A sphere is defined as a completely round geometrical object in a three-dimensional space just like a round ball. To be geometrical, a sphere is a set of points that are equidistant from a point in space. The distance between the outer point and centre of the sphere is called the radius, denoted by r and the maximum straight distance between any two sides of the sphere through the centre is known as the diameter, denoted by d.

A hemisphere is exactly half of a sphere which can only be obtained when a sphere is split from the middle. The biggest circle of a sphere is a circle that has the same centre and radius of a sphere. A great circle of the sphere is a circle that has the same radius and centre as the sphere itself. In this article, let us discuss how to derive the equation of a sphere along with the surface area and the volume of the sphere in detail.

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## How to Derive the Equation of a Sphere?

The equation of a circle of radius r is given by:

x^{2} + y^{2} = r^{2}

You can relate it to the algebraic method of starting the Pythagoras theorem.

The point (x,y) lies on the circle only when the right triangle has sides of length |x| and |y| and hypotenuse of length r, which can be written as:

x^{2} + y^{2} = r^{2}

Pythagoras theorem can be used twice for the equation of a sphere. In the below figure, O is the origin and P(x,y,z) is a point in three-space. P is on the sphere with radius r only when the distance from O to P is r.

Since OAB is a right angle triangle, x^{2} + y^{2} = s^{2}. The triangle OBP is another right triangle and therefore, s^{2} + z^{2} = r^{2}. Hence, the distance between O and P can be expressed by:

x^{2} + y^{2} + z^{2} = |OP|^{2}

Hence, we can conclude that (x,y,z) lies on the sphere with radius r only if,

**x ^{2} + y^{2} + z^{2} = r^{2}**

which is called the equation of a sphere.

If (a, b, c) is the centre of the sphere, r represents the radius, and x, y, and z are the coordinates of the points on the surface of the sphere, then the general equation of a sphere is (x – a)² + (y – b)² + (z – c)² = r²

## Volume of a Sphere Equation

The formula to calculate the volume of a sphere is given by the equation:

The volume of the sphere = \(\frac{4}{3} \pi r^{3}\)

Where r is the radius of the sphere.

## Surface Area of a Sphere Equation

The formula to calculate the surface area of the sphere is given by:

The Surface area of the sphere= \( 4 \pi r^{2}\) square units.

### Equation of a Sphere Example

**Example: **

Write the equation of the sphere in the standard form where the centre and radius of the sphere are given as (11, 8, -5) and 5 cm respectively.]

**Solution:**

Given: Centre = (11, 8, -5) = (a, b, c)

Radius = 5 cm

We know that the equation of the sphere in the standard form is written as:

(x-a)^{2} + (y-b)^{2} + (z-c)^{2} = r^{2}

Now, substitute the given values in the above form, we get:

(x-11)^{2} + (y-8)^{2} + (z -(-5))^{2} =5^{2}

(x-11)^{2} + (y-8)^{2} + (z +5)^{2} = 25

Thus, the equation of the sphere is (x-11)^{2} + (y-8)^{2} + (z +5)^{2} = 25

To know more about the properties of spheres and along with operations and problems, you can visit BYJU’S – The Learning App.