UnitMath Example: ( Geometric Solids )

Last Modified: 9/10/2000

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The following can be found from the lengths of the sides of a box.

`
volume: length * width * height;
SurfaceArea: 2 * ( length * width + length * height + width * height );
diagonal: sqrt( length^2 + width^2 + height^2 );
`

volume: length * width * height;

SurfaceArea: 2 * ( length * width + length * height + width * height );

diagonal: sqrt( length^2 + width^2 + height^2 );

length: 2 feet; height: 18 inches; width: yard;

volume as cu feet; = 9 cu feet

SurfaceArea as sq feet; = 27 sq feet

diagonal as feet; ~ 3.91 feet

Find the volume, surface area, and diagonal of a room; given, length: 30 feet, height: 8 ft, width: 16 ft;

length: 30 feet; height: 8 ft; width: 16 ft;

volume as cu yd; = ( 142 + 2/9 ) cu yd

SurfaceArea as sq yd; = ( 188 + 4/9 ) sq yd

diagonal as yd; ~ 11.64 yd

The following can be found from the radius and height of a can.

`
volume: pi * radius^2 * height;
SurfaceArea: 2 pi radius ( radius + height );
`

volume: pi * radius^2 * height;

SurfaceArea: 2 pi radius ( radius + height );

height: ( 4.5 to 5) inches;

diameter: (2.5 to 3 ) inches;

radius: diameter/2;

volume as ml; ~ ( 344. to 462. to 579. ) ml

volume as fl_oz ; ~ ( 12. to 20. ) fl_oz

SurfaceArea as sq inches; ~ ( 44. to 61. ) sq inches

SurfaceArea as sq cm; ~ ( 282. to 395. ) sq cm

Find the volume, and surface area, of a cylindrical tower; given, height: 50 feet, diameter: 30 feet.

height: 50 feet; diameter: 30 feet;

volume as gallons; ~ 264,383. gallons

volume as cu yards; ~ 1,309. cu yards

SurfaceArea as sq ft; ~ 6,126. sq ft

SurfaceArea as sq m; ~ 569. sq m

The volume of the above figure is given by:.

`
volume: Length * ( acos( ( Radius - Height )/ Radius ) * Radius^2 - sqrt( Height * ( Diameter - Height ) ) * ( Radius - Height ) );`

Length: 5 inches;

Diameter: 3 inches;

Radius: Diameter / 2;

fullVolume: pi sq Radius * Length as fl_oz; ~ 19.58 fl_oz

Height: 0 inch; volume as fullVolume = 0 fullVolume

Height: Diameter/8; volume as fullVolume ~ 0.072 fullVolume

Height: Diameter/4; volume as fullVolume ~ 0.196 fullVolume

Height: Diameter/2; volume as fullVolume = ( 1/2 ) fullVolume

Height: 3/4 Diameter; volume as fullVolume ~ 0.804 fullVolume

Height: 7/8 Diameter; volume as fullVolume ~ 0.928 fullVolume

Height: Diameter; volume as fullVolume = fullVolume

Find the volume of liquid, in a horizontal coke can; given, Length: ( 4.5 to 5) inches, Diameter: (2.5 to 3 ) inches, and Height: 1.0 inch. Note: these are rough estimates as I don't have a coke can at hand to measure.

volume: Length * ( acos( ( Radius - Height )/ Radius ) * Radius^2 - sqrt( Height * ( Diameter - Height ) ) * ( Radius - Height ) );

Length: ( 4.5 to 5) inches;

Diameter: (2.5 to 3 ) inches;

Radius: Diameter / 2;

Height: 1.0 inch;

volume as fl_oz ~ ( 4.2 to 6.1 ) fl_oz

Find the volume of liquid, in a horizontal tank; given, Length: 20 ft, Diameter: 5 ft, and Height: 2 ft.

