BIZOMAT MAKEOS · Geometry
Volume and Surface of Solids
What volume, surface area and mass does this solid have?
Volume, surface area and mass of spheres, hollow spheres, spherical caps, zones, sectors and wedges, cones, pyramids, cylinders, cylindrical hoofs, paraboloids, barrels, tori and ellipsoids.
Open the calculatorRuns in your browser. No account needed.
How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Solid | choice | Solid type; the option text names each dimension. |
| Dimension A | mm | First dimension as described. |
| Dimension B | mm | Second dimension where needed. |
| Dimension C | mm | Height or length where needed. |
| Wedge angle | ° | Included angle of a spherical wedge. |
| Density (for mass) | kg/m³ | Material density to give a mass (0 to skip). |
What each input means
Open an input to see what it is, its units, the worked example and the common mistakes, with a small diagram.
What this means: Solid
- What it is
- The solid whose volume and surface you want. Each option names what Dimensions A, B and C and the wedge angle mean for that solid, so read it before filling them in.
- Units and format
- Choose one of 16 options from the list: Sphere (A = diameter); Hollow sphere (A = outer dia, B = inner dia); Spherical cap or segment (A = base radius, C = height); Spherical zone (A, B = end radii, C = height); Spherical sector (A = sphere radius, C = cap height); Spherical wedge (A = radius, angle); Cone (A = base dia, C = height); Square pyramid (A = base side, C = height); Cylinder (A = dia, C = length); Cylindrical hoof, cut through the base diameter (A = radius, C = height); Paraboloid (A = base dia, C = height); Paraboloidal segment (A, B = end dias, C = height); Barrel, parabolic sides (A = bung dia, B = head dia, C = length); Torus (A = centre-line dia, B = tube dia); Ellipsoid (A, B, C = full axes); Cube (A = side).
- Example
- The form opens with “Spherical cap or segment (A = base radius, C = height)” selected as a worked example. Choose the option that matches your case.
- Get it right
- Some solids take diameters (sphere, cone, cylinder) and others radii (cap, sector, wedge, hoof); check the option text.
What this means: Dimension A
- What it is
- The main size of the chosen solid: a diameter (sphere, cone, cylinder, paraboloid, barrel bung, torus centre line), a radius (cap base, zone end, sphere for sector or wedge, hoof), a full axis of an ellipsoid, or the side of a cube or pyramid base.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 10000000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 100 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Radius and diameter are easy to swap; the option text for each solid says which one A is.
What this means: Dimension B
- What it is
- The second size where the solid has one: the inner diameter of a hollow sphere, the other end radius of a zone, the other end diameter of a paraboloidal segment, the head diameter of a barrel, the tube diameter of a torus or the second axis of an ellipsoid.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0 mm to 10000000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 60 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- For a hollow sphere it must be smaller than A. Solids that do not name B ignore it.
What this means: Dimension C
- What it is
- The height or length of the solid along its axis: the height of a cap, zone, cone, pyramid, hoof or paraboloid, the length of a cylinder or barrel, the cap height of a spherical sector, or the third axis of an ellipsoid.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0 mm to 10000000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 30 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Measure square to the base, not along a sloping side. For a spherical sector the cap height cannot exceed the sphere diameter.
What this means: Wedge angle
- What it is
- The angle at the axis between the two flat faces of a spherical wedge, like the angle of an orange segment. Only the wedge uses it.
- Units and format
- Enter a number in °: degrees of angle. The form accepts 0.001° to 360°. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 90° as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- The wedge volume rises in direct proportion to the angle.
- Get it right
- Give the full included angle in degrees; 360 gives the whole sphere.
What this means: Density (for mass)
- What it is
- The density of the material the solid is made of, used only to turn the volume into a mass. Take it from the material supplier’s data sheet.
- Units and format
- Enter a number in kg/m³: kilograms per cubic metre. The form accepts 0 kg/m³ to 50000 kg/m³. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 7850 kg/m³ as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- The mass is directly proportional to density; volume and surface are not affected.
- Get it right
- Set 0 to skip the mass. Use kg/m³: a value in g/cm³ must be multiplied by 1000.
Method and formulas
- Sphere πd³/6; cap πh(3a² + h²)/6; zone πh(3a² + 3b² + h²)/6; sector 2πr²h/3; wedge (θ/360) × 4πr³/3.
- Cone and pyramid ⅓ × base × height; hoof 2r²h/3; paraboloid πr²h/2; paraboloidal segment πh(R² + r²)/2.
- Barrel with parabolic staves πL(8D² + 4Dd + 3d²) ÷ 60 (D at the bung, d at the heads); torus 2π²Rr²; ellipsoid πabc ÷ 6 with full axes.
- Example: a spherical cap of 100 mm base radius and 30 mm height holds 485.38 cm³.
Limitations
- Ideal solids; the barrel formula assumes parabolic staves.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Find volumes and masses of turned and formed parts
- Estimate tank and vessel capacities
- Check CAD volumes
Who it is for
- Design engineers
- Toolmakers
- Estimators
What you get
- Volume
- Surface area (where a simple formula exists)
- Mass
Interpreting the result
- For hollow parts, subtract the inner volume or use the hollow shape options.
References
- Machinery’s Handbook, 29th edition (Industrial Press, 2012), printed pp. 82–88. Topic reference; no source tables or recommended settings are reproduced.
BIZOMAT tools are screening and decision-support utilities. Verify important financial, engineering, safety, legal or operational decisions using appropriate professional standards and your organisation’s approved data.