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# How to calculate the volume of a tapered hopper

Hoppers are containers used to hold, transport and distribute granular materials. They are frequently used in both agriculture and construction. A tapered hopper will often be in the shape of an inverted cone or rectangular pyramid, with a wide top and narrower bottom. You can calculate the volume of a regularly-shaped hopper by using the formula for the geometric volume of a cone or pyramid. The bottom of the hopper will not form a point however. It is essentially a truncated cone or pyramid with the tip missing. You will therefore have to account for the volume of this "missing" part in the formula.

## Calculating volume for a conical hopper

Measure the diameter of the circle that forms the top of the hopper or inlet. This is the total width of the circle passing through the centre and will be its widest point. Call this diameter "D."

Measure the diameter of the circle at the bottom of the hopper or outlet. Call this diameter "d."

Measure the height of the hopper from the inlet to the outlet. This is the theoretical line that would pass through the centre of the two circles, not the length of the outside surface. Call this height "H."

Calculate the Volume (V) by using the measurements in place of the variables in the following formula:

V = Pi * H / 3 * (R^2 + (R * r) + r^2)

## Calculating volume for a pyramidal hopper

Measure the dimensions of the rectangle that forms the top of the hopper. Call the length "X" and the width "Y."

Measure the dimensions of the rectangle at the bottom of the hopper. Call the length "x" and the width "y."

Measure the height of the hopper from the centre of the two rectangles. Call this height "H."

Calculate the Volume (V) by using the measurements in place of the variables in the following formula:

V = H/6 * (Xy + xY + 2 * (XY+xy))

#### Warning

The above formulas can only be applied to regular conical or pyramidal tapered hoppers.

• Tape measure
• Calculator