Skewed Triangular Prism Calculator

Name: Skewed Triangular Prism - Geometry Calculator
Author: Jürgen Kummer

A triangular prism or right wedge is a right prism with triangles as bases. Cutting a base off by an inclined plane results in a skewed triangular prism.
Enter the lengths of the even base triangle and the three heights. Choose the number of decimal places, then click Calculate.

Formulas:

a^{'} = \sqrt{a^{2} + {(h_{c} - h_{b})}^{2}}

b^{'} = \sqrt{b^{2} + {(h_{c} - h_{a})}^{2}}

c^{'} = \sqrt{c^{2} + {(h_{b} - h_{a})}^{2}}

p = a + b + c

p^{'} = a^{'} + b^{'} + c^{'}

A_{1} = \sqrt{\frac{p}{2} \cdot (\frac{p}{2} - a) \cdot (\frac{p}{2} - b) \cdot (\frac{p}{2} - c)}

A_{2} = \sqrt{\frac{p^{'}}{2} \cdot (\frac{p^{'}}{2} - a^{'}) \cdot (\frac{p^{'}}{2} - b^{'}) \cdot (\frac{p^{'}}{2} - c^{'})}

A_{3} = \frac{a \cdot (h_{b} + h_{c})}{2}

A_{4} = \frac{b \cdot (h_{a} + h_{c})}{2}

A_{5} = \frac{c \cdot (h_{a} + h_{b})}{2}

A = A_{1} + A_{2} + A_{3} + A_{4} + A_{5}

V = \frac{A_{1} \cdot (h_{a} + h_{b} + h_{c})}{3}

Lengths and heights have the same unit (e.g. meter), the areas have this unit squared (e.g. square meter), the volume has this unit to the power of three (e.g. cubic meter). A/V has this unit ^-1.

A straight cut simply yields a smaller triangular prism. With an oblique cut, resulting in two or three different heights, the slanted triangular face of the skewed triangular prism has a larger area than the triangle forming the base. In the case of an oblique cut, the three lateral trapezoids are either all of different sizes, or two are identical if two of the heights are equal.
The severed portion of the original triangular prism is itself another skewed triangular prism, provided the cut does not pass through a vertex. If, however, the cut passes through a vertex, the severed part is a pentahedron with a right trapezoid as its base and four triangles as lateral faces, essentially an oblique, irregular pyramid.

A skewed triangular prism can also be conceptualized as a linear height interpolation over a triangle. Its top surface is uniquely determined by the heights of the three vertices above the base. Linear height interpolation is a method frequently employed in land surveying, terrain modeling, geology, hydrology and computer graphics. This process often involves subdivision into triangular sections, as any three non-collinear points define a plane. Triangulation, the method used to decompose a surface into triangles, is the standard approach for such applications and has long been established. If the area of interest extends beyond the surface itself to include the underlying volume down to a specific depth, the result is a collection of skewed triangular prisms.

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