Sphere

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.

Sphere Definition

Sphere

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”.

Equation of a Sphere

In analytical geometry, the sphere with radius “r”, the locus of all the points (x, y, z) and centre (x0, y0, z0), then the equation of a sphere is given as

(x -x0)2 + (y – y0)2 + (z-z0)2 = r2

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) = 4πr2 Square units

Where “r” is the radius of the sphere.

Volume of a 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 πr3 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 πr3 Cubic Units

V = (4/3)× (22/7) ×53

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) = 4πr2 Square units

SA = 4× (22/7)× 72

SA = 4 × 22 × 7

SA = 616 cm2

Therefore, the surface area of a sphere = 616 square units.

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Practise This Question

With 200 roses, a person starts walking from his home to the temple, which is 50 m far from his home.  After walking each meter, he drops 2 roses. Form a series for the number of roses the person has after walking each meter.