`m = "mD" * "n" * "h" /12 * ( "b" ^2 + "b" * "B" + "B" ^2) * cot(pi/ "n" )`

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The Mass or Weight of the Frustum of a Pyramid calculator computes the mass or weight of a regular pyramid frustum based on the dimensions of the polygon base, the height of the frustum and the density of material.

INSTRUCTIONS: Choose units and enter the following:

• (n) Number of Polygon Sides
• (b) Length of Sides on Top
• (B) Length of Sides on Bottom
• (h) Height of Pyramid
• (mD) Density of Material

Pyramid Frustum Mass or Weight (m): The calculator returns the mass in kilograms, and the volume in cubic meters. However, these can be automatically converted to compatible units via the pull-down menu.

The Math / Science

This Weight of the Frustum of a a Polygon based Pyramid calculator computes the weight or mass of a right pyramid frustum based on the dimensions of the frustum, a height (h) and a (mD) mean density.  The base area (A) is defined by a polygon of (n) sides each of length (b).

The formula for the mass of a regular pyramid frustum is:

`m = mD * n*h/12 * (b^2 + b*B + B^2) * cot(pi/n)`

where:

• m = mass of the pyramid frustum
• mD = density of material
• h = height of pyramid
• b = length of top sides
• B = length of base sides
• n = number of sides

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Common Mean Densities

Fluids Pure Water - 1,000 kg/m³  Seawater - 1,022 kg/m³  Milk - 1,037 kg/m³  Olive Oil - 860 kg/m³ Cement Slurry - 1,442 kg/m³  Fuels Diesel Fuel - 885 kg/m³  Crude Oil - 870 kg/m³  to 920 kg/m³  Fuel Oil -  890 kg/m³  Ethanol - 789 kg/m³  Gasoline (petrol) - 737 kg/m³  Propane - 493 kg/m3 Liquid Natural Gas - 430 to 470 kg/m3 Market-Ready Grains Corn - 56 lb/bu (721 kg/m3) Wheat - 60 lb/bu (772 kg/m3) Barley - 48 lb/bu (618 kg/m3) Oats - 32 lb/bu (412 kg/m3) Rye - 56 lb/bu (721 kg/m3) Soybean - 60 lb/bu (772 kg/m3)

Metals Aluminum - 2700 kg/m³ Brass - 8530 kg/m³ Bronze - 8150 kg/m³ Chromium - 7190 kg/m3 Cobalt - 8746 kg/m3 Copper - 8920 kg/m³ Gallium - 5907 kg/m3  Gold - 19300 kg/m³ Iridium - 22560 kg/m3 Iron - 7847.0 kg/m³  Lead - 11340 kg/m³ Molybdenum - 10280 kg/m3 Nickle - 8908  kg/m3  Palladium -  12160 kg/m³ Platinum - 21450.0 kg/m³ Rhodium - 12410 kg/m3 Steel - 7850 kg/m³ Silver - 10490 kg/m³ Titanium - 4500 kg/m³ Tungsten - 19600 kg/m³ Zinc - 7135 kg/m³ Zirconium - 6570 kg/m³

Mean Density is the average amount of mass within a volume for a substance.  Note, volume of a material is often highly subject to the temperatures, since materials expand as they warm.  For that reason, mean densities of substances are often cited with a set of nominal conditions such as temperature and barometric pressure. 

The formula for mean density is:

μD = V / m

where:

• μD = mean density
• V = Volume in units like cubic meters or cubic inches
• m = Mass in units like kilograms or pounds

Mean density is also often indicated as the Greek symbol rho (ρ).

Mass and Weight

Density is a function of mass.  However, converting from mass to weight is trivial under the right conditions.  Fortunately those conditions are generally true anywhere on the surface of the Earth, so the conversions built into the vCalc engine can be assumed to be fairly accurate unless you require weight at very high altitudes or in space.

CLICK HERE  for a pop-up function that provides the mean density (mD or μD) of many common substances, elements, liquids and materials.

Mean Density Units

Mean density is scientifically volume divided by mass.  There are  various unit for density adopted by cultures and industries. Common units for density included the following:

• kilograms per cubic meter (kg/m³)
• grams per cubic centimeter (g/cm³)
• grams per liter (g/L)
• pounds per cubic feet (lb/ft³)
• ounces per cubic inch (oz/in³)
• pounds per barrel (lb/bbl)
• pounds per bushel (lb/bu)

vCalc provides for automatic conversions between density units via the pull-down menus.