General analytical solution: angle between surfaces of a pyramid/polyhedronOtherwise called “compound saw angles”A cabinet-maker wanted to find the angle to chamfer the edges of two or more adjoining pieces of wood to make them neatly fit together, forming a regular pyramid. My High School maths teacher Mr. Stephen Bell posed this challenge to me in 1994, from his colleague (Mr Kennedy the craft teacher), who had been unable to solve the problem analytically. I was unable to solve the problem using classical geometry / trigonometry at age sixteen, but two years later, while studying for the A-Level Further Maths qualification, we were equipped with the tools necessary to answer this question (a combination of vector products: scalar product and cross product). First, the cross-product is used to obtain surface-normal vectors for adjacent upward-facing surfaces of the pyramid, and then these normal vectors are compared using scalar-product to determine the angle between the surfaces. These workings are accurate for regular pyramids (i.e. those comprising similar equilateral or isosceles triangles atop a polygon-shaped base). A similar method might be applied to general compound saw angle type problems.
The chamfer angle (for the solution to the cabinet maker's problem) may be found by dividing by angle Theta by two (since two surfaces meet at that point). Now using “p” and “q” to mean something different to what they meant before:The inscribed radius and perpendicular height of an n-sided pyramid may be found using the following formulae:
Therefore Theta may be expressed purely in terms of
Pyramid dimensioning calculator:Caveats: Distance measurements are in arbitrary units (calculations are valid regardless of the units, provided that all the measurements are supplied in the same units of distance measurement. The results will be in these same units.) In other words, you may calculate according to a unitary-sized pyramid and scale the distance-measurements accordingly. (Remember to scale area factor according to the square of the scale factor, and volume or mass according to the cube of the scale factor.)
If you enter all input measurements, the calculator will produce an scaled isometric projection of your pyramid, so that you can verify (approximately) that the measurements are as you intended. |
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