Length: 20 feet;

Diameter: 5 ft;

Radius: Diameter / 2;

Height: 2 ft;

volume as gal ~ 1,097.279 gal

Length: 20 feet pm inch/8;

Diameter: 5 ft pm inch/8;

Radius: Diameter / 2;

Height: 2 ft pm inch/8;

volume as gal ~ ( 1,087.6 to 1,107.0 ) gal

A Circular Spiral is made by moving along a turning circular cylinder at a steady rate. Since Circular Spirals are often used to make electrical solenoids and springs ( both made of wires ), the variables are wireLength and wireArea.

` volume: wireLength * wireArea;`

Where

`
wireLength: sqrt( rate^2 + radiusLarge^2 ) * length / rate;
wireArea: pi radiusSmall^2;
`

`
spacing: 1 / trunsPerLength;
rate: 2 pi spacing;
`

`
volume: wireLength * wireArea;
wireLength: sqrt( rate^2 + radiusLarge^2 ) * length / rate;
wireArea: pi radiusSmall^2;
spacing: 1 / trunsPerLength;
rate: 2 pi spacing;
`

`
length: 5 feet;
radiusSmall: 1/32 inch;
radiusLarge: 6 inches;
trunsPerLength: 10 /inch;
volume as cu in ~ 1.77 cu in
volume as cc ~ 28.96 cc
`

`
`

`
length: 5 inches;
radiusSmall: 5/1000 inch;
radiusLarge: 1/2 inches;
trunsPerLength: 100 /inch;
volume as cu in ~ 3.15e-3 cu in
volume as cu mm ~ 51.61 cu mm
`

The following can be found from the radius and height of a cone.

`
volume: 1/3 pi radius^2 * height;
slantHeight: sqrt( radius^2 + height^2 );
SurfaceArea: pi radius * slantHeight;
`

volume: 1/3 pi radius^2 * height;

slantHeight: sqrt( radius^2 + height^2 );

SurfaceArea: pi radius * slantHeight;

height: ( 4.5 to 5) inches; diameter: (2.5 to 3 ) inches;

radius: diameter/2;

volume as ml; ~ ( 115. to 193. ) ml

volume as fl_oz ; ~ ( 4. to 5. to 7. ) fl_oz

SurfaceArea as sq inches; ~ ( 18. to 25. ) sq inches

SurfaceArea as sq cm; ~ ( 115. to 159. ) sq cm

Find the volume, and surface area, of a conical tower; given, height: 50 feet, diameter: 30 feet.

height: 50 feet; diameter: 30 feet;

volume as gal; ~ 88,128. gal

volume as cu m ; ~ 334. cu m

SurfaceArea as sq yards; ~ 273. sq yards

SurfaceArea as sq m; ~ 229. sq m

An oblate spheroid is formed by the rotation of an ellipse about its minor axis. The Earth is an example of an oblate spheroid. The volume and surface area of a oblate spheroid is:.

`
volume: 4/3 pi a^2 b;
ecc: sqrt( sq a - sq b ) / a;
surfaceArea: 2 pi a^2 + pi b^2 / ecc * ln( ( 1 + ecc ) / ( 1 - ecc ) );
`

ecc: sqrt( sq a - sq b ) / a;

surfaceArea: 2 pi a^2 + pi b^2 / ecc * ln( ( 1 + ecc ) / ( 1 - ecc ) );

`
a: ( 6378.533 pm 0.437 ) km;
b: ( 6356.912 pm 0.437 ) km;
volume as cu miles; ~ 2.60 e11 cu miles
surfaceArea as sq miles; ~1.97e8 sq miles
`

A prolate spheroid ( shown below ) is formed by the rotation of an ellipse about its major axis.

The volume and surface area of a prolate spheroid is:.

