Calculate the volume of a spherical cap (dome-shaped segment) using radius and height. Useful for tank design, arch construction, and geometry problems.

Volume: 41.89 cubic units

Formula: V = (π × h² × (3r − h)) / 3 — Valid when 0 < h ≤ 2r (cap height cannot exceed diameter).

What is a Spherical Cap?

A spherical cap is a portion of a sphere cut off by a plane. If the plane passes through the sphere's center, the cap is a hemisphere. The cap is defined by two parameters: the radius of the original sphere (r) and the height of the cap (h), measured from the base plane to the top of the cap.

V = (π × h² × (3r − h)) / 3

Where:

• V — volume of the spherical cap
• r — radius of the sphere
• h — height of the cap (0 < h ≤ 2r)

The formula is derived from integral calculus by rotating the circle segment around the axis of symmetry. This shape appears frequently in real-world applications:

• Tank design — spherical storage tanks and domes
• Architecture — dome roofs and vaulted ceilings
• Engineering — pressure vessels and satellite dishes
• Geometry — problems involving spherical segments

For example, a dome with r = 5 units and h = 2 units has a volume of about 41.89 cubic units. The total sphere volume would be V = (4/3)πr³ = 523.6 cubic units, so this cap represents about 8% of the full sphere.

The cap's base radius (a) can also be calculated as a = √(2rh − h²), though this calculator focuses on volume.

How to Use This Calculator

1. Enter Sphere Radius: Input the radius of the original sphere in the first field.
2. Enter Cap Height: Input the height of the spherical cap in the second field. This is the distance from the flat base to the highest point of the cap.
3. Calculate: Click the “Calculate Volume” button or press Enter on your keyboard.
4. Reset: Use the “Reset” button to restore default values (r = 5, h = 2).
5. Interpretation: The result shows the volume in cubic units. For example, if radius = 5 and height = 2, the volume is 41.89 cubic units.
6. Validation: The cap height must be between 0 and 2r (diameter). If h > 2r, the calculator will show an error.