What is a Sphere?
A sphere is a geometrical shape in 3-dimensional space that is equidistant from a fixed point and does not have any vertex. In simple words, a sphere is a round shaped 3-dimensional figure. The fixed point is known as the center of the sphere. The fixed distance from the center to its surface is called the radius of that sphere. The geometrical figure of a sphere is as in the image.
Some real life examples of spheres are basketballs, globes, cheese balls, and many more.
Properties of Spheres
- A sphere is symmetrical in shape.
- A sphere is not a polyhedron.
- All the points that lie on the surface of the sphere are at the same distance from the center.
- The sphere is curved in shape.
- A sphere has no vertex.
Hemi-Sphere
If we divide the sphere into two equal parts, then each of these parts are known as hemispheres.
Surface Area of Sphere
The surface area of a sphere is the total area covered by the sphere in space. If r is the radius of the sphere then surface area of sphere is \(4\pi r^2\).
Surface area of sphere = \(4\pi r^2\).
Volume of the Sphere
The volume of a sphere is the amount of space that has been occupied by the sphere. If r is the radius of the sphere then volume of sphere is \(\frac{4}{3}\pi r^3\).
Volume of sphere = \(\frac{4}{3}\pi r^3\).
Surface Area and Volume of Similar Spheres
All spheres are similar in shape. Also there is a relationship between the volume and surface area of two spheres.
Both the spheres shown here are similar, and the radius of the first and second sphere are r and R respectively.
The relationship between the surface area of these two spheres is,
\(\frac{Surface~Area~of~1st~Sphere}{Surface~Area~of~2nd~Sphere}=\left(\frac{r}{R}\right)^2\)
The relationship between the volume of the two spheres is,
\(\frac{Volume~of~1st~Sphere}{Volume~of~2nd~Sphere}=\left(\frac{r}{R}\right)^3\)
Solved Examples on Sphere
Example 1: Find the volume of a sphere whose radius is 22 cm.
Solution:
The radius of the sphere is 22 cm [Given]
Volume of a sphere = \(\frac{4}{3}\pi r^3\) [Volume formula of a sphere]
= \(\frac{4}{3}\pi (22)^3\) [Substitute r = 22]
= 44602.19 [Simplify]
Hence, the volume of the given sphere is 44602.19 \(cm^3\).
Example 2: Find the surface area of a sphere whose radius is 7 cm.
Solution:
The radius of the sphere is 7cm [Given]
Surface area of Sphere = \(4 \pi r^2\) [Surface area formula of a sphere]
= \(4 \pi (7)^2\) [r = 7]
= 4310.27 [Simplify]
So, the surface area of given sphere is 4310.27 \(cm^2\).
Example 3: Sam has a basketball whose surface area is 225 \(cm^2\). He also has a tennis ball. The radius of the basketball is 3 times the radius of the tennis ball. What is the surface area of the tennis ball?
Solution:
The surface area of the basketball is 225 \(cm^2\)
Let the radius of the tennis ball be r.
So, the radius of the basketball is 3r.
Since the basketball and tennis ball are similar. So,
\(\frac{Surface~Area~of~basketball}{Surface~Area~of~tennis~ ball}=\left(\frac{3r}{r}\right)^2\)
\(\frac{225}{Surface~Area~of~tennis~ ball}=\left(\frac{3}{1}\right)^2\)
Surface area of tennis ball = \(\frac{225}{9}\)
Surface area of tennis ball = 25
Therefore, the surface area of tennis ball is 25 \(cm^2\).
A sphere is a geometrical shape in 3-dimensional space that is equidistant from a fixed point and does not have any vertex.
The surface area of a sphere is the total area of the outer surface of the sphere.
The volume of a hemisphere is half of the volume of a sphere.
The circle is a 2-dimensional closed round shape and a sphere is a 3-dimensional closed round shape.