Bent Pyramid Calculator

Name: Bent Pyramid - Geometry Calculator
Author: Jürgen Kummer

Calculations at a bent pyramid. The lower part of the bent pyramid is a square frustum, the upper part is a square pyramid. Mathematically, the bent pyramid is equal to the obelisk.
Enter both side lengths a and b and two of the three heights. Choose the number of decimal places, then click Calculate.

Formulas:

h = i + j

A = a^{2} + (a + b) \cdot \sqrt{{(a - b)}^{2} + 4 i^{2}} + b \cdot \sqrt{4 \cdot j^{2} + b^{2}}

V = \frac{i}{3} \cdot [(a^{2} + b^{2}) + \sqrt{a^{2} \cdot b^{2}}] + \frac{j}{3} \cdot b^{2}

Lengths and heights 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 most famous bent pyramid was built nearly 5000 years ago by Pharaoh Sneferu in ancient Egypt. It was originally intended to be the first "normal" pyramid, a regular square pyramid. However, for structural reasons, the angle of inclination had to be changed during construction. The initial inclination was too steep, so at a height of about 46 meters, slightly less than halfway to its total height, a different angle was introduced, resulting in the bend. Without this bend, the pyramid would have been taller, but it now has a unique shape. This bend was likely considered a flaw, as Sneferu commissioned the construction of another pyramid, the Red Pyramid, which is flatter and has no bend.

The calculation of the height, surface area, and volume of a bent pyramid is, of course, based on the two parts it comprises: the truncated pyramid and the pyramid itself. The total volume is simply the sum of the volumes of these two parts, and the same applies to the heights. The total surface area is the surface area of the two individual parts minus twice the square base area of the pyramid, which is also the upper surface, i.e., the small base of the truncated pyramid.

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