Quick Formula Reference Cheat Sheet
All conical solids are grounded in fundamental Euclidean geometry and integral calculus. Below is an executive reference of the governing equations across every calculator:
| Calculator Type | Primary Equation | Key Variables |
|---|---|---|
| Standard Cone Volume | $$V = \frac{1}{3}\pi r^2 h$$ | $r$: base radius, $h$: vertical height |
| Volume from Diameter | $$V = \frac{\pi D^2 h}{12}$$ | $D$: base diameter ($2r$) |
| Volume from Circumference | $$V = \frac{C^2 h}{12\pi}$$ | $C$: base perimeter ($2\pi r$) |
| Cone Surface Area (TSA) | $$\text{TSA} = \pi r (l + r)$$ | $l$: slant height ($\sqrt{r^2 + h^2}$) |
| Truncated Frustum | $$V = \frac{\pi h}{3}(R^2 + Rr + r^2)$$ | $R$: bottom radius, $r$: top radius |
| Hollow Cone Shell | $$V_{\text{mat}} = \frac{\pi h}{3}(R^2 - r^2)$$ | $R$: outer radius, $r$: inner void radius |
| Partial Fill (Apex down) | $$V_{\text{fill}} = V_{\text{tot}}\left(\frac{d}{h}\right)^3$$ | $d$: liquid depth from bottom apex |
| Elliptical Base Cone | $$V = \frac{\pi a b h}{3}$$ | $a, b$: semi-major and semi-minor axes |
| Bisected Half Cone | $$V_{1/2} = \frac{\pi r^2 h}{6}$$ | Flat cut area = $r \times h$ |