed to make a conical tent whose base radius is 17.5 m and height 6 m?
8. There are two cones. The curved surface area of one is twice that of the other. The slant height of later is twice that of the
former. Find the ratio of their radii.
9. If the radius of the base of a circular cone is halved, keeping the height same, what is the ratio of the volume of the reduced
cone to that of the original cone?
10. A conical vessel whose internal radius is 5 cm and height 24 cm is full of water. The water is emptied into a cylindrical vessel
with internal radius 10 cm. Find the height to which the water rises in the cylindrical vessel.
11. A conical tent is to accommodate 11 persons. Each person must have 4 m2 of the space on the ground and 20 m3 of air to
breathe. Find the height of the cone.
12. A cone of maximum size is carved out of a cube of edge 14 cm. Find the volume of the cone and of the remaining material.
13. A hollow cylindrical pipe 50 cm long, whose external diameter is 7 cm and the internal diameter is 5 cm, is melted and recast
into a right circular cone, whose base radius is 10 cm. Calculate the height of the cone.
14. A tent of height 8.25 m is in the form of a right circular cylinder with diameter of base 30 m and height 5.5 m, surmounted
by a right circular cone of the same base. Find the cost of the canvas of the tent at the rate of Rs 44 per m2.
15. The interior of a building is in the form of a cylinder of base radius 12 m and height 3.5 m surmounted by a cone of equal
base and slant height 13 m. Find the internal curved surface area and the capacity of the building.
16. From a cuboidal solid of metal 42 cm × 30 cm × 20 cm, a conical cavity of base radius 14 cm and height 20 cm is drilled
out. Find :
(i) the surface area of the remaining solid
(ii) the volume of the remaining cavity
(iii) the weight of the conical cavity if the metal weighs 7 gm per cm3.
17. A right triangle with sides 3 cm and 4 cm is revolved around its hypotenuse. Find the volume and surface area of the double
cone thus generated.
18. What quantity of canvas 1.25 m wide will be required to make a conical tent whose radius is 21 m and slant height 30 m?

19. The height of a cone is 30 cm. A small cone is cut off at the top by a plane parallel to its base. If its volume be
of the

