Equation of A Sphere

General Equation of Sphere

A sphere is a geometrical object in three-dimensional space that resembles the surface of a ball. Similar to a circle in two-dimensional space, a sphere can be mathematically defined as the set of all points that are at the same distance from a given point. This given point is called the center of the sphere. The distance between the center and any point on the surface of the sphere is called the radius, represented by r.

Let us suppose there are two points  A and B on the opposite sides of the sphere. If we draw a line joining these two points, passing through the center, the maximum distance measured by it will be called the diameter of the sphere, represented by d.

In the figure given below, r is the radius while d is the diameter.

The diameter divides the sphere into two equal halves, known as hemispheres. It is very similar to cutting a ball in two halves. The general equation of the sphere is x2 + y2 + z= rand in this article, we will learn about deriving the equation of a sphere along with its volume and surface area.

Deriving the Equation of A Sphere

For a circle, whose center lies at the origin (0,0), the equation is given by:

x2 + y2 = r2

In the above figure, the point whose coordinates are represented by (x,y) lies on the circumference of the circle. The figure marked in blue, is actually a right-angled triangle, with the base and perpendicular being denoted by x and y respectively, while the radius r is the hypotenuse.

Applying the Pythagoras Theorem here, we get

x2 + y2 = r2

However, if the center of the circle is not the origin of the coordinate axes, and is represented by any random point, say (h,k), then the equation of the circle becomes:

(x-h)2 + (y-k)2 = r2

In order to arrive at the equation of a sphere, we need to apply the Pythagorean theorem twice.

In the figure given above, P is a point whose coordinates are represented by (x,y,z).

MON is a right-angled triangle, whose legs are x and y, while t is the hypotenuse.

x2 + y2 = t2               ------- Eq. 1

PON is also a right-angled triangle whose legs are t and z, while r is the hypotenuse.

t2 + z2 = r2         ------- Eq. 2

Now, if we substitute the value of t2 in the second equation from the first equation, we get,

x2 + y2 + z= r2

This is called the equation of a sphere, also known as the general equation of a sphere or the equation of a sphere through the circle.

Here, the distance between the points O and P is the radius, so r in the above equation can also be written as OP.

Then, the equation would become:

x2 + y2 + z= OP2

This general equation of sphere would only stay valid if we take the origin as the center of the sphere. In case, any random point  (a,b,c) in three-dimensional space is taken as the center of the sphere, the equation of sphere would be written as,

(x-a)2 + (y-b)2 + (z-c)= r2

Volume of a Sphere:

The volume of any figure is defined as the amount of three-dimensional space occupied by it.

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

V= 4/3πr3

where r is the radius of the sphere.

Surface Area of a Sphere:

The surface area of a solid object is a measure of the total area that the surface of the object occupies.

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

A= 4πr2

where r is the radius of the sphere.

Solved Examples

Example 1

What would be the equation of a sphere through a circle in the standard form whose center and radius are given by (-1,-2,3) and 4 cm respectively?

Solution:

Given: Centre- (-1,-2,3)

These are the values of a,b,c respectively.

As we have learnt above, the general equation of a sphere in standard form is given by

(x-a)2 + (y-b)2 + (z-c)= r2

Now, substituting the values of x,y,z and r by -1,-2 and 3 and 4 respectively, we obtain

(x-(-1))2 + (y-(-2))2 + (z-3)= 42

(x+1)2 + (y+2)2 + (z-3)= 16

Example 2

The diameter of a sphere has its endpoints at (4,1,2) and (10,3,4). Find its equation.

Solution:

Since we know the endpoints of the diameter, we can calculate its length using the distance formula.

So, length of the diameter =  $\sqrt{(10-4)^{2} + (3-1)^{2} + (4-2)^{2}}$

$\sqrt{6^{2} +2^{2} + 2^{2} }$

=  $\sqrt{44}$

So, radius = 1/2  $\sqrt{44}$

= $\sqrt{11}$

Therefore, the coordinates of the centre will be ((10+4)/2, (3+1)/2, (4+2)/2) i.e. (7,2,3)

Hence, the equation becomes  (x-7)2 + (y-2)2 + (z-3)= 11

Did you know?

The shortest possible distance between any two points on a sphere is known as a geodesic.

Great circle, also known as orthodrome, is an important concept related to the sphere. It is a circle on the surface of a sphere which lies in a plane passing through the sphere’s center. The minor arc of a great circle between two points is the shortest surface path between them.

Fun Fact:

This concept is used to determine the shortest route that an aircraft or a ship can take.