Surface area of sphere = 4πr²
How to find out the surface area of sphere. Before starting to explain the equation, we will see how a sphere is forming. Please consider below figures.
The FIG-1a is a circle cut it from a sheet of paper. Please fix a string on the top side of the circular paper as shown in the fig-1b. And if we rotate the string we can see our circular paper as shown in the fig-1b. Now rotate the string much more speed. And we can see our circular paper as a sphere as shown in fig-1c.
Consider the center of the circle is C in the fig1-a. So the center of the circle C will become the center of sphere in fig-1c. Please remember, the circle is a closed figure whose every point lies at constant distance (called radius). From a fixed point (called center). So we can now say, A sphere is a three dimensional figure (solid figure) .Made up with all points in the space which lie in equal distance (called radius of the sphere). from a fixed point (called center of the sphere).
Surface area of the sphere
Now we can see how to find out the surface area of sphere. Consider a rubber ball (nothing but a sphere). And fixe a nail on it surface as shown in the fig-2a. Now take a long string and start to wind it on the ball from the nail without any space and complete the all surface of the ball with string. cut the remaining string.Now we can unwind the string and keep it in some place.
Assigning the Area of the sphere in to circle
Cut four circular paper with the same radius of the ball. Now start to fill the four circles, one by one with the string which is used to wind on the ball. After filling each circle cut the string and use remaining string for the next circle as shown in fig-2b. After filling four circle, we can see that the entire string is used to cover all the surface area of the four circle. Therefore we can say that the surface area of the sphere is equal to the surface area of four circle with the same radius of the sphere.
As we know the area of the circle is πr²
so the surface area of a sphere is 4πr²
Where r is the radius of the sphere.
Now all ready we studied how to find out the surface area of sphere. As we see a sphere have only and only one surface. And that is a curved surface area. The surface area of the sphere can be calculated with equation 4πr² . If cut the sphere through it diameter we can get two supprate figure as shown in the fig-3.
This figure is nothing but a half sphere. It can also be called as hemisphere. But in this case we can see that the hemisphere have two surfaces. One of it surface is curved surface.and the other one is flat circle. The curved surface area of the hemisphere is the half surface area of the sphere. (as it is half of the sphere). There for the curved surface area of a hemisphere is 2πr²
And the flat circle surface area of the hemisphere is πr²
So the total surface area of the hemisphere = 2πr²+πr² = 3πr²
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