volume of the given cone, at what height above the base is the section made?
20. A cone of radius 10 cm is divided into two parts by drawing a plane through the mid-point of its axis, parallel to its base.
Compare the volume of the two parts.
ANSWERS
1. 10.5 cm, 1386
5. 1925
9. 1 : 4
cm2,
14. Rs 54450
cm2
2. 1005.71 cm3
cm3
9724.46
10. 2 cm
15. 754
17. 30.17 cm3, 52.8 cm2
6. 56556 cm3
11. 15 m
2 2
2
m , 2338 m3
7
7
18. 1584 m
3. 3 : 1
4. 10 cm, 26 cm
7. 814 m
8. 4 : 1
3
12. 718.7 cm , 2025.3 cm3
13. 9 cm
16. (i) 5857.6 cm2 (ii) 21093.33 cm3 (iii) 147.65 kg
19. 20 cm
1
20. 1 : 7
ASSIGNMENT - 2
1. The given figure represents a hemisphere surmounted by a conical block of wood. The diameter of their
bases is 6 cm each and the slant height of the cone is 5 cm. Calculate :
(i) the height of the cone.
(ii) the volume of the solid.
2. The surface area of a solid metallic sphere is 616 cm2. It is melted and recast into smaller spheres of diameter 3.5 cm. How
many such spheres can be obtained?
3. A metallic sphere of radius 10.5 cm is melted and then recast into small cones, each of radius 3.5 cm and height 3 cm. Find
the number of cones thus obtained.
4. A piece of butter 3 cm by 5 cm by 12 cm is placed in a hemispherical bowl of diameter 6.5 cm. Will the butter overflow when
it melts completely?
5. A vessel is in the form of an inverted cone. Its height is 11 cm and the radius of its top which is open is 2.5 cm. It is filled
with water upto the rim. When lead shots, each of which is a sphere of radius 0.25 cm and dropped into the vessel,
2
of the
5
water flows out. Find the number of lead shots dropped into the vessel.
6. A hollow sphere of internal and external diameters 4 cm and 8 cm respectively, is melted into a cone of base diameter 8 cm.
Find the height of the cone.
7. The surface area of a solid metallic sphere is 1256 cm2. It is melted and recast into solid right circular cones of radius 2.5
cm and height 8 cm. Calculate :
(i) the radius of the solid sphere
(ii) the number of cones recast.
(Take = 3.14)
8. The figure shows the cross-section of an ice-cream cone consisting of a cone surmounted
by a hemisphere. The radius of the hemisphere is 3.5 cm and the height of the cone is 10.5
cm. The outer shell ABCDEF is shaded and is not filled with ice-cream. AF = DC = 0.5 cm,
AB || FE and BC || ED. Calculate :
(i) the volume of the ice-cream in the cone (the unshaded portion including the hemisphere) in cm3.
(ii) the volume of the outer shell (the shaded portion) in cm3. Give your answer correct to
the nearest cm3.
ANSWERS
1. (i) 4 cm (ii) 94.29
cm3
2. 64
3. 126
4. Yes 5. 440 6. 14 cm
2
7. (i) 10 cm (ii) 80 8. (i) 175 cm3 (ii) 50 cm3
ASSIGNMENT - 3
1. The surface area of a sphere is 1256
hemispheres. Calculate :
cm2.
It is cut into two
16. A toy is in the form of a cone mounted on a hemisphere
with the same radius. The diameter of the base of the
conical portion is 12 cm and its height is 8 cm. Determine
the surface area and volume of the toy. ( = 3.14)
(i) radius of the sphere
(ii) total surface area of a hemisphere
17. A cone, a hemisphere and a cylinder stand on equal bases
and have the same height, the height being equal to the
radius of the circular base. Find the ratio of their whole
surfaces.
(iii) volume of a hemisphere correct to 2 decimal places.
2. If the number of square centimetres on the surface of a
sphere is equal to the number of cubic centimetres in its
volume, find the diameter of the sphere.
18. A building is in the form of a cylinder surmounted by a
3. The total surface area of a hemisphere is 462 cm2. Find
the diameter and volume of the hemisphere.
19 3
m of air.
21
If the internal diameter of the building is equal to its total
height above the floor; find the height of the building.
hemispherical vaulted dome and contains 41
4. How many balls each of radius 1 cm can be made by
melting a bigger ball whose diameter is 8 cm?
5. A copper sphere having a radius of 6 cm is melted and
then drawn into a cylindrical wire of radius 2 mm.
Calculate the length of the wire.
6. A hemispherical bowl of radius 3 cm is full of water. The
water is emptied into a cylindrical can of radius 2 cm.
Find the depth of water in the can.
7. What is the ratio of the volume of a cube to that of a
sphere which will fit inside the cube?
19. In the figure, a cylinder is surmounted by a cone at one
end and a hemisphere at the other end. Given that common radius = 3.5 cm, the height of the cylinder = 6.5 cm
and the total height = 12.8 cm, calculate the volume of the
solid correct to the nearest integer.
8. The external and internal diameters of a hemispherical
bowl are 17 cm and 15 cm respectively. Find the cost of
polishing it all over at 25 paise per cm2.
20. From a sphere of radius 10 cm, a right circular cylinder of
base diameter 12 cm is carved out. Calculate the volume
of the right circular cylinder correct to 2 decimal places.
9. The largest sphere is carved out of a cube of edge 7 cm.
Find the volume of the sphere.
10. The radius of a sphere and the base radius of a cone are
equal, each being 8 cm. If the volumes of these two solids
are also equal, find the slant height of the cone.
21. Some lead spheres each of diameter 6 cm are dropped
into a beaker containing some water and are fully
submerged. The diameter of the beaker is 18 cm. How
many lead spheres are dropped into it, if the water level
rises by 40 cm?
11. A cylindrical vessel 60 cm in diameter is partially filled
with water. A sphere of diameter 36 cm is dropped into it
and is fully submerged in water. Find the increase in the
level of water in the vessel.

 without

changing the shape, find the per cent increase in the
surface area.
22. If the volume of a sphere is increased by 
12. A cone and a hemisphere have equal bases and equal
volumes. Find the ratio between height of the cone and
radius of the hemisphere.
23. The figure is obtained by removing two hemispheres from
a solid right cylinder whose area of base is 154 cm2. If the
height of the cylinder is 24 cm, find the volume of the
solid.
13. Find the ratio of the volumes of a cylinder, a cone and a
sphere, if each has the same radius and same height.
14. The radius of a sphere is increased by 50%. Find the
increase per cent in its volume.
15. The curved surface of a solid cone with base radius 3 cm
330
cm2. Three such cones are melted and recast into
7
a sphere. Find the radius and surface of the sphere.
Assume that there is no loss of metal in melting and
recasting.
is
3
24. The figure shows a sphere which
circumscribes a cylinder whose
height is 8 cm and base radius is 3
cm. Find the ratio of the volumes of
the sphere and the cylinder.
25. A toy is made up of a right circular cylinder with hemispherical ends. The radius of the cylinder and that of each
hemisphere are same. If the volume of the toy is twice the
sum of the volumes of the hemispherical ends, find the
ratio of the height and the radius of the cylinder.
ANSWERS
1. (i) 10 cm (ii) 942 cm2 (iii) 2093.33 cm3
2. 6 cm
3. 14 cm, 718.67 cm3
4. 64
7. 6 :
8. Rs 214.50
10. 32.98 cm
5. 90 m
11. 8.64 cm
6. 4.5 cm
12. 2 : 1
e
16. 414.48 cm2, 753.6 cm3
17.
21. 90
23. 5133
22. 56.25%
13. 3 : 1 : 2
j
2 1 :3:4

cm3

18. 4 m
24. 125 : 54
19. Here, OA = 30 cm, AB = R, CD = r, OC = h
OAB ~ OCD
4
14. 237.5%
19. 376 cm3
9. 179
15. 3 cm, 113
20. 1810.29 cm3
25. 4 : 3
2
cm3
3
1
cm2
7