Spheres
Spheres
The figure below shows a sphere.
r
The point O is called the centre of the sphere.
The distance between any point on the surface of the sphere
and its centre (e.g. OA, OB and OC) is called the radius (r) of
the sphere.
a hemisphere
centre of the sphere
a hemisphere
If a sphere is divided into two parts by a plane which passes
through its centre, the two parts are identical and each part is
called a hemisphere.
Volumes of Spheres
For any spheres,
4 3
volume of a sphere   r
3
Refer to the figure on the right.
3 cm
Volume of the sphere
4
    3 3 cm 3
3
 36 cm 3
r = 3 cm
Follow-up question
Find the volume of a hemisphere with diameter 12 cm.
Give your answer in terms of  .
Solution
Radius of the hemisphere  12 cm
2
 6 cm
1 4
2 3
 144  cm3


The required volume       63  cm3
Example 14
The radius of a hemisphere is 6 m. Find the volume of the hemisphere
in terms of .
Example 15
The volume of a football is 2300 cm3. Find the radius of the football.
(Give your answer correct to 3 significant figures.)
Example 16
A capsule is composed of a cylinder and two
hemispheres as shown in the figure. Find the
volume of the capsule in terms of .
Example 14
The radius of a hemisphere is 6 m. Find the volume of the hemisphere
in terms of .
Solution
1 4
    63 m 3
2 3
 144 m 3
Volume of the hemisphere 
Example 15
The volume of a football is 2300 cm3. Find the radius of the football.
(Give your answer correct to 3 significant figures.)
Solution
Let r cm be the radius of the football.
∵ Volume of the football  2300 cm3
∴
4 3
r  2300
3
r 3  1725
r  12.0 (cor. to 3 sig. fig.)
∴
The radius of the football is 12.0 cm.
Example 16
A capsule is composed of a cylinder and two
hemispheres as shown in the figure. Find the
volume of the capsule in terms of .
Solution
1 4
Volume of the hemisphere      33 mm 3
2 3
 18 mm 3
Height of the cylinder  (8  3  3) mm  2 mm
Volume of the cylinder    3 2  2 mm 3
∴
 18 mm 3
Volume of the capsule
 2  volume of the hemisphere  volume of the cylinder
 (2  18  18 ) mm 3
 54 mm 3
Example 17
The figure shows a cylindrical vessel
which contains some water. When 4
identical metal spheres are put into the
vessel and totally submerged in the
water, the water level rises by 2 cm.
Find the radius of each metal sphere
correct to 3 significant figures.
Solution
Volume of the rise in the water level    52  2 cm 3
 50 cm 3
Let r cm be the radius of each metal sphere.
∵ Total volume of 4 metal spheres  volume of the rise
in the water level
4
4  r 3  50
∴
3
r 3  9.375
r  2.11 (cor. to 3 sig. fig.)
∴
The radius of each metal sphere is 2.11 cm.
Surface Areas of Spheres
For any spheres,
surface area of a sphere  4 r 2
Refer to the figure on the right.
Surface area of the sphere
 4    9 2 cm 2
 324 cm 2
r = 9 cm
9 cm
Follow-up question
Find the radius of a hemisphere with curved surface area
200  cm2.
Solution
Let r cm be the radius of the hemisphere.
∵ Curved surface area of the hemisphere  200 cm 2
∴
1
 4 r 2  200
2
r 2  100
r  10
∴ The radius of the hemisphere is 10 cm.
Example 18
Find the total surface area of a solid hemisphere of radius 14 cm
correct to 3 significant figures.
Example 19
It is given that painting a spherical wooden ball requires
0.3 litre of paint and one litre of paint can paint an area of 13 m2.
(a)
Find the radius of the wooden ball.
(b) Find the volume of the wooden ball.
(Give your answers correct to 3 significant figures.)
Example 20
The figure shows a solid which is made up of
a right circular cone and a hemisphere both of
radii 3 cm.
If the volume of the solid is 30 cm3, find
(a) the height of the cone,
(b)
the total surface area of the solid.
(Give your answer in terms of .)
Example 18
Find the total surface area of a solid hemisphere of radius 14 cm
correct to 3 significant figures.
Solution
1
Curved surface area of the hemisphere   4    14 2 cm 2
2
 392 cm 2
Area of the flat surface of the hemisphere    14 2 cm 2
 196 cm 2
Example 19
It is given that painting a spherical wooden ball requires
0.3 litre of paint and one litre of paint can paint an area of 13 m2.
(a) Find the radius of the wooden ball.
(b) Find the volume of the wooden ball.
(Give your answers correct to 3 significant figures.)
Solution
(a)
Surface area of the wooden ball  0.3  13 m 2  3.9 m 2
Let r m be the radius of the wooden ball.
4r 2  3.9
r  0.557 09
 0.557 (cor. to 3 sig. fig.)
∴ The radius of the wooden ball is 0.557 m.
(b)
4
   0.557 09 3 m 3
3
 0.724 m 3 (cor. to 3 sig. fig.)
Volume of the wooden ball 
Example 20
The figure shows a solid which is made up of
a right circular cone and a hemisphere both of
radii 3 cm.
If the volume of the solid is 30 cm3, find
(a) the height of the cone,
(b) the total surface area of the solid.
(Give your answer in terms of .)
Solution
(a)
Let h cm be the height of the cone.
∵ Volume of the solid  volume of the cone 
volume of the hemisphere
1
1 4
2


30
3




h

    33
∴
3
2 3
30  3h  18
3h  12
h4
∴
The height of the cone is 4 cm.
(b)
Slant height of the cone  3 2  4 2 cm (Pyth. theorem)
 5 cm
Curved surface area of the cone    3  5 cm 2
 15 cm 2
1
Curved surface area of the hemisphere   4    32 cm 2
2
 18 cm 2
∴
2
Total surface area of the solid  (15  18 ) cm
 33 cm 2
ID10 (P.175)
3.4 cm
ID12 (P.177)
ID11 (P.176)
6.6 cm
ID13 (P.178)
3.4 cm
Do Ex.10C
(9,13)
36 cm3
39 cm2
Do Ex.10C
(8,18,20)
2145 cm3
2094 cm3
254 cm2
603 cm2