Question

What is the total surface area of a sphere?

Answer 1

Hint: Sphere is a three-dimensional object symmetric in all directions. Surface areas of the sphere do not include the height term within it. Centre is a point inside the sphere from which the distance of the set of all points on the surface of the sphere is the same and known as the radius of the sphere.

Complete Step-by-Step solution:
As we know that a sphere is a perfectly round 3-dimensional object. It can be characterized as a set of all points located at distance r away from a given point, where distance r and point inside the sphere are termed as radius of sphere and centre of sphere respectively. It is perfectly symmetrical, and has no edges or vertices. So, a sphere is a geometrical object in three-dimensional space.
As we read about circles in two-dimensional similarly, we read about spheres in a three- dimensional space. Same terminologies that are radius, distance and diameter are used for a circle and for a sphere as well.
Now, we know about surface area that it measures the total area that the surface of the object occupies in a (3-D shape) object. It is useful because it tells us how much material is required in order to cover the object.
So, we know the sphere has only one characteristic with measurements that is radius. So, total surface area of sphere will be given in terms of radius only and hence, that would be given as
Total Surface area of sphere $=4\pi r^2$
Here, the value of $\pi$ is ${\displaystyle \frac{22}{7}}$ and r is the radius of the sphere. IT can be proved using the integration in two directions by taking a small circle in 2-D and hence integrate it from 0 to r. But we are not required to prove it as per the question. So, the total surface area of the sphere is $4\pi r^2$.
Where value of $\pi ={\displaystyle \frac{22}{7}}$ and $r=radius$

Note: Don’t confuse the lateral and total surface for the sphere. Both will be the same as there is not any lateral/vertical surface in the sphere as the length of the radius is the same in all directions. So, it is symmetric in 3-Dimensional.
Remember that sphere has only one measure that is its radius. Don’t include height factor with the surface area of the sphere and don’t confuse it with the areas of cylinder, cone, cuboid or any other geometrical shape as they include height factor as well. So, remembering the total surface area of the sphere be $4\pi r^2$ for future references.