The Derivation of Surface Area of Sphere

The sphere is a 3-D geometric representation of a circle. The sphere is different from other 3-D geometrical shapes as it does have an edge or vertices. The point of difference between a circle and a sphere is that the former is a 2-D figure whereas the latter is the 3-D figure. We can easily find various objects shaped spheres from our daily life such as football, bubbles, globes, etc. As with any other type of 3 –D sphere geometric figure, we can find out the surface area of sphere.

The surface area of the sphere represents the area covered by its outer space. The surface area of a sphere is always denominated in terms of square units. For a better understanding of how to calculate the sphere’s surface area, one must understand the sphere’s essential properties. These include:

• The sphere is totally symmetrical from all the points and directions.
• Sphere has only one type of surface area i.e. curved surface area.
• The sphere is only a 3-D geometrical shape with no edge or vertices.
• When all the shapes of the same surface area are considered, the sphere will have the largest volume.

Hence these are some of the unique properties of the sphere that makes it different from the rest of the 3-D sphere geometric shapes. The understanding of the properties of the sphere will help in the calculation of the surface area of the sphere.

The surface area of the sphere was derived by Archimedes with the help of the lateral surface area of the sphere. According to him, the surface area of the sphere = lateral surface area of the sphere.

• Therefore the surface area of sphere = 2πrh

(However the height of the cylinder i.e. h = diameter of the sphere which thereby can be represented as 2r.)

• Thus the final formula for the surface area of sphere = 2πr (2r) = 4πr² square units.

Thus from this, we can conclude that the surface area of a sphere is fully dependent on the radius of the sphere. One must follow certain steps in the calculation of the surface area of the sphere to avoid any chance of a mistake. These steps are mentioned below:

• As the surface area of a sphere is dependent on its radius, the very step is to know the radius of the sphere. Radius is nothing but a line segment that is drawn from any point of the sphere to its center. Assume O represents the center of the circle and A is any point on the sphere. Thus the line segment OA represents the radius of the sphere. In many situations, one may come across the diameter of the sphere which is nothing but the line segment passing through the center that’s connected to opposite points on the sphere. One can find the radius of the sphere through the diameter by dividing the diameter by 2.
• The next step is to square the radius of the sphere by multiplying by itself.
• After the square of the radius, multiply with 4.
• After this, the product of the above-mentioned is to be multiplied with an approximate value of π. This is the final step in the calculation of the surface area of the sphere. One must not forget to denominate it in terms of square units.

Hence with the help of the above-mentioned steps, now calculation of the surface area of the sphere is no longer a challenge. Besides the surface area, the volume of sphere is also calculated. The volume of the sphere represents the space that is occupied by the sphere. It is represented in terms of cubic units. The volume of a sphere is given by the formula (4/3) πr3.

Following are some of the very basic real-life examples of the sphere shapes:

1. The basketball with which kids play
2. The world globe which has been utilised in terms of teaching social studies to the kids
3. The marbles in the game of kids
4. The planets
5. The moon and several other kinds of other real-life examples are very easily prevalent in the human life.

It is also very much important for the kids to be clear about different kinds of formulas associated with the sphere shape. Being clear about the surface area formula and volume formula is very much important for the kids so that they end up making the right decisions in the real life. Apart from this it is also very much important for the kids to be clear about hemisphere because it will be the exactly half of the sphere and will be having a curved surface along with flat surface.

With the proper understanding and practice, one will face no difficulty in the calculation of the surface area of the sphere. For this, one must only trust Cuemath as they make learning of all the concepts very interesting.