Spherical Cap Calculator

What Is the Spherical Cap Calculator?

The Spherical Cap Calculator computes the volume, curved surface area, and total surface area of a spherical cap (dome-shaped region of a sphere cut by a single plane). It also solves for the full sphere radius from a known cap volume and height. The calculator handles automatic unit conversion and provides step-by-step derivations for every result. For full sphere calculations, try our Sphere Calculator and Spherical Segment Calculator.

Spherical Cap Formulas

A spherical cap is defined by the full sphere radius $R$ and the cap height $h$ (the perpendicular distance from the cutting plane to the top of the dome). All key quantities derive from this pair:

$$V = \frac{\pi h^2}{3}(3R - h)$$

$$S_{\text{curved}} = 2\pi R h$$

$$S_{\text{total}} = \pi h(4R - h)$$

$$R = \frac{V}{\pi h^2} + \frac{h}{3}$$

$$a = \sqrt{h(2R - h)}$$

Where:

• $V$ is the volume of the spherical cap
• $S_{\text{curved}}$ is the curved dome surface area (excluding the flat base)
• $S_{\text{total}}$ is the total surface area (curved dome plus flat base disk)
• $R$ is the full sphere radius
• $h$ is the cap height ($0 < h \le 2R$)
• $a$ is the base radius of the cap (radius of the circular cross-section)

How to Use the Calculator

Select the quantity you want to solve for from the dropdown menu. Enter the known values and choose appropriate units. The result updates in real time with a complete step-by-step breakdown. The supplementary base radius and base area are always shown alongside the primary result.

Example Calculation

Problem: A spherical pressure tank has interior radius $R = 5$ m and is filled with liquid to a depth of $h = 2$ m measured from the bottom. Compute the liquid volume, wetted curved surface area, and the radius of the free-surface circle.

Solution: Select "Solve for Volume (V)", enter $R = 5$ m and $h = 2$ m:

$$V = \frac{\pi \times 2^2}{3}(3 \times 5 - 2) = \frac{4\pi}{3} \times 13 = \frac{52\pi}{3} \approx 54.45 \text{ m}^3$$

$$S_{\text{curved}} = 2\pi \times 5 \times 2 = 20\pi \approx 62.83 \text{ m}^2$$

$$a = \sqrt{2(10 - 2)} = \sqrt{16} = 4 \text{ m}$$

Special Cases

• Hemisphere ($h = R$): The cap is exactly half a sphere. $V = \frac{2}{3}\pi R^3$, $S_{\text{total}} = 3\pi R^2$.
• Full sphere ($h = 2R$): The cap becomes the entire sphere. $V = \frac{4}{3}\pi R^3$ and the base radius collapses to zero.
• Shallow dome ($h < R$): Typical for contact lenses, shallow domes, and large-radius tank fills.

Applications

• Storage tanks: Liquid volume in partially filled spherical tanks as a function of fill depth
• Architecture: Dome roofs, planetariums, gas-holder caps -- volume and surface area for HVAC sizing
• Optics: Contact lens curved area, corneal cap geometry
• Aerospace: Spherical-cap nose cone heat-shield surface area