Spherical Cap Calculator
Find the volume and surface of a spherical cap from the sphere radius and cap height, recover the minor height from the base radius, or recover the sphere from height and base. The formulas are a = √(h(2R − h)), V = (1/3)πh²(3R − h), and curved area 2πRh.
Curved 62.8318530718 · total S = 113.097335529
- Base radius a = √(h(2R − h)) = √(2 × 8) = 4.
- Volume V = (1/3)πh²(3R − h) = (1/3) × 3.14159265359 × 4 × 13 = 54.4542726622.
- Curved surface = 2πRh = 62.8318530718. Total including the flat base = curved + πa² = 113.097335529.
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How to Use the Spherical Cap Calculator
Choose the two measurements you know — sphere radius and cap height, sphere radius and base radius, or height and base — then Calculate. The rest come from a = √(h(2R − h)), V = (1/3)πh²(3R − h), and curved area 2πRh.
- Select From R and h, From R and a, or From h and a.
- Enter non-negative numbers. Height must satisfy 0 ≤ h ≤ 2R. From R and a this page returns the minor cap, so 0 ≤ h ≤ R.
- Read the volume (or the recovered h or R), plus the curved dome and the total including the flat base.
A dome whose height equals the radius belongs on the Hemisphere Calculator (V = (2/3)πR³). The full ball belongs on the Sphere Calculator (V = (4/3)πR³). A band with two circular cuts belongs on the Spherical Zone Calculator (V = (1/6)πh(3a² + 3b² + h²)). The flat circular face alone belongs on the Circle Calculator (A = πa²). Converting cubic units belongs on the Volume Converter.
Volume and Surface
A spherical cap is the solid cut from a ball by one plane. The sphere has radius R. The cap has height h measured from the cutting plane to the pole. The cut is a circle of radius a. Pythagoras in the right triangle from the sphere’s center to the rim gives a² + (R − h)² = R², which rearranges to a = √(h(2R − h)).
Volume is V = (1/3)πh²(3R − h). The curved dome (a zone of one base) is 2πRh — it depends on R and h, not on a separately. The solid including the flat circular lid has total surface 2πRh + πa². Solving the same Pythagoras for the sphere gives R = (a² + h²) / (2h) when h > 0. From R and a there are two caps with the same base: the minor height h = R − √(R² − a²) and the complementary major height 2R − h. This page returns the minor cap. Enter h > R in From R and h when you want the deeper slice.
Spherical Cap Calculator Examples
Radius 5, height 2
a = √(2 × 8) = 4. V = (1/3)π × 4 × 13 = 52π/3 ≈ 54.4542726622. Curved = 20π ≈ 62.8318530718. Base = 16π ≈ 50.2654824574. Total = 36π ≈ 113.097335529.
Hemisphere, h = R = 5
a = 5, the great-circle face. V = (1/3)π × 25 × 10 = 250π/3 ≈ 261.799387799, the same volume as the Hemisphere Calculator. Curved = 50π. Total = 75π.
Full sphere, h = 2R = 10
a = 0 — the “cut” is a point at the opposite pole. V = (1/3)π × 100 × 5 = 500π/3 ≈ 523.598775598, the Sphere Calculator volume. Curved = 100π, the whole skin. The flat base vanishes.
From R = 5 and a = 4
Minor h = 5 − √(25 − 16) = 5 − 3 = 2, the first example. The major cap on the same base has height 8.
From h = 2 and a = 4
R = (16 + 4) / 4 = 5. The same cap, recovered from the slice without being told the sphere first.
Height past the diameter
R = 5 and h = 12 is impossible: the ball is only 10 units across. The calculator rejects it instead of inventing a larger sphere.
Hemisphere and Full Sphere
Set h = R. Then a = √(R(2R − R)) = R, V = (1/3)πR²(3R − R) = (2/3)πR³, curved = 2πR², total = 3πR². Those are the hemisphere formulas. A hemisphere is the cap whose cutting plane passes through the center.
Set h = 2R. Then a = 0, V = (4/3)πR³, and the curved area is 4πR² — the whole sphere. Set h = 0 and the cap collapses to a point. Between those ends, a shallower slice is a genuine different solid: you cannot swap in (2/3)πR³ and expect the right volume.
Common Spherical Cap Mistakes
Using (2/3)πR³ for a shallow slice
That is a hemisphere, on the Hemisphere Calculator, and only when h = R. A cap of height 2 on a radius-5 ball is 52π/3, not 250π/3.
Using (4/3)πR³ for a slice
That is the full ball, on the Sphere Calculator. Use (1/3)πh²(3R − h) unless h really is 2R.
Letting h exceed 2R
A cap cannot be taller than the sphere. Check 0 ≤ h ≤ 2R before substituting.
Taking the major cap from R and a
The same base circle sits on two caps. This page returns the minor one (h ≤ R). Enter the larger height in From R and h if the problem wants the rest of the ball.
Treating a zone (two bases) as a cap
A spherical zone or segment of two bases is a band with two circular cuts. That solid belongs on the Spherical Zone Calculator.
Converting liters on this page
V is already in the cube of your input unit. Use the Volume Converter for liters, gallons, and cups.
Spherical Cap Calculator FAQs
What is a spherical cap?
The solid cut from a ball by one plane. It has sphere radius R, height h, and a circular base of radius a = √(h(2R − h)).
What is the volume of a spherical cap?
V = (1/3)πh²(3R − h). For R = 5 and h = 2, V = 52π/3 ≈ 54.4542726622 cubic units.
What is the surface area of a spherical cap?
The curved dome is 2πRh. The total including the flat base is 2πRh + πa². For R = 5 and h = 2 those are 20π ≈ 62.832 and 36π ≈ 113.097.
How do you find the sphere radius from h and a?
R = (a² + h²) / (2h), for h > 0. For h = 2 and a = 4, R = 5.
Is a hemisphere a spherical cap?
Yes — the special case h = R. Use the Hemisphere Calculator when that is all you need. This page also covers shallower and deeper slices.
Are my numbers sent to a server?
No. The calculation runs locally in your browser.
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Reviewed by Prof. Dr. Khalil Mudassar (PhD) · Last updated: [mc_modified_date]
Sources
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Disclaimer
This calculator is intended for general educational and practical use. Review important results before relying on them in professional work.