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

### 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.

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.