# Sphere

on . Posted in Solid Geometry

• Sphere (a three-dimensional figure) has all points equally spaces from a given point of a three dimensional solid.
• Lune of a sphere is the space occupied by a wedge from the center of the sphere to the surface of the sphere.
• Sector of a sphere is the space occupied by a portion of the sphere with the vertex at the center and conical boundary.
• Segment and zone of a sphere is the space occupied by a portion of the sphere cut by two parallel planes.
• Sperical cap is the space occupied by a portion of the sphere cut by a plane.
• See Artical Link  -  Moment of Inertia of a Sphere

## Circumference of a Sphere formula

$$\large{ C = 2 \; \pi \; r }$$     (Circumference of a Sphere)

$$\large{ r = \frac{ C }{ 2 \; \pi } }$$

### Solve for r

 circumference, C

Symbol English Metric
$$\large{ C }$$ = circumference $$\large{ in }$$ $$\large{ mm }$$
$$\large{ \pi }$$ = Pi  $$\large{3.141 592 653 ...}$$
$$\large{ r }$$ = radius $$\large{ in }$$ $$\large{ mm }$$
$$\large{ d }$$ = diameter $$\large{ in }$$ $$\large{ mm }$$

## Diameter of a Sphere formula

$$\large{ d = 2\;r }$$     (Diameter of a Sphere)

$$\large{ r = \frac{d}{2} }$$

### Solve for r

 diameter, d

Symbol English Metric
$$\large{ d }$$ = diameter $$\large{ in }$$ $$\large{ mm }$$
$$\large{ r }$$ = radius $$\large{ in }$$ $$\large{ mm }$$

## RADIUS of a sphere formula

$$\large{ r = \sqrt{ \frac{ S }{ 4 \; \pi } } }$$

$$\large{ r = \sqrt{ \frac{ 3 }{ 4 } \; \frac{ V }{ \pi } } }$$

Symbol English Metric
$$\large{ r }$$ = radius $$\large{ in }$$ $$\large{ mm }$$
$$\large{ \pi }$$ = Pi  $$\large{3.141 592 653 ...}$$
$$\large{ S }$$ = surface area $$\large{ in^2 }$$ $$\large{ mm^2 }$$
$$\large{ V }$$ = volume $$\large{ in^3 }$$ $$\large{ mm^3 }$$

## Surface Area of a sphere formula

$$\large{ S = 4\; \pi \;r^2 }$$
Symbol English Metric
$$\large{ S }$$ = surface area $$\large{ in^2 }$$ $$\large{ mm^2 }$$
$$\large{ \pi }$$ = Pi  $$\large{3.141 592 653 ...}$$
$$\large{ r }$$ = radius $$\large{ in }$$ $$\large{ mm }$$

## Volume of a sphere formula

$$\large{ V = \frac{4}{3} \; \pi \;r^3 }$$

$$\large{ V = \frac{ \pi \; d^3 }{ 6 } }$$

Symbol English Metric
$$\large{ V }$$ = volume $$\large{ in^3 }$$ $$\large{ mm^3 }$$
$$\large{ \pi }$$ = Pi  $$\large{3.141 592 653 ...}$$
$$\large{ r }$$ = radius $$\large{ in }$$ $$\large{ mm }$$

## Surface Area of a sphere Cap formula

$$\large{ S = 2 \; \pi \; r \; h }$$
Symbol English Metric
$$\large{ S }$$ = surface area $$\large{ in^2 }$$ $$\large{ mm^2 }$$
$$\large{ h }$$ = height $$\large{ in }$$ $$\large{ mm }$$
$$\large{ \pi }$$ = Pi  $$\large{3.141 592 653 ...}$$
$$\large{ r }$$ = radius $$\large{ in }$$ $$\large{ mm }$$

## VOLUME of a sphere Cap formula

$$\large{ V = \frac {1}{3} \; \pi\;h^2 \left( 3 r - h \right) }$$

$$\large{ V = \frac {1}{6} \; \pi\;h \left( 3 r_1{^2} + h^2 \right) }$$

Symbol English Metric
$$\large{ V }$$ = volume $$\large{ in^3 }$$ $$\large{ mm^3 }$$
$$\large{ h }$$ = height $$\large{ in }$$ $$\large{ mm }$$
$$\large{ \pi }$$ = Pi  $$\large{3.141 592 653 ...}$$
$$\large{ r }$$ = radius $$\large{ in }$$ $$\large{ mm }$$
$$\large{ r_1 }$$ = radius $$\large{ in }$$ $$\large{ mm }$$

## Surface Area of a sphere Segment formula

$$\large{ S = 2 \; \pi \; r \; h }$$
Symbol English Metric
$$\large{ S }$$ = surface area $$\large{ in^2 }$$ $$\large{ mm^2 }$$
$$\large{ h }$$ = height $$\large{ in }$$ $$\large{ mm }$$
$$\large{ \pi }$$ = Pi  $$\large{3.141 592 653 ...}$$
$$\large{ r }$$ = radius $$\large{ in }$$ $$\large{ mm }$$

## VOLUME of a sphere Segment formula

$$\large{ V = \frac {1}{6} \; \pi\;h \left( 3 r_1{^2} + 3 r_2{^2} + h^2 \right) }$$
Symbol English Metric
$$\large{ V }$$ = volume $$\large{ in^3 }$$ $$\large{ mm^3 }$$
$$\large{ h }$$ = height $$\large{ in }$$ $$\large{ mm }$$
$$\large{ \pi }$$ = Pi  $$\large{3.141 592 653 ...}$$
$$\large{ r }$$ = radius $$\large{ in }$$ $$\large{ mm }$$
$$\large{ r_1 }$$ = radius $$\large{ in }$$ $$\large{ mm }$$
$$\large{ r_2 }$$ = radius $$\large{ in }$$ $$\large{ mm }$$

## Surface Area of a sphere WEDGE formula

$$\large{ S = \frac{ \theta }{ 360 } \; 4 \; \pi \; r^2 }$$
Symbol English Metric
$$\large{ S }$$ = surface area $$\large{ in^2 }$$ $$\large{ mm^2 }$$
$$\large{ \theta }$$ = angle $$\large{ deg }$$ $$\large{ rad }$$
$$\large{ \pi }$$ = Pi  $$\large{3.141 592 653 ...}$$
$$\large{ r }$$ = radius $$\large{ in }$$ $$\large{ mm }$$

## Volume of a sphere WEDGE formula

$$\large{ V = \frac{ \theta }{ 2 \; \pi } \; \frac{ 4 }{ 3 } \; \pi \; r^2 }$$
Symbol English Metric
$$\large{ V }$$ = volume $$\large{ in^3 }$$ $$\large{ mm^3 }$$
$$\large{ \theta }$$ = angle $$\large{ deg }$$ $$\large{ rad }$$
$$\large{ \pi }$$ = Pi  $$\large{3.141 592 653 ...}$$
$$\large{ r }$$ = radius $$\large{ in }$$ $$\large{ mm }$$

## VOLUME of a sphere SECTOR formula

$$\large{ V = \frac {2}{3}\; \pi \; r^2\;h }$$
Symbol English Metric
$$\large{ V }$$ = volume $$\large{ in^3 }$$ $$\large{ mm^3 }$$
$$\large{ h }$$ = height $$\large{ in }$$ $$\large{ mm }$$
$$\large{ \pi }$$ = Pi  $$\large{3.141 592 653 ...}$$
$$\large{ r }$$ = radius $$\large{ in }$$ $$\large{ mm }$$
$$\large{ r_1 }$$ = radius $$\large{ in }$$ $$\large{ mm }$$ 