A hemisphere is exactly half a sphere, cut through the centre. This hemisphere volume calculator takes a radius, a diameter or a target volume and returns the volume, the curved surface area, the area of the flat circular face and the true total surface area, plus capacity in litres and US gallons for bowls, domes and tank ends.
Arb Digital built this page around one detail that a surprising number of competing pages get wrong: the total surface area of a solid hemisphere is 3πr², not 2πr². Halving a sphere creates a new flat face that did not exist before, and that face has to be counted. The calculator reports the curved area and the flat face separately so you can see the difference and choose the right one for your job.
What This Hemisphere Calculator Does
Everything about a hemisphere follows from its radius, so the calculator accepts whichever measurement you happen to have. Radius is the textbook input. Diameter is how domes, bowls, dishes and tank heads are actually specified in catalogues, and the tool halves it for you. Volume works backwards, taking a capacity you need to hit and returning the radius that delivers it.
The results split surface area into three figures. Curved area is the dome itself, 2πr². The flat circular face is the disc created by the cut, πr². Total surface area is the sum, 3πr², which applies when the hemisphere is a closed solid with the flat side exposed. Keeping the three apart is the whole point, because different jobs need different ones.
Volume is reported in cubed units with litres and gallons underneath. For mixing bowls, hemispherical tank ends, domed lids and half-round basins, capacity is the number that matters far more than the abstract cubic figure.
How to Use It
- Choose radius, diameter or volume depending on what you already know. The field label changes to match.
- Enter the value. In volume mode the unit is treated as cubed rather than linear.
- Pick your unit. Areas come back squared, volume cubed, with capacity underneath.
- Choose your surface area figure. Curved for an open dome or bowl, total for a closed solid with the flat face exposed.
- Read the working line to confirm the substitution matches your intended numbers.
The Formula: How a Hemisphere Is Calculated
For a hemisphere of radius r:
- Volume: V = ⅔πr³, exactly half the sphere formula. With r = 6 cm: ⅔ × π × 216 = 452.39 cm³, or about 0.452 litres.
- Curved surface area: 2πr², exactly half the sphere's 4πr². Here 2 × π × 36 = 226.19 cm².
- Flat circular face: πr² = 113.10 cm².
- Total surface area: 3πr² = 339.29 cm².
- Height of the dome = r, and the base diameter = 2r, so a dome is always exactly twice as wide as it is tall.
Working backwards from a volume, the radius is the cube root of 3V divided by 2π. For a 452.39 cm³ target that returns 6 cm, closing the loop. The standard relations are set out at Wolfram MathWorld's hemisphere entry.
The Flat Face Catch: Why Total Area Is 3πr²
Cutting a sphere in half does two things at once. It halves the curved surface, from 4πr² down to 2πr², and it creates a brand new flat circular face of πr² that was interior material a moment earlier. A solid hemisphere therefore has 3πr² of surface in total, not half of the sphere's figure.
The consequence is that a hemisphere does not have half a sphere's surface area. It has three quarters of it: 3πr² against 4πr². Volume halves cleanly, surface area does not. That asymmetry is exactly what the square-cube law predicts, and it is why cutting an object in half always increases the total exposed surface of the pieces.
This matters commercially. If you are plating, painting, anodising or coating solid half-round components and you priced the job at half a sphere's surface area, you have under-ordered by 50% on the material. If you are computing heat loss from a solid hemisphere sitting flat side up, the exposed flat disc conducts and radiates too. Only when the flat face is bonded, buried or sitting on a floor can you legitimately drop it and use 2πr² alone.
When to Use the Curved Area Alone
Plenty of real hemispheres are open shells rather than solids, and for those the curved figure is the correct one. A garden dome, a geodesic-style canopy, a hemispherical light diffuser, a bowl being glazed on the outside, a half-round roof: none of these has a flat face to finish, because the flat side is the opening.
Bowls are the case that needs care, because a bowl has two curved surfaces, an inside and an outside, and their radii differ by the wall thickness. Glazing both faces of a thin bowl is roughly twice the curved area, plus the narrow rim. Treating a bowl as a solid hemisphere and using 3πr² would be wrong in both directions at once.
