Equation of A Sphere

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 + z2 = r2  and 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 + z2 = 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 + z2 = OP2

This general equation of the sphere will 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)2 = 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: Center- (-1,-2,3) 

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

Radius= 4 cm

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)2 = 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)2 = 42

(x+1)2 + (y+2)2 + (z-3)2 = 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 = \[\sqrt{(10 - 4)^{2}(3 -1)^{2}(4 -2)^{2}}\] (10−4)2+(3−1)2+(4−2)2

\[\sqrt{6^{2} + 2^{2} + 2^{2}}\]

=\[\sqrt{44}\]   44  

So, radius = 1/2 \[\sqrt{44}\] 44 

=\[\sqrt{11}\]  11

Therefore, the coordinates of the center 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)2 = 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 an 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. 

Archimedes discovered that the surface area of a sphere was the same as the lateral surface area of a cylinder with the base radius equal to that of the sphere and the height equal to the diameter of the sphere by deriving the surface area of a sphere formula.

Important Elements of a Sphere

• Radius: The distance between the sphere's center and any point on its surface measured in line segments. If 'O' is the sphere's center and A is any point on its surface, then OA is the radius of the sphere.

• Diameter: The diameter of a sphere is defined as the length of a line segment going through the center of the sphere from one point on the surface to the opposite point. The diameter of the sphere is exactly half the length of the sphere's radius.

• Circumference: The circumference of a sphere is the distance or length around the border or outside surface of the sphere.

• Volume: A sphere takes up space in the same way that every other three-dimensional object does. The volume of the sphere is the amount of space it takes up. Cubic units are used to express it.

• Surface Area: The surface area of a sphere is the area occupied by its surface. It is based on square units of measurement.

Properties of a Sphere

A sphere is a three-dimensional object with all of its points on its outside surface equidistant from the center. The following spherical features make it simple to recognize a sphere:

• A sphere is symmetrical in every direction.

• A sphere has only a curved surface area.

• The sphere's surface points are all at the same distance from the center.

• Because it lacks vertices, edges, and flat faces, a sphere is not a polyhedron. A polyhedron is an object that must have all of its faces be flat.

• Because the sphere has the smallest surface area, air bubbles adopt the form of a spherical sphere.

• The sphere would have the most volume out of all the forms with the same surface area. 4/3 r3 is the formula for sphere volume.

• A flat two-dimensional surface (plane) that goes through the center of a sphere can divide it into two equal hemispheres.