Triangular Obelisk: Calculate Angle & Height

[markmdesigns]

Hi all

I am new to the forum. I'm a designer for theatre and I'm having some trouble figuring out a scenic piece I want to build.

The piece is essentially an obelisk with a triangular base. The three sides of the solid are trapezoids with 4' wide base, 2'-10" wide top and an overall height of 14' (dimensions shown in the image). The base is an equilateral triangle with 4' sides and the top is an equilateral triangle with 2'-10" sides.

Each of the sides must slope back slightly in order for their edges to meet. So the two questions are:
1) What is that angle? (shown as angel n)
2) What is the overall height of the solid?

Thanks so much for your help!

~M.

 

[adriank]

Welcome!

I'm assuming the height of each trapezoidal face is 14', and you want the total height. I'm also assuming the angle you want (which I will refer to by θ) is the angle between the vertical and one of the side faces, not one of the edges.

Let h be the total height of the solid (which is to be found), a be the width of a trapezoidal piece at the bottom (4'), b the width at the top (2' 10"), and c the height of each piece (14'). Looking at the base of the solid, let d be the (perpendicular) distance from the centre of the base to one of the edges of the base; similarly, let e be the distance from the centre of the base to one of the edges of the top. Trigonometry gives d = a√(3)/6 and e = b√(3)/6, so c sin θ = d - e; also, h² = c² - (d - e)². Then θ = 1.378° and h = 13' 11.95" (which I'm guessing is close enough to 14' for your purpose).

If you want the angle between an edge and the vertical, put c sin θ = 2(d - e) above to obtain θ = 2.657°.

 

[markmdesigns]

Thanks Ardiank.

I was able to find the angle I was looking for as = 1.38 using my CAD software, and subsequently the height. I just had trouble figuring out how to rotate the object to the correct angle and needed the exact angle to input.

Your description and formulas point out the exact thing I was overlooking: the distance from the center of the base to one edge which forms your right triangle with the tilted face and allows you to run the calculations.

While I use geometry every day to compute simple stuff (how much paint will I need to cover this drop, etc.) I rarely need to use trigonometry and since this object is all triangles, it a rare shape for me to deal with. Now I just have to figure out how to build it!