Sphere is a solid bounded by closed surface every point of which is equidistant from a fixed point called the center.
 

Sphere with small and great circles shown

 

Properties of a Sphere

  • Every section in the sphere made by a cutting plane is a circle. If the cutting plane passes through the center of the sphere, the section made is a great circle; otherwise the section is a small circle.
  • For a particular circle of a sphere, the axis is the diameter of the sphere perpendicular to the plane of the circle.
  • The ends of the axis of the circle of a sphere are called poles.
  • The nearer the circle to the center of the sphere, the greater is its area.
  • The largest circle in the sphere is the great circle.
  • The radius (diameter) of the great circle is the radius (diameter) of the sphere.
  • All great circles of a sphere are equal.
  • Every great circle divides the sphere into two equal parts called hemispheres.
  • Two intersecting spheres form a circular sectionThe intersection of two spherical surfaces is a circle whose plane is perpendicular to the line joining the centers of the spheres and whose center is on that line. (See figure to the right.)
  • A plane perpendicular to a radius at its extremity is tangent to the sphere.

 

Volume and Surface Area of a Sphere

Surface Area, A
The surface area of a sphere is equal to the area of four great circles.
$A = 4\pi R^2$

$A = \pi D^2$

 

Volume, V

$V = \frac{4}{3}\pi R^3$

$V = \frac{1}{6}\pi D^3$

 

Spherical Sector

A spherical sector is a solid generated by revolving a sector of a circle about an axis which passes through the center of the circle but which contains no point inside the sector. If the axis of revolution is one of the radial sides, the sector thus formed is spherical cone; otherwise, it is open spherical sector.
 

Open Spherical Sector and Spherical Cone

 

Properties of Spherical Sector

  • Spherical sector is bounded by a zone and one or two conical surfaces.
  • The spherical sector having only one conical surface is called a spherical cone, otherwise it is called open spherical sector.
  • The base of spherical sector is its zone.

 

Volume and Surface Area of Spherical Sector

Total surface area, A
The total surface area of a spherical sector is equal to the area of the zone plus the sum of the lateral areas of the bounding cones.

Surface area = zone + lateral area of bounding cones
$A = A_{zone} + A_1 + A_2$

$A = 2\pi Rh + \pi aR + \pi bR$

$A = \pi R(2h + a + b)$

 

Note that for spherical cone, b = 0 and the equation will reduce to

$A = \pi R(2h + a)$

 

Volume, V
The volume of spherical sector, either open spherical sector or spherical cone, is equal to one-third of the product of the area of the zone and the radius of the sphere. This is similar to the volume of a cone which is Vcone = (1/3) Abh. In spherical sector, replace Ab with Azone and h with R.

$V = \frac{1}{3} A_{zone} R$

$V = \frac{2}{3} \pi R^2 h$

 

Spherical Segment

Spherical segment is a solid bounded by two parallel planes through a sphere. In terms of spherical zone, spherical segment is a solid bounded by a zone and the planes of a zone's bases.
 

Spherical segment of two bases

 

Properties of Spherical Segment
spherical-segment-one-base.jpg

  • The bases of a spherical segment are the sections made by the parallel planes. The radii of the lower and upper sections are denoted by a and b, respectively. If either a or b is zero, the segment is of one base. If both a and b are zero, the solid is the whole sphere.
  • If one of the parallel planes is tangent to the sphere, the solid thus formed is a spherical segment of one base.
  • The spherical segment of one base is also called spherical cap and the two bases is also called spherical frustum.
  • The altitude of the spherical segment is the perpendicular distance between the bases. It is denoted by h.

 

Formulas for Spherical Segment

Area of lower base, A1
$A_1 = \pi a^2$

 

Area of upper base, A2
$A_2 = \pi b^2$

 

Area of the zone, Azone
$A_{zone} = 2\pi Rh$

 

Total Area, A
The total area of segment of a sphere is equal to area of the zone plus the sum of the areas of the bases.

