Graphics UnitMath Example: ( Geometric Solids )
Last Modified: 9/10/2000

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The following examples show how to use UnitMath with geometric solids.

Box ( Rectangular Parallelepiped ) TOP

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 );

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Example Calculations for Boxes

Find the volume, surface area, and diagonal of a box; given, length: 2 feet, height: 18 inches, width: yard;

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


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Can ( Right Circular Cylinder ) TOP

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

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

Example Calculations for Cylinders

Find the volume, and surface area, of a coke can; given, height: ( 4.5 to 5) inches, diameter: (2.5 to 3 ) inches. Note: these are rough estimates as I don't have a coke can at hand to measure.

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


Horizontal Cylinder ( Partly Filled ) TOP

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The volume of the above figure is given by:.

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

Test Cases

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


Example Calculations

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


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Circular Spiral TOP

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;

Example Calculations for a circular spirals

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


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Cone ( Right Circular Cone ) TOP

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;

Example Calculations for Cones

Find the volume, and surface area, of an ice cream cone; given, height: ( 4.5 to 5) inches, diameter: (2.5 to 3 ) inches. Note: these are rough estimates as I don't have an ice cream cone at hand to measure.

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


Graphics

Oblate Spheroid TOP

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 ) );

Example Calculation for Oblate Spheroid

Find the volume and surface area of the Earth. Given the equatorial radius is ( 6378.533 pm 0.437 ) km and the polar radius is ( 6356.912 pm 0.437 ) km.

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 ) );

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


Prolate Spheroid TOP

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

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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 );

Example Calculation for Prolate Spheroid

Find the volume and surface area of a American ( National Football League ) foot ball; which, approximates a prolate spheroid. The dimensions used below are from John Lord's book " Sizes" page 105. I had to do a bit of work to get the major semi axis as the circumference was given not the minor axis, but "circumference = 2 pi radius, so radius (b) = circumference / ( 2 pi )".

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 );

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


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Lune TOP

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;

Example Calculations for Lune

Find the surface area of lune on the Earth for the following angles: 1 deg, 5 deg, & 10 deg.

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


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Sphere TOP

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;

Example Calculations for Spheres

Find the volume and surface area of the Earth; given, Earth's radius: (6371.315 pm 0.437) km

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


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Zone and Segment of One Base TOP

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;

Example Calculations for a Zone and Segment of One Base

Find the volumes and surface areas of the Earth above various latitudes.

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


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