For the flat disc on its own, whether you are cutting a base plate or a lid, the circle calculator gives the area and circumference from the same radius. If you need to turn an area into a quantity of coating at a given spread rate, the paint calculator takes it from there.
Domes, Tank Ends and Partial Fill
Hemispherical ends are standard on pressure vessels and many storage tanks because a curved end handles internal pressure far better than a flat plate. A cylindrical tank with two hemispherical ends holds the cylinder volume plus one full sphere's worth from the two halves, which is a neat shortcut worth remembering: two hemispheres of the same radius equal one sphere exactly. The cylindrical body can be computed with the cylinder volume calculator and simply added.
Partial fill in a bowl-shaped hemisphere follows the spherical cap formula rather than a simple proportion. Filling a bowl to half its depth does not give half its capacity; it gives a good deal less, because the bowl is much narrower near the bottom. Anyone reading depth as a proportion of contents in a hemispherical vessel will overestimate what is left.
A related fact worth knowing for stability work: the centroid of a solid hemisphere sits at three eighths of the radius above the flat face, not halfway up. That is why a solid half-round object is far more stable resting on its flat side than intuition suggests, and it is the basis of self-righting hull shapes. To move any of these capacity results between units, use the volume converter.
Hemisphere or Full Sphere: Which Page You Want
This page covers the half-solid case: a dome, bowl, tank head or half-round component, with the flat face treated explicitly. If your object is a complete sphere, use our sphere volume calculator instead, which handles the full solid from a radius, diameter or circumference. The relationship between the two pages is simple and exact: a hemisphere holds half the volume and has three quarters of the surface area of the sphere with the same radius.
All unit conversions used on this page follow the definitions published by NIST, and cubed factors are cubed rather than applied linearly, so one cubic metre is 1,000,000 cubic centimetres.
Arb Digital designs, writes and ships free calculator pages built to rank, complete with schema markup and the technical SEO underneath them.
Browse All Free Tools Talk To Arb DigitalCommon Mistakes to Avoid
- Halving the sphere's surface area. A solid hemisphere has 3πr², which is three quarters of a sphere's area, not half.
- Counting the flat face when it is not there. An open dome or shell has only the curved 2πr².
- Entering a diameter as a radius. Volume depends on the cube of the radius, so that slip makes the answer eight times too large.
- Treating bowl depth as proportional to contents. A hemispherical bowl filled halfway holds well under half its capacity.
- Ignoring wall thickness in a shell. The inside and outside of a bowl have different radii, so their areas and volumes differ.
Related Free Tools From Arb Digital
For the full solid, use the sphere volume calculator. For tanks with hemispherical ends, add the body from the cylinder volume calculator. The circle calculator covers the flat face on its own, and the volume converter moves any capacity between units. Browse the full free online tools hub for everything else.
Frequently Asked Questions
Volume is two thirds of pi times the radius cubed, exactly half a sphere. A hemisphere of radius 6 cm holds 452.39 cubic centimetres, or about 0.452 litres.
Three pi r squared. The curved dome contributes 2πr² and the flat circular face created by the cut contributes another πr². For a 6 cm radius that is 339.29 square centimetres.
Because cutting a sphere in half creates a new flat face that did not previously exist. Volume halves cleanly, but surface area comes to three quarters of the sphere's, not half.
Use the curved 2πr² for open shells such as domes, canopies and diffusers, and whenever the flat side is bonded down, buried or resting on a floor so it is not an exposed surface.
Multiply the volume by three, divide by two pi, and take the cube root. A 452.39 cubic centimetre hemisphere returns a radius of 6 cm.
Its height equals its radius, so a dome is always exactly twice as wide as it is tall. A 6 cm radius hemisphere stands 6 cm high on a 12 cm base.
No. The bowl narrows toward the bottom, so filling to half the depth holds considerably less than half the volume. Partial fill follows the spherical cap formula rather than a simple proportion.
Results are mathematical values rounded for display. Real domes and bowls have wall thickness, rims and manufacturing tolerance, so treat these figures as nominal.
