Till now you have been studying about the plane and flat figures that are usually known as 2-D figure. Now in higher grade you will be studying about 3-dimensional figure that will come with new concepts and terms. So here we would learn about the surface area of sphere and the volume of the sphere in an easier way.
WHAT IS A SPHERE?
A sphere is just like a ball but it is not a circle. The difference between a sphere and a circle is that a circle is a 2-D shape but a sphere is a solid three dimensional figure that has all the points in the spaceπ, which lie at the constant distance (called radius) from the fixed point known as the centre of the sphere. Though just like a circle, a sphere is mathematically defined in respect to radius, r and the centre. Sphere is actually a solid shape with no edges or vertices.
SURFACE AREA OF SPHERE:
SURFACE AREA of a sphere object is a measure of the total area occupied by all the outer surface of a sphere. This suggests that the surface area of any spherical figure is 4 times the area of the circle of same radius as a sphere or as CUEMATH defines the surface area of a sphere as the total area of the faces surrounding it.
This means,Surface Area of a Sphere= 4* the area of a circle of radius r= 4*πr^2
Where r is the radius of the sphere (i.e. the common distance from centre to any point on the surface of the sphere)
*NOTE: The surface area or any sort of area of any kind of shape is always written in square units (inshort sq. units)
DERIVATION(how to quickly learn the formula)
A well-known scientist and mathematician Archimedes found that if the radius of cylinder and sphere is “r”, then the surface area of the sphere is equal to the lateral surface of the cylinder. Hence the relation comes out to be,
Surface area of sphere = Lateral surface area of cylinder
We know that,Surface area of cylinder = 2πrh
Height of cylinder = diameter of sphere= 2r
Thus Surface area of sphere is 2πrh= 2πr*2r= 4πr^2 sq. units
In terms of diameter it becomes S= 4π (d/2) ^2
Where d is the diameter of the respective sphere.
VOLUME OF THE SPHERE:
Volume actually is the measurement of space any shape can occupy.
Volume of sphere is defined as the capacity of the sphere or the cubic units that will exactly fill a sphere or the storage capacity of the sphere. The volume of a sphere depends on the radius of the sphere and hence the change in radius results in the change of volume. Thesphere studied usually are of two types. They are: solid sphere and hollow sphere. Both of them have different volumes.
VOLUME of a Solid Sphere: If the radius is r and volume is assumed to be V, then
.Volume of sphere= (4/3) πr^3
VOLUME of a Hollow Sphere: If the radius of outer sphere is R and that of inner sphere is r and volume of sphere is V then
VOLUME of Hollow Sphere, V= Volume of outer sphere, R–Volume of inner sphere
= (4/3) ΠR^3- (4/3) πr^3= (4/3) π(R^3-r^3)
*Note: The volume of a sphere is given as cubic units or (units) ^3.
Derivation of volume of sphere:
According to Archimedes, if radius of a sphere, cone and a cylinder is “r” and the cross-section area is same then their volumes are in the ratio of 1:2:3. Thus, the relation between volume of sphere, volume of cone and volume of cylinder is given as:
Volume of cylinder= volume of cone+ Volume of sphere
Volumeof sphere= Volume of cylinder-volume of cone
We know that
Volume of cylinder = πr^2h
Volume of cone= (1/3) πr^2h
So Volume of sphere = πr^2 h- (1/3) πr^2h= (2/3) πr^2h
Here, height of cylinder=diameter of sphere= 2r
Hence Volume of Sphere= (2/3) πr^2(2r) = 4/3πr^3
With the above explanation the understanding of SURFACE AREA and VOLUME of a SPHERE becomes easier.