Cylinder, Cone & Sphere Volume and Surface Area Calculator

Formed by translating a circle of radius $r$ vertically through space along an orthogonal height axis $h$. It features two identical parallel flat circular disks (top and base) connected by a rectangular sheet rolled into a cylindrical lateral tube of area $2\pi rh$.

Formed by connecting every point on a circular base of radius $r$ to a singular apex point positioned at perpendicular height $h$. The straight distance from the apex to any base boundary point forms the slant height $s = \sqrt{r ^ 2 + h ^ 2}$.

Defined as the complete 3D locus of all points in Euclidean space situated at constant radius $r$ from an origin focus point $(0,0,0)$. A sphere has zero flat edges or vertices, maximizing volumetric containment per unit of enclosing surface area.

1. Cone Volume via Disk Integration

Place a cone of base radius $R$ and height $H$ along the x-axis with apex at the origin. The radius of a disk at distance $x$ is $r(x) = \frac{R}{H}x$. The differential volume of each thin circular disk is $dV = \pi [r(x)]^2 dx$. Integrating from $0$ to $H$:

This mathematically proves the exact $1/3$ ratio for all right circular cones.

2. Sphere Volume via Disk Slicing

A sphere centered at $(0,0,0)$ obeys $x^2 + y^2 + z^2 = R^2$. A horizontal cross-sectional disk at height $z$ has radius $r(z) = \sqrt{R ^ 2 - z ^ 2}$. Integrating disk area $A(z) = \pi(R^2 - z^2)$ from $z = -R$ to $z = R$:

Differentiating volume with respect to radius yields surface area: $d/dr[(4/3)\pi r^3] = 4\pi r^2$.

• 1. Calculate Slant Height (s):
• s = \sqrt{r^2 + h^2} = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10.0000\text{ m}
• 2. Calculate Total Volume (V):
• V = \frac{1}{3}\pi r^2 h = \frac{1}{3}\pi(36)(8) = 96\pi \approx 301.5929\text{ m}^3
• 3. Calculate Lateral Surface Area (LSA):
• LSA = \pi r s = \pi(6)(10) = 60\pi \approx 188.4956\text{ m}^2
• 4. Calculate Total Surface Area (TSA):
• TSA = LSA + \pi r^2 = 188.4956 + 36\pi = 96\pi \approx 301.5929\text{ m}^2
• 5. Liquid Storage Capacity:
• \text{Capacity} = 301.5929 \times 1000 = 301,592.9\text{ Liters}

• 1. Determine Radius from Diameter:
• r = d / 2 = 14 / 2 = 7.0000\text{ m}
• 2. Calculate Enclosed Volume (V):
• V = \frac{4}{3}\pi r^3 = \frac{4}{3}\pi(343) = \frac{1372}{3}\pi \approx 1436.7550\text{ m}^3
• 3. Calculate Total Surface Area (TSA):
• TSA = 4\pi r^2 = 4\pi(49) = 196\pi \approx 615.7522\text{ m}^2
• 4. Circumference (Equatorial):
• C = 2\pi r = 2\pi(7) = 14\pi \approx 43.9823\text{ m}
• 5. Liquid Gallon Capacity:
• \text{Capacity} = 1436.7550 \times 264.172 = 379,550.6\text{ Gallons}