### Simplified Equation For Lower Basal Wedge - Improved Model

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**Thu Sep 21, 2017 7:41 pm**Attached is a simplified version of the equation for the lower slanted basal frustum below high side ground level and ground perimeter. As demonstrated in the graphic, using the 1/2 frustum model is an over estimate, especially for trees on a steep slope. Bob Leverett came up with a proof that shows this upper wedge is always smaller than the lower wedge. Maybe he'll post that here ?

For example, we have a lower trunk of a tree with high side of ground level radius, as measured by tape wrap, at 5 feet ("r" in the attached formula"). The height differential of the low side of base and high side is 3 feet ("H" in the attached formula) and the trunk off-set from high side to low side in the horizontal direction is 1 foot ("d" in the attached formula). This would give a volume for the lower slant frustum of 123.51 cubic feet. If you model the lower basal wedge using the 1/2 frustum model, you get 142.94 cubic feet. This shows the old model gives an over-estimate by around 15% for this example. For trees on steeper slopes, this over-estimate becomes even greater.

For example, we have a lower trunk of a tree with high side of ground level radius, as measured by tape wrap, at 5 feet ("r" in the attached formula"). The height differential of the low side of base and high side is 3 feet ("H" in the attached formula) and the trunk off-set from high side to low side in the horizontal direction is 1 foot ("d" in the attached formula). This would give a volume for the lower slant frustum of 123.51 cubic feet. If you model the lower basal wedge using the 1/2 frustum model, you get 142.94 cubic feet. This shows the old model gives an over-estimate by around 15% for this example. For trees on steeper slopes, this over-estimate becomes even greater.