Eleven solid models organize the workbench by geometry
Prisms and cylinders repeat a constant base area through a perpendicular height, so their volume is B·h. Pyramids and cones taper to a point and use B·h/3. Spheres and ellipsoids scale three perpendicular radii. Capsules, spherical caps, frustums, and tubes are assembled from or subtracted among simpler solids. A formula works only when the measured object matches those assumptions.
This page is designed for switching among sphere, cone, cube, cylinder, rectangular prism, capsule, spherical cap, conical frustum, ellipsoid, square pyramid, and tube workflows. When the object is definitely one right circular cylinder, the cylinder volume calculator removes unrelated shape choices and focuses on radius, perpendicular height, and output volume.
A rain-storage vessel combines a cylinder and a rounded cap
Consider a tank modeled as a vertical cylinder of radius 0.8 m and straight height 1.9 m topped by a hemisphere of the same radius. This is a composite solid, not a single cylinder.
- Calculate the cylindrical body
V₁=πr²h=π(0.8)²(1.9)=1.216π≈3.820 m³. Height must be perpendicular to the circular base.
- Calculate half a sphere
A hemisphere has V₂=(1/2)(4πr³/3)=2π(0.8)³/3≈1.072 m³. Using the full sphere formula would double-count the cap.
- Add, convert, and label assumptions
The geometric capacity is about 4.893 m³, or 4,893 L. Wall thickness, fittings, freeboard, tilt, and usable fill level are not included, so this should not be presented as guaranteed operating capacity.
Output units reveal whether the task is geometry or capacity
Cubic units preserve spatial scale
Dimensions measured in feet naturally produce ft³; metric dimensions can produce mm³, cm³, or m³. A factor of 100 in length becomes a factor of one million in volume.
Liters are a named cubic relationship
NIST identifies 1 L with 1 dm³ and 1 mL with 1 cm³. That exact bridge supports container-capacity reporting, but a U.S. gallon conversion requires an explicit system. When a vessel is specified by its circular boundary instead of its radius, the circumference calculator can recover the base radius before any cubic formula is applied.
Surface coverage remains separate
Paint, lining, and wrapping decisions usually need surface area, not interior volume. Use the area calculator for plane pieces and an appropriate surface model for a solid’s exterior.
Composite-solid estimates fail at their seams
- Check whether listed dimensions are internal or external; capacity and displaced material can differ by wall thickness.
- Convert each length before applying a formula, especially when the workbench permits independent units for different dimensions.
- At junctions, add only nonoverlapping regions and subtract bores or inner radii once; a tube is an outer cylinder minus an inner cylinder.
- Keep a diagram with the model and label every approximation, because rounded corners or sloped bottoms may not match any ideal solid exactly.