Cone Volume Calculator

Compute the volume, slant height, and surface area of any right circular cone with step-by-step arithmetic.

▲ Cone Volume Calculator

Formula: V = (1/3)πr²h

Base Radius (r) *

Radius of the circular base

Cone Height (h) * Perpendicular height from apex to base

Decimal Precision Display precision for result

Calculated Cone Volume

56.5487 cm³

Step-by-Step Mathematical Breakdown:

1. Radius \( r = 3\text{ cm} \), Height \( h = 6\text{ cm} \)

2. Volume Formula: \( V = \frac{1}{3}\pi r^2 h \)

3. Slant Height: \( l = \sqrt{r^2 + h^2} = \sqrt{9 + 36} = \sqrt{45} \approx 6.7082\text{ cm} \)

4. Volume Calculation: \( V = \frac{1}{3} \times \pi \times 9 \times 6 = 18\pi \approx \mathbf{56.5487\text{ cm}^3} \)

5. Total Surface Area: \( A = \pi r(r + l) \approx \pi(3)(3 + 6.7082) \approx \mathbf{91.4984\text{ cm}^2} \)

Geometric Derivation of Cone Volume

The volume of a cone is derived from the principle that any pyramid or cone occupies exactly one-third of the prism or cylinder having identical base and height dimensions:

V = ⅓ × \text{Base Area} × h = ⅓ π r² h

Compare this with our flagship Sphere Volume Formula: a sphere has volume \(\frac{4}{3}\pi r^3\), which equals exactly four cones with base radius \(r\) and height \(r\)!