Hemispherical Frustum (Spherical Cap) Volume and Surface Area Calculator

Enter Calculation

Sphere radius r

Height h (h ≤ r)

Enter r and h in the same length unit. Constraint: 0 < h ≤ r (at most a hemisphere).

Formula

Show calculation steps (2)

1. Total Surface Area

   

   Curved (cap) area plus flat circular base area; base radius a = sqrt(h(2r - h))

2. Base Radius

   

   Radius of the flat circular base of the cap

Results

Volume V

0.654498

cubic length units (length³)

Total surface area S	5.497787 (length²)
Curved (dome) area	3.141593
Flat base area	2.356194
Base circle radius a	0.866025

What is a hemispherical frustum (spherical cap)?

A spherical cap is the solid you get when a sphere of radius r is sliced by a single horizontal plane and you keep the dome-shaped piece above (or below) that plane. Its height h is measured from the flat cut face up to the top of the sphere. This tool restricts h to be no greater than r, so the largest possible solid is exactly a hemisphere. The flat circular cut has radius a, where \(a^2 = h(2r - h)\).

A spherical cap is the region of a sphere of radius r cut off by a single plane, with cap height h.

How to use the calculator

Enter the sphere radius r and the cap height h in the same length unit (centimeters, inches, meters — your choice; results follow as that unit cubed and squared). Make sure 0 < h ≤ r. The calculator returns the volume, the total surface area (curved dome plus flat base), and useful intermediate values: the dome area, the base disk area, and the base circle radius a.

The formulas explained

The cap volume is $$V = \frac{\pi h^2}{3}(3r - h)$$ The curved spherical surface is the spherical zone $$2\pi r h$$ a neat result from Archimedes. The flat base is a circle of area $$\pi a^2 = \pi h(2r - h)$$ Adding these gives the total surface area $$S = 2\pi r h + \pi h(2r - h) = \pi h(4r - h)$$

Key quantities: sphere radius r, cap height h, the curved (spherical) surface and the flat circular base.

Worked example

For \(r = 1\) and \(h = 0.5\): $$a = \sqrt{0.5 \times 1.5} = \sqrt{0.75} \approx 0.8660$$ $$V = \pi \times \frac{0.25}{3} \times 2.5 = \pi \times 0.20833 \approx 0.65450$$ $$\text{Curved area} = 2\pi \times 1 \times 0.5 = \pi \approx 3.14159$$ $$\text{Base area} = 0.75\pi \approx 2.35619$$ $$\text{Total } S = \pi \times 0.5 \times 3.5 = 1.75\pi \approx 5.49779$$

FAQ

Why is h limited to r? The original tool models "at most a hemisphere," so it caps the height at the sphere radius. Mathematically a cap can have h up to 2r, but this version stays within h ≤ r.

Does the surface area include the flat disk? Yes. The reported total surface area is the curved dome plus the flat circular cut. If you only need the dome, use the curved-area row.

What happens at h = r? You get a perfect hemisphere: \(V = \frac{2}{3}\pi r^3\), \(\text{dome} = 2\pi r^2\), \(\text{base} = \pi r^2\).