Sphere Definition
A sphere is an object that is an absolutely round geometrical shape in three-dimensional space. In this article, let us look at the sphere definition, properties and sphere formulas like surface area and volume of a sphere along with examples in detail.
Like a circle in 2D space, a sphere is a three-dimensional shape and it is mathematically defined as a set of points from the given point called “centre” with an equal distance called radius “r” in the three-dimensional space or Euclidean space. The diameter “d’ is twice the radius. The pair of points that connects the opposite sides of a sphere is called “antipodes”. The sphere is sometimes interchangeably called “ball”.
You will study the following important topics about Sphere:
- Equation
- Properties
- Formula
- Surface Area
- Volume
- Examples
Equation of a Sphere
In analytical geometry, the sphere with radius “r”, the locus of all the points (x, y, z) and centre (x_{0}, y_{0}, z_{0}), then the equation of a sphere is given as
(x -x_{0})^{2} + (y – y_{0})^{2} + (z-z_{0})^{2} = r^{2}
Properties of a sphere
The important properties of the sphere are:
- A sphere is perfectly symmetrical
- It is not a polyhedron
- All the points on the surface are equidistant from the centre.
- It does not have a surface of centres
- It has constant mean curvature
- It has a constant width and circumference.
Sphere Formula
We know that the radius is twice the radius, the diameter of a sphere formula is given as:
The diameter of a sphere, D = 2r units
Since all the three-dimensional objects have the surface area and volume, the surface area and the volume of the sphere is explained here.
Surface Area of a Sphere
The surface area of a sphere is the total area of the surface of a sphere, then the formula is written as,
The Surface Area of a Sphere(SA) = 4πr^{2} Square units
Where “r” is the radius of the sphere.
Volume of a Sphere
The amount of space occupied by the object three-dimensional object called a sphere is known as the volume of the sphere. According to the Archimedes Principle, the volume of a sphere is given as,
The volume of Sphere(V) = 4/3 πr^{3} Cubic Units
Sphere Examples
Example 1:
Find the volume of the sphere that has a diameter of 10 cm?
Solution:
Given, Diameter, d = 10 cm
We know that D = 2 r units
Therefore, the radius of a sphere, r = d / 2 = 10 / 2 = 5 cm
To find the volume:
The volume of sphere = 4/3 πr^{3} Cubic Units
V = (4/3)× (22/7) ×5^{3}
Therefore, the volume of sphere, V = 522 cubic units
Example 2:
Determine the surface area of a sphere having a radius of 7 cm
Solution:
Given radius = 7 cm
The Surface Area of a Sphere(SA) = 4πr^{2} Square units
SA = 4× (22/7)× 7^{2}
SA = 4 × 22 × 7
SA = 616 cm^{2}
Therefore, the surface area of a sphere = 616 square units.
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