Surface Area of a Sphere
The total curved surface around the sphere is called the
total surface area (TSA) of the sphere. Archimedes (6th century BC)
discovered that the total surface area of a spherical object like tennis ball,
football etc. is equal to the curved surface area of the circumscribing
cylinder.
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The figure given above is the cylinder circumscribing a
sphere. Here diameter of the sphere AB is equal to the diameter PQ of the
cylinder and diameter of sphere is equal to the height QR of cylinder.
We know that the curved surface area (CSA) of a cylinder
= 2πrh
Now, by Archimedes principle on TSA of sphere,
Total surface area of sphere = curved surface area of the
circumscribing cylinder
= 2πrh
= 2πr × d [∵ h = d]
= 2πr × 2r
[∵ d = 2r]
= 4πr2
∴ Total surface
area of a sphere = 4πr2
Surface area of a hemisphere
Hemisphere is one of the half part of a
sphere when it is divided into two equal parts. It has two surfaces, one circular
face (great circle) and another curved surface.
The radius of hemisphere is equal to the
radius of the sphere. Since hemisphere is half part of sphere,
Curved surface area of hemisphere = ½ of
curved surface area of sphere
=
½ × 4πr2
=
2πr2
Total surface area of hemisphere = CSA of hemisphere + area of circular surface
= 2πr2 + πr2
= 3πr2
∴ TSA of hemisphere = 3πr2
Workout Examples
Example 1: Find the total surface area of a sphere of diameter 21cm.
Solution: Here,
Diameter
of sphere (d) = 21cm
∴ radius of sphere (r) = 21/2 = 10.5cm
Now,
TSA of sphere =
4πr2
= 4 × 22/7 × (10.5)2
= 88/7 × 110.25
= 1386cm2
Thus, the total surface area of
sphere is 1386cm2.
Example 2: Find the total surface area of a sphere if the
circumference of the great circle is π cm.
Solution: Here,
Circumference
of great circle = π cm
i.e. 2πr = π
or, r = π/2π
or, r = ½
or, r = 0.5cm
Now,
TSA of sphere =
4πr2
= 4 × π × (0.5)2
= 4 × π × 0.25
= π cm2
Thus, the total surface area of
sphere is π cm2.
Example 3: Find the total surface area of two hemispheres when a
sphere of radius 7cm is cut into two halves.
Solution: Here,
Radius
of hemisphere = radius of sphere = 7cm
Now,
The
total surface area (TSA) of a hemisphere = 3πr2
= 2 × 22/7 × 72
= 2 × 22 × 7
= 308cm2
∴ Total surface area of two hemisphere = 2 × 308cm2
= 616 cm2
Thus, the total surface area of two
hemisphere is 616 cm2.
You can comment your questions or problems
regarding the TSA of sphere and hemisphere here.
The area of a great circle of a sphere is a cm^ 2 What is the surface area of the sphere?
ReplyDeleteArea of greater circle = πr^2 = a cm^2
DeleteSurface area of sphere = 4πr^2 = 4a cm^2