$A = A_{zone} + A_1 + A_2$

$A = 2\pi Rh + \pi a^2 + \pi b^2$

$A = \pi(a^2 + b^2 + 2Rh)$

 

Volume, V
The volume of spherical segment of two bases is given by
$V = \frac{1}{6}\pi h(3a^2 + 3b^2 + h^2)$

 

The volume of spherical segment of one base is given by
$V = \frac{1}{3}\pi h^2 (3R - h)$

 

Figure for the derivation of formula of spherical segment of one base from segment of two basesThe formula for the volume of one base can be derived from volume of two bases with b = 0. Consider the following diagram:
$a^2 + (R - h)^2 = R^2$

$a^2 + (R^2 - 2Rh + h^2) = R^2$

$a^2 = 2Rh - h^2$
 

Substitute a2 = 2Rh - h2 and b = 0 to the formula of spherical segment of two bases
$V = \frac{1}{6}\pi h(3a^2 + 3b^2 + h^2)$

$V = \frac{1}{6}\pi h \, [ \, 3(2Rh - h^2) + 3(0^2) + h^2 \, ]$

$V = \frac{1}{6}\pi h \, [ \, 6Rh - 3h^2 + h^2 \, ]$

$V = \frac{1}{6}\pi h \, [ \, 6Rh - 2h^2 \, ]$

$V = \frac{1}{6}\pi h (2h) [ \, 3R - h \, ]$

$V = \frac{1}{3}\pi h^2 (3R - h)$       (okay!)
 

Note also that the volume of segment of a sphere of altitude h and radii a and b is equal to the volume of a sphere of radius h/2 plus the sum of the volumes of two cylinders whose altitudes are h/2 and whose radii are a and b, respectively.
 

Spherical Wedge and Spherical Lune

A spherical wedge is a solid formed by revolving a semi-circle about its diameter by less than 360°. Spherical Lune is the curve surface of the wedge, it is a surface formed by revolving a semi-circular arc about its diameter by less than 360°.
 

Spherical Wedge and Spherical Lune

 

Formula for Spherical Wedge and Spherical Lune

The formula for spherical wedge and Lune can be found by ratio and proportion, meaning, spherical wedge is similar to the sphere and spherical Lune is similar to spherical surface.
 

Volume of wedge, Vwedge
Volume of wedge / central angle = Volume of sphere / 1 revolution

$\dfrac{V_{wedge}}{\theta_{deg}} = \dfrac{\frac{4}{3}\pi R^3}{360^\circ}$

$V_{wedge} = \frac{1}{270} \pi R^3 \theta_{deg}$

$V_{wedge} = \frac{2}{3} R^3 \theta_{rad}$

 

Area of Lune, Alune
Area of Lune / central angle = Area of sphere / 1 revolution

$\dfrac{A_{lune}}{\theta_{deg}} = \dfrac{4\pi R^2}{360^\circ}$

$A_{lune} = \frac{1}{90} \pi R^2 \theta_{deg}$

$A_{lune} = 2 R^2 \theta_{rad}$

 

Spherical Zone

A zone is that portion of the surface of the sphere included between two parallel planes.
 

Spherical zone of one base and two bases

 

Properties of Spherical Zone

  • The bases of the zone are the circumference of the sections made by the two parallel planes.
  • The altitude of the zone is the perpendicular distance between these two parallel planes.
  • If one of the bounding parallel planes is tangent to the sphere, the surface bounded is a zone of one base.

 

Area of the Zone

The area of any zone (one base or two bases) is equal to the product of its altitude h and the circumference of the great circle of the sphere.

$A_{zone} = \text{circumference of great circle } \times \text{ altitude}$

$A_{zone} = 2\pi Rh$

Note that when h = 2R, the area of the zone will equal to the total surface area of the sphere which is 4πR2.
 

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