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Volume of Soil

Date: 02/16/99 at 09:05:17
From: Amy Atteberry
Subject: Volume of soil needed for a trapezoidal bed

I am building some planting beds in a trapezoidal shape. The beds will 
be 8" deep. The dimensions of the trapezoidal bed are 2 x 6 x 4 x 4' 
by 8". Therefore, I need to be able to calculate how many cubic feet 
of soil will be needed to fill these beds.

I look forward to your reply.

Date: 02/16/99 at 12:41:14
From: Doctor Rick
Subject: Re: Volume of soil needed for a trapezoidal bed

Okay, your 3-dimensional shape might be called a "right prism with a
trapezoidal base." Your question might still need more clarification. 
I am guessing that the dimensions of the trapezoid you give are the 
lengths of the sides, and in this order:

 4 /       \ 4
  /         \

and that the 2' and 6' sides are parallel.

The formula for the area of a trapezoid is

  A = (L1 + L2) * H / 2

where L1 and L2 are the lengths of the two parallel sides, and H is 
the height, that is, the perpendicular distance between the parallel 
sides. We need to find the height. If you draw a line from the left 
corner of the 2' side, perpendicular to the 6' side, you get a right 
triangle with hypotenuse 4' and bottom leg 2'. Using the Pythagorean 
Theorem, we find that the height (the other side of the triangle) is

  H = sqrt(4^2 - 2^2) = sqrt(12) = 3.464'

Now the area of the trapezoid is

  A = (2' + 6') * 3.464' / 2
    = 13.856 ft^2

To get the volume of the prism, we multiply the area of the base 
by the height; this time the height is the truly vertical dimension, 
8" = 2/3 foot. So the volume is

  V = A * h
    = 13.856 ft^2 * 2/3 ft
    = 9.237 ft^3

That's how many cubic feet of soil you need, if I interpreted your
description correctly. If not, perhaps I have given you enough 
information to figure out the volume yourself.

- Doctor Rick, The Math Forum   
Associated Topics:
High School Geometry
High School Higher-Dimensional Geometry
High School Practical Geometry

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