Frequently Asked Questions About Cone Volume & Geometry

A cone has exactly one-third the volume of a cylinder with the same base radius and height. This was proven by Archimedes in 250 BC and can be demonstrated experimentally by pouring liquid from three identical full cones into one cylinder. In calculus, integrating circular disk cross-sections leads to $\int z^2 dz = \frac{z^3}{3}$, producing the factor of $1/3$.

If you are given the diameter ($d$), divide it by 2 to obtain the radius ($r = d / 2$), then substitute into $V = \frac{1}{3}\pi r^2 h$. Alternatively, use the direct diameter formula: $$V = \frac{1}{12}\pi d^2 h \approx 0.2618 \cdot d^2 h$$

Since the radius ($r$), vertical height ($h$), and slant height ($l$) form a right-angled triangle, apply the Pythagorean theorem: $$h = \sqrt{l^2 - r^2}$$ Once you calculate $h$, substitute it into the volume formula $V = \frac{1}{3}\pi r^2 h$.

Rearrange the volume formula to solve for height: $$h = \frac{3V}{\pi r^2}$$ Multiply the known volume by 3, and divide by the base area ($\pi r^2$).

Rearrange the volume formula to solve for radius: $$r = \sqrt{\frac{3V}{\pi h}}$$ Multiply volume by 3, divide by $\pi h$, and take the positive square root.

A conical frustum (a cone with its top sliced off parallel to the base) with bottom radius $R$, top radius $r$, and vertical height $h$ has volume: $$V = \frac{1}{3}\pi h (R^2 + r^2 + Rr)$$

Yes. By Cavalieri's principle, if two three-dimensional solids have equal vertical heights and equal cross-sectional areas at every height level, they have equal volumes. An oblique cone with base radius $r$ and vertical altitude $h$ has volume $V = \frac{1}{3}\pi r^2 h$, identical to a right cone.

Vertical height ($h$, or altitude) is the straight perpendicular distance from the apex to the center of the base. Slant height ($l$) is the distance along the outer angled surface from the apex to any point on the base perimeter. Slant height is always longer than vertical height ($l > h$).

1 Liter = 1,000 cm³ (mL). To convert cm³ to liters, divide by 1,000. 1 US liquid gallon = 3,785.41 cm³ (approx. 3.785 L). To convert cm³ to US gallons, divide by 3,785.41.

The semi-vertical apex angle $\theta$ is the angle between the central vertical axis and the slant surface: $\tan(\theta) = \frac{r}{h}$. The total apex opening angle is $2\theta$.

The lateral (curved) surface area is $A_{lat} = \pi r l$. The base area is $B = \pi r^2$. The total surface area including the base is: $$A_{tot} = \pi r^2 + \pi r l = \pi r (r + l)$$