# Volume of Trapezoidal Prism

For a plot of land of 100 m × 80 m, the level is to be raised by spreading the earth from a stack of a rectangular base 10 m × 8 m with vertical section being a trapezium of height 2 m. The top of the stack is 8 m × 5 m. By how many centimetres can the level be raised?

I know the approach needed to solve this problem. I only confusion I have about this problem is the calculation of the volume of the stack which I believe is the trapezoidal prism (or truncated (right) rectangular prism or frustum of (right) rectangular prism). I also assume a prism is the same thing as a pyramid for geometrical purposes.

A trapezoidal prism is a 3D figure made up of two trapezoids that is joined by four rectangles.

I saw online different methods giving different answers to this question. I am confused what is the correct approach.

Method I

Volume of Right rectangular pyramid

Method II

Identify the parallel sides of the base (trapezoid) to be $$b_{1}$$ and $$b_{2}$$ and the perpendicular distance between them to be $$h$$ and find the area of the trapezoid using the formula:
Area of the trapezoid $$=\frac{1}{2}\left(b_{1}+b_{2}\right) \times h$$
Identify its height / length of the prism (the vertical distance between two bases).
Multiply the base area and the height to find the volume.

Method III

$$\text { Volume of truncated pyramid/prism }=\frac{1}{3}\left(A_{1}+A_{2}+\sqrt{A_{1} A_{2}}\right) h$$ $$A_{1} = 8X5 = 40 \mathrm{m}^{2}$$ $$A_{2} = 10X8 = 80 \mathrm{m}^{2}$$ $$\Rightarrow\text { Volume of truncated pyramid/prism }=\frac{1}{3}\left(40+80+\sqrt{40X80}\right) 2$$ $$\Rightarrow\text { Volume of truncated pyramid/prism }= 58.85 \mathrm{m}^{3}$$

• Method II is the right one: a prism is NOT a pyramid. Dec 28, 2021 at 17:45
• @Intelligentipauca In the original problem statement, it is not specified whether to consider prism or pyramid. How to know which shape to select? Dec 28, 2021 at 17:49
• They say: "with vertical section being a trapezium of height 2 m". If all sections are equal, it's a prism. And even if they weren't equal, it's enough they have the same area. Dec 28, 2021 at 17:51
• @Andrei If it were a pyramidal frustum then it couldn't have all its vertical sections formed by a trapezium of height 2 m. Dec 28, 2021 at 17:53
• @Intelligentipauca you are right. It looks more like the mansard roof in this image Dec 28, 2021 at 18:06

The pyramid-based answers do not work because the trapezoidal prism is not actually part of a pyramid: the non-horizontal edges do not meet in a single point. Furthermore the question might be ambiguous whether the $$8m$$ edge of the top face is parallel or perpendicular to the $$8m$$ edge of the bottom face, and this affects the final result.

In case the $$8m$$ on top and bottom are parallel, you have a trapezium prism, with trapezium area $$(10m+5m)/2 \times 2m$$ and "height" $$8m$$ (perpendicular to the trapezium), resulting in a volume of $$120 m^3$$.

If the top and bottom faces of the stack are laid out as hinted in the question, with the bottom $$10m$$ parallel to the top $$8m$$ and the bottom $$8m$$ parallel to the top $$5m$$, it is neither a trapezium prism nor a truncated pyramid, because the non-horizontal edges do not intersect in a single point. Assuming the faces are still plane, the cross-section at height $$x$$ (measured in $$m$$) is given by $$(10-x)\times(8-\frac 32 x)$$, and the volume can be determined by integration to yield $$V = \int_0^2(10-x)(8-\frac 32 x)dx = 118 m^3$$.

In general the formula to compute such a shape with height $$h$$, top rectangle $$a\times b$$ and bottom rectangle $$c \times d$$, with the $$a$$ side parallel to the $$c$$ side, is $$\frac16 h(2ab+2cd+ad+bc)$$. I'm not sure whether you are expected to know this though.

Here is a Blender model of the second case, demonstrating how it is not a frustum:

• The second case must be discarded, because all vertical sections of the stack are formed by a trapezium of height 2 m. And, by the way, your second case IS a pyramidal frustum. Dec 28, 2021 at 18:00
• The question says a vertical section of the stack is such a trapezium, not all of them (in which case the posed scenario would be impossible in both cases). Dec 28, 2021 at 18:03
• @Intelligentipauca I have added an image to demonstrate that the second case is indeed not a pyramidal frustum. Dec 28, 2021 at 18:04
• You are right about the frustum, but the text reads "a stack with vertical section being a trapezium of height 2 m" and not "a stack with a vertical section being a trapezium of height 2 m". Dec 28, 2021 at 18:07
• @Intelligentipauca I did not do any modifications from my end. That is the original text. Dec 28, 2021 at 18:14