🌊 Partially Filled Sphere Calculator
Calculate volume, radius, or fill height of spherical tanks and caps
Enter Sphere & Fill Level
Fill: 40% (h = 8 cm)
📊 Results
Sphere Radius (r)
10 cm
Fill Height (h)
8 cm
Fill Percentage
40%
Fill Volume
1,675.5 cm³
Total Capacity
4,188.8 cm³
Empty Space
2,513.3 cm³
Surface Width
12 cm
Cap Surface Area
502.7 cm²
📝 Step-by-Step Solution
Given: r = 10 cm, h = 8 cm (fill height from bottom)
Spherical cap formula: V = (πh²/3)(3r - h)
V = (π × 8² / 3)(3 × 10 - 8) = (π × 64 / 3)(22)
V = 1675.52 cm³ = 1.676 liters
Fill percentage: 1675.52 / 4188.79 × 100 = 40%
📐 Partially Filled Sphere Formulas
Spherical Cap Volume: V = (πh²/3)(3r - h)
Alternative: V = πh(3a² + h²) / 6 (where a = surface radius)
Full Sphere: V_total = (4/3)πr³
Surface Width: w = 2√(h(2r - h))
Cap Surface Area: A = 2πrh
Understanding Partial Sphere Fill
🌊 Spherical Cap
Portion of sphere cut by a plane. Also called spherical segment. Volume depends on height h and sphere radius r.
📏 Fill Height
Measured from bottom. h = 0 means empty, h = 2r means full. At h = r (halfway), volume is exactly 50%.
⚖️ Non-Linear Fill
Volume doesn't match height linearly. Half height ≠ half volume. The spherical cap formula accounts for curvature.
🏭 Applications
Spherical tanks, domes, balls. Used in propane tanks, pressure vessels, water towers, and storage spheres.
Frequently Asked Questions
What is the formula for partial sphere volume?
V = (πh²/3)(3r - h). Where h is fill height from bottom, r is
sphere radius. This is the spherical cap formula.
Is half height equal to half volume?
Yes, exactly 50%. When h = r (fill height equals radius), volume
is exactly half. This is a special property of spheres.
How do I find fill height from volume?
Requires cubic equation solving. Our calculator uses numerical
methods (Newton-Raphson) to find h from V.
What is a spherical cap?
Dome-shaped portion of sphere. Created when a plane cuts through a
sphere. The curved surface above the cut plane.
How accurate for real tanks?
Mathematically exact for perfect spheres. Real tanks may have
supports, fittings, or slight deformations affecting actual capacity.
What if h > 2r?
Invalid input. Maximum fill height is diameter (2r). Calculator
caps h at 2r automatically.
How do I calculate surface width?
w = 2√(h(2r - h)). The diameter of the circular liquid surface at
height h.
What is cap surface area?
A = 2πrh. The curved surface area of the spherical cap. Not the
flat circular base.
Can I use diameter instead of radius?
r = d/2. Convert diameter to radius first. Or use V = (πh²/6)(3d -
2h) with diameter.
How do I convert to gallons?
1 gallon = 3785.41 cm³. Our calculator can output in gallons,
liters, or other units automatically.
What is percent from height?
Use fill volume / total volume × 100. Not simply h/(2r) × 100 due
to non-linear relationship.
How to find height for X% fill?
Numerical solution required. Use our "Percent Filled" mode to
enter percentage and get required height.
What about spherical segment (two caps)?
Different formula. For a slice between two parallel planes: V =
(πh/6)(3a² + 3b² + h²).
Why use spherical tanks?
Maximum volume for surface area. Spheres efficiently store
pressure. Common for LNG, propane, water storage.
How does fill rate compare to height?
Faster near middle, slower at top/bottom. Volume change per height
unit is maximum at h = r.
What is the wetted surface area?
A = 2πrh. This equals the cap surface area contacted by liquid.
Important for heat transfer.
Can I use this for hemispheres?
Yes, set h ≤ r. A hemisphere is half a sphere. Maximum fill height
equals the radius.