Surface Area of a Sphere

Introduction :

The surface area is defined as the sum of the areas of the closed surface. A three- dimensional shape is a solid that has depth and height. A prism is a three-dimensional shape that has non-curved sides. A cube is a prism, but a sphere is not because the sphere has curved surfaces. A prism has a pair of congruent sides, called bases (like the cube, triangular prism and the rectangular prism). A prism is different from a pyramid because a pyramid has one base.

 

In other words, the surface area is a total of all the areas of each side of a closed surface. Consider the shape of a birthday present. In this case, the wrapping paper is its surface area. Now if we take off the wrapping paper cautiously and add up all the sides’ area, it would give the total area. Thus the total surface area is the sum of the area of all the surfaces.

 

Definition of the Surface Area of the Sphere:

A sphere is a perfectly round three-dimensional geometrical shape. Its properties are analogous to circle. The point drawn from at equal distance from the origin in space is a sphere. Most of the things around us are spherical. The study of spheres is basic to terrestrial geography and is one of the principal areas of  Euclidean geometry and elliptic geometry. The formula of surface area was discovered over two thousand years ago by Archimedes. The Greek mathematician Archimedes discovered that the surface area of a sphere is the same as the lateral surface area of a cylinder having the same radius as the sphere and height the length of the diameter of the sphere. The surface area of the sphere is defined as the number of square units required to cover the surface. 

 

Terms of a Sphere:

Radius: Usually it is denoted by ‘r’. It is the distance from the centre of the sphere to any point on the surface of the sphere.

 

Diameter: The diameter is the longest line on the sphere that touches the ends of the sphere and passes from the centre of the sphere. It is twice the radius of the sphere.

Pi is denoted by \[\pi\] and has a value as 3.14 or \[\frac{22}{7}\]. It is a constant.

 

The surface area of sphere formula:

Area = 4\[\pi\]r

where ‘r’ is the radius of the sphere.

 

Interesting Facts:

  • The sphere is the shape that has the smallest area for a given volume. Due to this reason bubbles, water droplets are of the shape of a sphere.

  • The area of a sphere is exactly equal to four times the area of a circle having the same radius.

  • A sphere is perfectly symmetrical around its center.

  • The diameter is the longest chord on the sphere which is twice the radius of the sphere.

  • If you cut the sphere from the middle you will get two equal hemispheres having the same radius as that of a sphere.

  • It has no edges.

  • It is not a polyhedron.

There are various formulas for finding the surface area of a cube, cuboid, prism, etc.

 

The Surface Area of a Cube:

The surface area of a cube is 6 times the area of one side that is 6 times the squared of one side as all the sides of the cube are equal.                                           

Area= 6 x a2

Where ‘a’ is the side of the cube.

The Surface Area of a Rectangular Prism:

It is the area of six rectangles that covers it. But we don't need to find the area of all the six because we know the area of the top is equal to area of bottom and area of the back is equals to the area of the front and left is equal to the right side.

Area = 2 (ab + bc + ca)

where ‘a’, ‘b’, and ‘c’ are the sides of the rectangular prism.

 

The Surface Area of a Cylinder:

In the case of a cylinder, let us consider a can. So for finding the area of a can, all the sides should be covered. Therefore, in case of a can having a circular top and a bottom and the labelled section is curved.

Area= area of top + area of bottom + area of curved surface

Area = \[\pi\] x r2 + \[\pi\] x r2 + 2 x \[\pi\] x r x h

Area = 2(\[\pi\]r2) + 2\[\pi\]rh

where ‘r’ is the radius of the cylinder and ‘h’ is the height of the cylinder.

 

Tip: 

  • You will only get the correct answer using this formula when the units are straight that is same.

 

For Example:

Find the area of a cube having sides of 2inches.

Solution:

Area = 6 x 22 = 24 inch2

  • Do the conversion of the units if required.

 

Example:

Find the surface area of the sphere whose radius is 22cm.

Solution:

Surface Area= 4\[\pi\]r2

4 x \[\pi\] x 22 x 22= 1936 \[\pi\] cm

 

Example :

Find the radius of the sphere if the surface area of the sphere is 220inch2.

Solution:

Since we know surface area of sphere =  4\[\pi\]r2

220 = 4 x \[\frac{22}{7}\]x r2

r2= 17.5

r= 4.18 inch

Hence the radius of the sphere is 4.18 inches.

 

Example:

If 64 identical small spheres are made out of a big sphere of diameter 8 cm, what is the surface area of each small sphere?

Solution:

Let the radius of small sphere be r cm

Hence, the volume of 64 small spheres is equal to volume if 1 big sphere 64 x \[\frac{4\pi{r^{3}}}{3}\] = \[\frac{4\pi \times 4^{3}}{3}\]

On solving we get,

64r3 = 64

r = 1 cm

The surface area of the small circle = 4\[\pi\]cm

 

Example:

If the ratio of the diameters of two spheres is 2: 10, then what is the ratio of their surface areas? \[\frac{\text{Surface area of 1}}{\text{Surface area of 2}}\] =

Solution:

\[\frac{4\pi(2x\times 2x)}{4\pi(10x\times 10x)}\] = \[\frac{1}{25}\]

Hence, the required ratio of their surface areas is 1:25.

 

Example :

Find the radius of the sphere if the surface area of the sphere is 300 inch2.

Solution:

Since we know the surface area of sphere =  4\[\pi\]r2

300 = 4 x \[\frac{22}{7}\]x r2

r2= 23.86

r= 4.88 inch

Hence the radius of the sphere is 4.88 inches.