Sphere
We see many solids like cricket ball, tennis ball, basket ball,
football, globe etc. All these are the examples of spheres. Let us know some
more facts about the sphere.
A sphere cab be described as the set of all those points in the space which are at equal distance from a fixed point. The fixed point is called centre and the constant distance is called the radius of the sphere.
In the given figure above, O is the centre of the sphere
and distances OA, OB, OC, OD, OE, OF are the radius of the sphere.
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A line segment passing through the centre of the sphere
with its end points on the sphere is called the diameter of the sphere. In the
given figure above AB, CD and EF are the
diameters of the sphere.
Surface Area of a Sphere
Take a cylinder of equal height and diameter and a sphere
equal diameter with the cylinder and wrap the sphere with a rope and with the
same rope the curved surface area of the cylinder can be wrapped.
∴ Surface area of the sphere = Curved sufrace
area of the cylinder
= 2πrh
= 2πr × 2r [∵ h = d = 2r]
= 4πr2
Thus, if r is the radius of a sphere then the
surface area of the sphere is 4πr2.
Volume of Sphere
Take a cylinder of equal height and
diameter and a sphere of equal diameter with the cylinder. Fill the cylinder
with water and drop the sphere into the cylinder, then two-third of water flows
out.
So we can say that,
Volume of sphere = 2/3 volume of the
cylinder
= 2/3 × πr2h
= 2/3 × πr2 ×
2r [∵ h = d = 2r]
= 4πr3/3
Thus, the volume of the sphere of radius r is 4πr3/3.
Workout Examples
Example 1: Find the total surface area and the volume of the given
cricket ball of radius 35mm.
Solution: Here,
Radius
of cricket ball (r) = 35mm
Now,
Total
surface area (TSA) = 4πr2
= 4 × 22/7 × (35mm)2
= 4 × 22/7 × 1225mm2
= 15400 mm2
Volume
(V) = 4πr3/3
= 4/3 × 22/7 × (35mm)3
= 88/21 × 42875mm3
= 179666.67mm3
Thus, the total surface area of cricket
ball is 15400 mm2 and the volume is 179666.67mm3.
Example 2: If the total surface area of a solid sphere is 616cm2,
what will be its radius?
Solution: Here,
Total
surface area of sphere (TSA) = 616cm2
i.e. 4πr2 = 616
or, 4 × 22/7 × r2 = 616
or, 88/7 × r2 = 616
or, r2 = 616 × 7/88
or, r2 = 49
or, r2 = 72
or,
r = 7cm
Thus, the radius of the solid sphere is
7cm.
Example 3: Find the total surface area of a sphere whose volume is
1372π/3 cm3.
Solution: Here,
Volume
of sphere = 1372π/3 cm3
i.e. 4πr3/3 = 1372π/3
or, 4r3 = 1372
or, r3 = 1372/4
or, r3 = 343
or, r3 = 73
or,
r = 7cm
Now,
Total
surface area (TSA) of sphere = 4πr2
= 4 × 22/7 × (7cm)2
= 4 × 22/7 × 49cm2
= 616cm2
Thus, the total surface area of the sphere
is 616cm2.
You can comment your questions or problems
regarding the area and volume of sphere here.
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