Kuzee
Vector Algebra and Analytic Geometry of Space

The Sphere

The Definition of the Sphere

Sphere

A sphere is the set of points whose distance from a fixed point in space (the centre) is a fixed length (the radius).

The Geometric Meaning of a Sphere

A sphere is the locus of the points at a fixed distance from the centre, and it represents the most fundamental symmetric surface in three-dimensional space.

The Standard Equation of a Sphere

The standard equation of a sphere

The equation of the sphere with centre (a,b,c)(a, b, c) and radius rr is:

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

Special Cases

  • the centre at the origin: x2+y2+z2=r2x^2 + y^2 + z^2 = r^2
  • the centre on a coordinate axis: for example the centre (a,0,0)(a, 0, 0) gives (x−a)2+y2+z2=r2(x - a)^2 + y^2 + z^2 = r^2

The General Equation of a Sphere

The general equation of a sphere

x2+y2+z2+Ax+By+Cz+D=0x^2 + y^2 + z^2 + Ax + By + Cz + D = 0

where A2+B2+C2−4D>0A^2 + B^2 + C^2 - 4D > 0.

Simplifying the Equation

Completing the square turns the general equation into the standard one:

x2+y2+z2+Ax+By+Cz+D=0x^2 + y^2 + z^2 + Ax + By + Cz + D = 0 (x+A2)2+(y+B2)2+(z+C2)2=A2+B2+C2−4D4(x + \frac{A}{2})^2 + (y + \frac{B}{2})^2 + (z + \frac{C}{2})^2 = \frac{A^2 + B^2 + C^2 - 4D}{4}

The Properties of a Sphere

Symmetry

A sphere has complete symmetry about its centre, and it is the surface with the highest degree of symmetry in three-dimensional space.

The Nature of the Sections

The sections of a sphere

The intersection of a sphere with a plane is a circle.

  • a plane through the centre: the section is a great circle
  • a plane not through the centre: the section is a small circle

The Tangent Plane

The tangent plane of a sphere

The tangent plane at a point of the sphere is perpendicular to the radius at that point.

The line joining the centre to the point of tangency is perpendicular to the tangent plane.

The Volume Formula

The volume of the sphere is V=43πr3V = \frac{4}{3}\pi r^3

The surface area of the sphere is S=4πr2S = 4\pi r^2

Applications of the Equation of a Sphere

Distance Problems

The equation of a sphere represents the set of the points at a fixed distance from a given fixed point.

Geometric Construction

Constructing the solid formed by the points equidistant from a given point in three-dimensional space.

Exercises

Exercise 1

Find the equation of the sphere with centre (1,2,3)(1, 2, 3) and radius 44.

the equation of a spherethe standard equationcentre and radius

Approach:

Use the standard equation formula for a sphere.

Steps:

  1. The centre: (1,2,3)(1, 2, 3)
  2. The radius: r=4r = 4
  3. The standard equation: (x−1)2+(y−2)2+(z−3)2=16(x - 1)^2 + (y - 2)^2 + (z - 3)^2 = 16

Answer: (x−1)2+(y−2)2+(z−3)2=16(x - 1)^2 + (y - 2)^2 + (z - 3)^2 = 16

Exercise 2

Find the centre and the radius of the sphere x2+y2+z2−2x−4y−6z+5=0x^2 + y^2 + z^2 - 2x - 4y - 6z + 5 = 0.

the equation of a spherethe general equationcompleting the square

Approach:

Complete the square to turn the general equation into the standard one.

Steps:

  1. Completing the square: (x−1)2+(y−2)2+(z−3)2=9(x - 1)^2 + (y - 2)^2 + (z - 3)^2 = 9
  2. The centre: (1,2,3)(1, 2, 3)
  3. The radius: 9=3\sqrt{9} = 3

Answer: the centre is (1,2,3)(1, 2, 3) and the radius is 33

Summary

Symbols Used in This Article

SymbolTypePronunciation/ExplanationMeaning in This Article
(a,b,c)(a, b, c)Centre coordinatescenter coordinatesThe coordinates of the centre
rrRadiusradiusThe radius of the sphere
(x−a)2+(y−b)2+(z−c)2=r2(x-a)^2 + (y-b)^2 + (z-c)^2 = r^2Sphere equationsphere equationThe standard equation of the sphere
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