Hoppers have many uses in the industrial and farming fields. A tapered hopper is generally the shape of a pyramid or cone turned upside down, with a large top that tapers down to a smaller bottom. When opened, gravity causes the material inside the hopper to feed out the bottom. The formula for the volume of a tapered hopper is based on the volume of a geometric pyramid or cone. The volume for a pyramid with any base is found by multiplying the area of the base by the height and dividing by 3. Because the hopper is essentially a truncated pyramid or cone with the tip cut off, its formula for volume uses similar triangle concepts and subtracts the missing part of the cone to determine the volume.

## Finding Volume of a Rectangular-based Tapered Hopper

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## Step 1

Measure the upper rectangular dimensions. The units of measurement must remain consistent throughout the entire process. Let "X" equal length and "Y" equal width. Use capital letters for the variables.

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Example: Length = 50 inches, Width = 30 inches. (X = 50, Y = 30)

## Step 2

Measure the lower rectangular dimensions. Again, let "x" equal length and "y" equal width. Use lower case letters for the variables.

Example: Length = 5 inches, Width = 3 inches. (x = 5, y = 3)

## Step 3

Measure the height from the upper base to the lower base. The height must be measured through the center, not down the slant sides.

Example: Height = 20 inches. (H = 20)

## Step 4

Calculate the volume by substituting in the values for the variables: V=(1/3)*H*[(X^2_Y-x^2_y)/(X-x)] where: H: Height between bases (shortest distance through middle of hopper) X: Length of upper rectangular base Y: Width of upper rectangular base x: Length of lower rectangular base y: Width of lower rectangular base

Example: V=(1/3)*20*[(50^2_30-5^2_3)/(50-5)] Calculations: V=(1/3)*20*[(2500_30-25_3)/45] V=(1/3)*20*[(75000-75)/45] V=(1/3)*20*[74925/45] The volume is 11,100 cubic inches.

## Finding Volume of a Tapered Hopper with a Circular Base

## Step 1

Measure the dimension of the upper circle. The unit of measurement must remain consistent throughout the entire process. Use upper case letters for the variable.

Example: Diameter = 12 feet. (D = 12)

## Step 2

Measure the dimension of the lower circle. Use lower case for the variable.

Example: Diameter = 4 feet. (d = 4)

## Step 3

Measure the height from the upper base to the lower base. The height must be measured through the center, not down the slant sides.

Example: Height = 15 feet. (H = 15)

## Step 4

Calculate the volume by substituting in the values for the variables: V=(1/12)*pi_H*[D^2+D_d+d^2] where: H: Height between bases D: Diameter of the upper circular base d: Diameter of the lower circular base

Example: V=(1/12)*pi_15*[12^2+12_4+4^2]

Calculations: V=(1/12)*pi_15*[144+48+16] V=(1/12)*pi_15*[208] V=(1/12)_3.14159_15*[208] The Volume is approximately 816.814 cubic feet.