Analytical Geometry-Three Dimensions-Equation of a Sphere

Introduction

In analytical geometry, also known as coordinate geometry, coordinate systems are used to study geometry. Geometry is the branch of mathematics that deals with the measurement of objects having different shapes, sizes, and angles. In analytical geometry in three dimensions, the equation of a sphere represents a sphere in its algebraic expression form.  The coordinate system represents the different shapes in the plane of coordinate points using lines and curves. The three dimensions-equation of a sphere is represented by centre points and the radius of the sphere in the coordinate system in three-dimensional space. The equation of the sphere in three dimensions is represented using the centre point and radius of the sphere. 

The three dimensions-equation of a sphere helps us represent the sphere in the equation form. The general equation of a sphere is,

(x-a)2+(y-b)2+(z-c)2=r2

Here, (a,b,c) represents the centre of the sphere, and r represents the radius of the sphere.

Analytical geometry is the method in which algebraic symbols and equations are used to solve the geometrical problem of the coordinate system. Analytical geometry is the study based on the modelling of geometry objects on the coordinate system. In analytical geometry, two-dimensional and three-dimensional spaces are used. Analytical geometry is the combined study of geometry and coordinate system.

Two-dimensional spaces

In analytical geometry, two-dimensional space is a plane shape geometry that has two-dimensional length and width. The two-dimensional space lies in the same plane; for example, rectangle, square, circle, etc.

Three-dimensional spaces

In analytical geometry, three-dimensional space is a solid geometry with three dimensions in the direction of longitude, latitude, and altitude. For example, pyramids, prisms, spheres, etc. Three dimensions are the linear arrangement of three independent vectors in space.

Sphere

A sphere is a three-dimensional solid in which each point on the surface is equidistant from the centre. A sphere is a solid object that is entirely round in shape. For example, a football and a cricket ball are examples of a sphere.

The Centre of a sphere

The centre of a sphere is a point inside the sphere such that all the points at the surface of a solid sphere are equidistant from that point.

The Radius of a sphere

The radius of a sphere is the line segment drawn between the centre of the sphere to any point at the surface of the sphere.

Three Dimensions-Equation of a sphere

The equation of a sphere is the representation of a sphere in the equation form.

The equation of a sphere represents the sphere by centre points and the radius of the sphere in the coordinate system. The general three dimensions-equation of a sphere is,

(x-a)2+(y-b)2+(z-c)2=r2

Here, (a,b,c) represents the centre of the sphere, and r represents the radius of the sphere.

Derivation for three dimensions-equation of a sphere

Suppose A is a constant point in three-dimensional space. Points (a,b,c) are the coordinate points of A. Let r be a positive real number in the space, and another point O, which is moving, has the coordinates as (x,y,z).

The distance OA is equal to the constant r.

OA=r

On squaring both sides of the equation, we get,

(OA)2=r2

According to the distance formula,

(x-a)2+(y-b)2+(z-c)2=r2

The above equation is the equation of a sphere with centre point O (x,y,z) and radius r.

When the centre of the sphere is at the origin of the coordinate system, then the coordinate of the centre of the sphere is O (0, 0, 0). The equation of the sphere for this is given as,

(x-0)2+(y-0)2+(z-0)2=r2

x2+y2+z2=r2

This is the three dimensions-equation of a sphere, when its centre is at the origin. The radius of the sphere for the same case is,

r=x2+y2+z2

Here, we have seen how the equation of the sphere is obtained for different values of centre point and radius. Let’s check out some examples/questions, which are solved using the equation of the sphere.

Examples

• What would be the equation of a sphere through a circle in a standard form whose coordinate points of the centre are (1,2,-3) and radius is 2 cm.

Solution:

Given; 

Centre point of the sphere (1,2,-3)

Radius of the sphere is 2 cm.

The standard form of equation of the sphere is,

(x-a)2+(y-b)2+(z-c)2=r2

Let’s put the values in the above equation.

(x-1)2+(y-2)2+(z-(-3))2=(2)2

(x-1)2+(y-2)2+(z+3))2=4

This is the equation of the sphere for the given centre point and radius.

• The endpoints of the diameter of a sphere are (8,2,4) and (20, 6, 8). Find the equation of the sphere.

Solution:

Given; 

The endpoint of the diameter of a sphere is (8,2,4) and (20, 6, 8)

From the distance formula the length of the diameter is,

d=(20-8)2+(6-2)2+(8-4)2

d=(12)2+(4)2+(4)2

d=(12)2+(4)2+(4)2

d=176

Thus, radius 

r=1/2 176

r=44

The coordinates of the centre are,

[(20+8)/2,(4+2)/2,(8+4)/2] which is (14, 4, 6)

The standard form of equation of the sphere is,

(x-a)2+(y-b)2+(z-c)2=r2

Let’s put the values in the above equation.

(x-14)2+(y-8)2+(z-6)2=(44)2

(x-14)2+(y-8)2+(z-6))2=44

This is the equation of the sphere.

• Write the equation of a sphere having a circle at a coordinated point (2,2,2) and radius 2 cm.

Solution:

Given; 

Centre point of the sphere (2,2,2)

Radius of the sphere is 2 cm.

The standard form of the three dimensions-equation of a sphere is,

(x-a)2+(y-b)2+(z-c)2=r2

Let’s put the values in the above equation.

(x-2)2+(y-2)2+(z-2)2=(2)2

(x-2)2+(y-2)2+(z-2))2=4

This is the equation of the sphere for the given centre point and radius.

Conclusion

This article has discussed analytical geometry and its type as two-dimensional and three-dimensional spaces. We also learnt about the various aspects, including the radius, centre, and equation of a sphere.

(x-a)2+(y-b)2+(z-c)2=r2

By the end of this article, we know how to use the three dimensions-equation of a sphere to solve geometric problems using a coordinate system. There are several more three-dimensional shapes like cylinders, cones, and rectangles whose equations play an important role in coordinate geometry.