`
volume: 4/3 pi a b^2;
ecc: sqrt( sq a - sq b ) / a;
surfaceArea: 2 pi b^2 + pi a * b / ecc * asin( ecc );
`

ecc: sqrt( sq a - sq b ) / a;

surfaceArea: 2 pi b^2 + pi a * b / ecc * asin( ecc );

`
a: ( 11 to 11 + 1/4 )/2 inch as in; = ( 5 + 5/8 ) in
circumference: ( (21 + 1/4) to (21 + 1/2) ) inches as in; ~ 21. in
b: circumference / ( 2 pi ) as in; ~ 3.4 in
ecc ~ 0.79 to 0.80
volume as cu in; ~ ( 269.51 to 275.89 ) cu in
surfaceArea as sq in; ~ ( 141.1 to 142.3 to 143.4 ) sq in
`

A Lune is formed by the rotating a half circumference of a sphere by an angle. The surface area of a Lune is: ` surfaceArea: 2 radius^2 angle; `

`
surfaceArea: 2 radius^2 angle;
`

`
radius: ( 6371.315 pm 0.437 ) km;
radius as miles ~ 3,959. miles
`

`
angle: 1 deg;
surfaceArea as sq miles ~ ( 547,026. to 547,176. ) sq miles
surfaceArea as sq km ~ ( 1,416,791. to 1,417,180. ) sq km
`

`
angle: 5 deg;
surfaceArea as sq miles ~ ( 2,735,131. to 2,735,882. ) sq miles
surfaceArea as sq km ~ ( 7,083,957. to 7,085,901. ) sq km
`

`
angle: 10 deg;
surfaceArea as sq miles ~ ( 5,470,263. to 5,471,764. ) sq miles
surfaceArea as sq km ~ ( 1.42 to 1.42 to 1.42 )e7 sq km
`

A sphere is made by rotating a circle around a diameter. The volume and surface area of a sphere is defined in terms of the sphere's radius.

`
volume: 4/3 pi cu radius;
SurfaceArea: 4 pi sq radius;
`

`
volume: 4/3 pi cu radius;
SurfaceArea: 4 pi sq radius;
radius: (6371.315 pm 0.437) km;
volume as cu miles; ~ 2.60 e11 cu miles
volume as cu m; ~ 1.08 e21 cu m
SurfaceArea as sq mile; ~ 1.97 e8 sq mile
SurfaceArea as sq m; ~ 5.10 e14 sq m
`

Find the volume and surface area of a tennis ball. The diameter used below is from John Lord's book " Sizes" page 295.

`
diameter: ( ( 2 + 1/2 ) to ( 2 + 5/8 ) ) inch;
radius: diameter / 2;
volume as cu inches; ~ ( 8.18 to 9.47 ) cu inches
volume as ml; ~ ( 134.07 to 155.20 ) ml
SurfaceArea as sq in; ~ ( 19.63 to 20.63 to 21.65 ) sq in
SurfaceArea as sq cm; ~ ( 126.68 to 139.66 ) sq cm
`

A Zone and Segment of One Base is made by cutting a sphere with a plane. The volume and surface area can be calculated from the following.

`
volume: 1/3 pi h^2 ( 3 radius - h );
volume: 1/6 pi h ( 3 a^2 + h^2 );
surfaceArea: 2 pi radius * h;
surfaceArea: pi p^2;
`

`
volume: 1/3 pi h^2 ( 3 radius - h );
surfaceArea: 2 pi radius * h;
`

`
Earth'sAveRadius: ( 6371.315 pm 0.437 ) km;
radius: Earth'sAveRadius;
`

`
h: radius ( 1 - sin( latitude ) );
`

`
latitude: 90 deg;
volume as cu miles; = 0 cu miles
surfaceArea as sq miles; = 0 sq miles
`

`
latitude: 89 deg;
volume as cu miles; ~ ( 4,520.7 to 4,522.6 ) cu miles
surfaceArea as sq miles; ~ 1.500e4 sq miles
`

`
latitude: 75 deg;
volume as cu miles; ~ 2.2 e8 cu miles
surfaceArea as sq miles; ~ ( 3,355,104.2 to 3,356,024.8 ) sq miles
`

`
latitude: 45 deg;
volume as cu miles; ~ 1.5e10 cu miles
surfaceArea as sq miles; ~ 2.9 e7 sq miles
`

`
latitude: 0 deg;
volume as cu miles; ~ 1.3 e11 cu miles
surfaceArea as sq miles; ~ 9.8e7 sq miles
`