Calculations at an isosceles antiprism. Unlike the regular antiprism, the sides of the isosceles antiprism are not equilateral triangles, but isosceles triangles. Therefore, the isosceles antiprism can have any height. Its base is also a regular n-gon.
Enter two of the three values of edge lengths and height, as well as the number of vertices of one base and choose the number of decimal places. Then click Calculate.
Formulas:
h = √ b² - a² / (4 * cos²(π/(2n)))
A = n * a² / (2 * tan(π/n)) + (n*a/2) * √ 4*b²-a²
V = [ n * a³ * cot(π/(2n)) * √ 4 * cos²(π/(2n)) - 1 / 12 ] * h / [ √ 1 - sec²(π/(2n)) / 4 * a ]
pi:
π = 3.141592653589793...
Lengths and height have the same unit (e.g. meter), the area has 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.
The volume of the isosceles antiprism is calculated by scaling the regular antiprism in the direction of its height. Hereby, the volume of the regular antiprism with side length a is multiplied by the height of the isosceles antiprism and then divided by the height of the regular antiprism. Multiplication and division by the lengths of the legs instead of the heights does not work, as these are not parallel to the direction of the scaling.
The most well-known building with a similar shape is probably One World Trade Center in New York City, which opened in 2015. This has a square base, and the section resembling an isosceles antiprism sits atop a cuboid-shaped plinth. However, the upper square base (with an edge length of 44 meters) is smaller than the lower one (with an edge length of 61 meters), so this calculation does not apply to this building. This is, of course, due to structural reasons, as buildings that taper towards the top are more stable than those that do not.
The volume of such a truncated oblique antiprism can be calculated using Cavalieri's principle and the formula for a general frustum.