Frusta Geometry

Backyard Metalcasting has really easy instructions for creating a nice burner flare. This page gets into a little more of the math for creating the outside surface of any kind of frustum (without the top and bottom).

Figure 1. Frustum

Figure 2. Surface area

Given: $r_1$, $r_2$, and $H$ where $r_2 > r_1 > 0$.
Problem: Find $R_1$, $R_2$, and $\theta$.
Solution:

\begin{displaymath} R_1 = \frac{r_1}{r_2 - r_1}\sqrt{(r_2 - r_1)^2 + H^2} \end{displaymath}

\begin{displaymath} R_2 = \sqrt{(r_2 - r_1)^2 + H^2} \end{displaymath}

\begin{displaymath} \theta = \frac{r_1}{R_1} 360^\circ \end{displaymath}

Proof

Figure 3. 		The relationships between the variables are shown in Figure 3. By by the Pythagorean Theorem:
(1)
In order to find $h_1$ in terms of known values, we make use of similar triangles to find this relationship:
Algebraic manipulation gives us $h_1$:
(2)
Plugging equation (2) into equation (1) gives
After a little algebraic manipulation we find
Finally, we get $R_1$ in terms of known values:
(3)
Next we need to find $R_2$. Again, from similar triangles:
Rearranging terms gives
We already know what $h_1$ is, so we just plug it in
(4)
Substituting equation (3) into equation (4) gives
Now we get $R_2$ in terms of known values

Figure 4. 		Finally, note from Figure 4 that
Therefore,

I used latex, metapost, and latex2html to generate the images for this page.

Home | Gingery Lathe Last modified at 2004/02/07 14:15:20 JST
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