Calculate Sphere Volume, Surface Area and Capacity
Find a sphere from its radius, diameter, circumference, surface area or known volume. You can also calculate a hollow spherical shell or the volume of a spherical cap and part-filled spherical tank.
Sphere Volume Calculator
Choose a sphere type, enter the known measurement and select your output units. All calculations run locally in your browser.
Sphere volume
Calculated from unrounded measurements
Your result will appear here.
Calculated measurements
| Measurement | Formula or basis | Result |
|---|---|---|
| Enter a valid measurement to see results. | ||
Calculation steps
- Choose a mode and enter the known measurement.
Recent calculations
How to Use This Sphere Volume Calculator
- Select a mode. Use solid for a complete sphere, hollow for shell material and cavity capacity, or cap for a partial sphere and spherical tank fill.
- Choose the known measurement. Solid mode accepts radius, diameter, great-circle circumference, total surface area or known volume. Hollow and cap modes use a length measurement.
- Enter the value and its unit. Decimal values, scientific notation and simple fractions such as 1/2 are accepted. Keep units in the unit menus.
- Add an inner size or cap height when requested. A hollow sphere needs a smaller inner radius. A cap height is measured upward from the sphere's lowest point.
- Select output units and precision. Length, area, volume and mass each have independent output settings.
- Add density only when you need mass. Use the average density of the actual material or liquid under the relevant conditions.
The calculator converts measurements to SI base units, solves with unrounded values, then converts the results to your chosen units. Use internal measurements for tank capacity. Exterior measurements include wall thickness and will overstate the usable space inside a vessel.
What the Calculator Finds
A sphere is the set of points in three-dimensional space that are the same distance from one center. That constant distance is the radius. A ball includes the space inside the spherical surface, so volume technically describes the ball. In everyday measurement, "sphere volume" remains the standard search term and this calculator follows that convention.
All outputs describe ideal geometry. Openings, necks, supports, seams, coatings, fittings and manufacturing tolerances require separate allowances. For a physical spherical tank, use verified internal dimensions and account for any non-spherical features.
Sphere Volume and Surface Area Formulas
Let r be radius. Sphere volume equals four-thirds of pi multiplied by the cube of the radius. Total surface area equals four times pi multiplied by the square of the radius.
Surface area: S = 4πr²
If diameter d is known, radius is d/2. Substitution gives V = πd³/6 and S = πd². If the great-circle circumference C is known, r = C/(2π), V = C³/(6π²) and S = C²/π.
The great-circle cross-sectional area is πr². It is the area of the largest flat circular slice through the center. A hemisphere has half the full sphere volume, curved area 2πr² and total area 3πr² when its flat circular base is included.
The calculator uses the full precision of pi supplied by the browser rather than the rounded value 3.14. Display rounding is applied only after all formulas and conversions are complete.
Hollow Sphere and Spherical Shell Formula
A hollow sphere has outer radius R and inner radius r. The material occupies the difference between the outer ball and the inner cavity. Direct subtraction of nearly equal cubes loses precision for a thin wall, so the calculator uses the factored difference of cubes.
Cavity capacity = (4π/3)r³
Wall thickness = R - r
Outer surface area is 4πR² and inner surface area is 4πr². Combined exposed area is their sum. This assumes a fully closed shell with a uniform concentric cavity. An opening removes material and exposed area, so a vessel with a neck or port needs a separate correction.
Material volume plus cavity capacity equals the outer sphere volume. This identity is useful for checking measurements. The inner dimension must be smaller than the outer dimension and both must use the same measurement type.
Spherical Cap and Part-Filled Sphere Formula
A spherical cap is the portion cut from a sphere by a plane. It also models liquid at the bottom of an ideal spherical tank. Let R be sphere radius and h be cap or liquid height measured from the lowest point. Height can range from zero to the full diameter 2R.
Base radius = √(h(2R - h))
Curved area = 2πRh
Free-surface area = πh(2R - h)
At h = R, the cap is a hemisphere and contains exactly half the sphere. At h = 2R, the cap equals the full sphere. Near a full tank, the calculator finds the small empty cap from 2R - h and subtracts it from full capacity. This stable approach avoids losing meaningful digits when two large, close values appear in the direct formula.
Free-surface area is the horizontal circular liquid surface. Curved area is the spherical wall in contact with the cap contents. The model assumes a level plane and a true sphere.
Reverse Calculation from Surface Area or Volume
Solid mode also works backward. If total surface area is known, divide it by 4π and take the square root. If volume is known, multiply by 3/(4π) and take the cube root.
From volume: r = ∛(3V/(4π))
Reverse solving is useful when a specification lists capacity but you need an equivalent diameter, or when coating area is known and you need the implied radius. It assumes the object is a perfect sphere. Small measurement errors matter more after powers are applied: a 1% radius error creates roughly a 2% surface-area error and a 3% volume error.
Worked Sphere Volume Examples
Example 1: solid sphere from radius
A sphere has radius 5 cm. Volume is (4/3)π × 5³ = 523.5988 cm³. Surface area is 4π × 5² = 314.1593 cm². Diameter is 10 cm, great-circle circumference is 31.4159 cm and hemisphere volume is 261.7994 cm³. The same full volume equals about 0.523599 litre.
Example 2: hollow steel shell
A shell has outer radius 5 cm and inner radius 4 cm. Material volume is (4π/3)(1)(25 + 20 + 16) = 255.5162 cm³. The cavity holds 268.0826 cm³. With density 7.85 g/cm³, estimated shell mass is about 2.006 kg. Openings and weld material are not included.
Example 3: spherical cap
A sphere has radius 10 cm and cap height 4 cm. Cap volume is π × 4² × (10 - 4/3) = 435.6342 cm³. Full sphere capacity is 4,188.7902 cm³, so the cap is 10.4% of the total. Its base radius is 8 cm and its free-surface area is 201.0619 cm².
Sphere Units, Conversions and Rounding
Length conversions must be applied before powers. If a length factor is f, the matching area factor is f² and the matching volume factor is f³. The calculator converts each input separately, so you can enter an outer radius in centimetres, an inner radius in millimetres and request cubic inches.
| Quantity | Useful relationship | Common use |
|---|---|---|
| Length | 1 in = 2.54 cm exactly | Radius, diameter, circumference, wall and cap height |
| Volume | 1 cm³ = 1 mL | Capacity and material volume |
| Volume | 1 m³ = 1,000 L | Large tanks and engineering work |
| Density | 1 g/cm³ = 1,000 kg/m³ | Mass estimate from geometric volume |
US gallons and Imperial gallons are different units, so the calculator lists them separately. Precision controls display digits, not the internal calculation. Very large and very small valid values switch to scientific notation instead of being rounded to an incorrect zero.
Sphere, Circle and Hemisphere Compared
| Shape | Dimension | Main measurement |
|---|---|---|
| Circle | Two-dimensional | Area = πr² |
| Sphere surface | Three-dimensional boundary | Surface area = 4πr² |
| Solid ball | Three-dimensional region | Volume = 4πr³/3 |
| Hemisphere | Half a sphere | Volume = 2πr³/3 |
| Spherical cap | Portion cut by a plane | Volume = πh²(R - h/3) |
A circle calculator handles a flat cross-section. This sphere calculator handles three-dimensional volume and curved surface area. Keep the difference clear when reading a drawing because a listed circular area is not the surface area of the sphere.
Accuracy, Assumptions and Good Measurement Practice
- Measure through the true center when taking diameter. An off-center chord is shorter than the diameter.
- Use internal dimensions for liquid capacity and external dimensions for the overall envelope.
- Confirm that a hollow shell has a concentric cavity before assuming uniform wall thickness.
- Use a calibrated density for the real substance. Density varies with composition, temperature, pressure and moisture.
- Keep extra digits during measurement and conversion. Round only the final result to the precision required.
- Do not treat a geometric mass estimate as a structural, pressure-vessel or shipping certification.
The formulas were reviewed against OpenStax sphere formulas. Unit relationships follow the NIST conversion guidance and the BIPM SI Brochure.
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Frequently Asked Questions
What is the formula for the volume of a sphere?
The volume formula is V = (4/3)πr³, where r is the radius. Cube the radius, multiply by pi, then multiply by four-thirds.
Is radius half the diameter of a sphere?
Yes. Radius is the center-to-surface distance and diameter passes through the center from one side to the other, so r = d/2.
How do I calculate sphere volume from diameter?
Use V = πd³/6, or divide the diameter by two and use V = (4/3)πr³. Both formulas give the same result.
Can I calculate sphere volume from circumference?
Yes. Find radius with r = C/(2π), then calculate volume. The direct formula is V = C³/(6π²).
Can the calculator find radius from area or volume?
Yes. Solid mode solves r = √(S/(4π)) from surface area or r = ∛(3V/(4π)) from a known volume.
How is hollow sphere volume calculated?
Subtract the inner ball volume from the outer ball volume. This tool uses the stable factored formula (4π/3)(R - r)(R² + Rr + r²).
What is a spherical cap?
A spherical cap is the portion of a sphere cut by a plane. Its volume is πh²(R - h/3), where R is sphere radius and h is cap height.
Which units does the sphere calculator support?
It supports common metric and US customary length, area, volume, capacity, density and mass units, including separate US and Imperial gallons.
How does density produce a mass estimate?
Mass equals density multiplied by the relevant volume. The calculator uses solid volume, shell material volume or cap volume according to the selected mode.
Why is my result slightly different from a manual calculation?
A manual result may use 3.14 for pi or round intermediate values. This tool uses browser precision and rounds only the displayed final values.
Measurement Disclaimer
This calculator provides geometric estimates for education and planning. Verify critical dimensions, density values, tolerances and engineering requirements with appropriate instruments and qualified professionals.