Hemisphere Volume Calculator
A hemisphere with a radius of 6 has a volume of about 452.39, exactly half the matching sphere's 904.78. This hemisphere volume calculator uses V = (2/3) pi r cubed and also reports the curved surface area, the flat base area, and the total surface area, the three measures that trip people up because a hemisphere gains a circular face a sphere never had. Enter a single radius and the calculator returns four figures at once: the volume the shape encloses, the curved dome-shaped surface you would paint or glaze, the flat circular base created by the cut, and the combined total surface area. That single-input design works because every dimension of a hemisphere scales from one radius, so there is nothing else to configure. It is useful for concrete dome footings, mixing bowls, storage tank caps, planetarium domes, and any half-sphere part in wood shop, 3D printing, or geometry homework where a plain sphere calculator only tells half the story.
Quick answer
A hemisphere is half a sphere, cut through the center, so its volume is half the sphere formula.
Volume
452.3893
Curved surface area
226.1947
Base area
113.0973
Total surface area
339.292
What this tells you
- •A hemisphere is half a sphere, cut through the center, so its volume is half the sphere formula.
- •Volume = (2/3) x pi x radius cubed.
- •The curved (dome) surface is 2 pi r squared, half the sphere's skin.
- •Total surface area adds the flat circular base: 3 pi r squared altogether.
- •Only one input, the radius, drives every result, since a hemisphere has no independent height or width to set.
- •Volume scales with the cube of the radius, so a hemisphere twice as wide holds eight times as much.
- •Surface area scales with the square of the radius, so doubling the radius quadruples both the dome and the base.
How to Use
- 1Enter the radius of the hemisphere.
- 2Read the volume as the main result.
- 3Check the curved, base, and total surface areas below.
- 4For a dome measured across its widest point, halve that diameter to get the radius first.
- 5Match the unit of your answer to the unit you entered. A radius in centimeters returns a volume in cubic centimeters and areas in square centimeters.
How It Works
Formula
V = (2/3) pi r^3, total surface = 3 pi r^2A full sphere has volume (4/3) pi r cubed, and slicing it in half through the center halves that figure to (2/3) pi r cubed, where r is the radius measured from the center of the flat cut to the dome's surface. For radius 6, that works out to (2/3) x pi x 216 = 452.39 cubic units. The surface splits into two pieces: the curved dome, 2 pi r squared, and the newly exposed flat circular base, pi r squared, which together give the total surface area of 3 pi r squared. The calculator assumes the cut passes exactly through the sphere's center, since any off-center slice turns the shape into a spherical cap instead, a different shape with its own height variable.
Calculation note: values are processed in the order shown above, using the current input units.
Worked Examples
Radius 6 hemisphere
Two thirds of pi times 216 gives 452.3893, which rounds to 452.39 cubic units. The total surface area works out to 339.29 square units, made up of a 226.19 curved dome and a 113.10 flat base. This is the calculator's default example and a useful baseline for checking that any other radius scales correctly from it.
A 10 cm mixing bowl (radius 5 cm)
A hemispherical mixing bowl with a 5 cm radius holds about 261.80 cubic centimeters, roughly 262 milliliters or 0.26 liters filled level with the rim. Bakers and kitchen designers use this kind of check to compare bowl capacity against a recipe's liquid volume before pouring. The bowl's rim, the flat circular base, measures 78.54 square cm if you need that figure for a lid or liner.
An 18 m greenhouse dome (radius 9 m)
A greenhouse dome spanning 18 meters across has a radius of 9 meters, giving a curved glazing surface of 508.94 square meters, the panel area a contractor would quote for glass or polycarbonate. The enclosed volume comes out to 1,526.81 cubic meters, useful for sizing heating or ventilation equipment. The flat base, 254.47 square meters, is the footprint the foundation ring needs to cover.
An 8 cm glass paperweight (radius 4 cm)
A hemispherical glass paperweight with a 4 cm radius contains about 134.04 cubic centimeters of material. Its flat base, where a label or engraving usually sits, measures 50.27 square centimeters. This kind of check confirms a mold or 3D print will use the expected amount of material before production starts.
A 20 cm salad bowl (radius 10 cm)
A wide hemispherical salad bowl with a 10 cm radius holds 2,094.40 cubic centimeters, close to 2.09 liters filled to the brim. Its total surface area, 942.48 square centimeters, covers the curved bowl wall (628.32 square cm) and the flat rim base (314.16 square cm). Doubling the radius from the 5 cm bowl example above multiplies the volume by 8, not 2, which is the scaling rule worth remembering for any hemisphere.
A small end cap (radius 3 cm)
A small hemispherical end cap with a 3 cm radius has a volume of 56.55 cubic centimeters and a total surface area of 84.82 square centimeters. Machinists and 3D-print designers use figures this small to estimate material weight or print time for a single part. At exactly this radius the curved surface area also happens to equal 56.55 square centimeters, a coincidence of the numbers at radius 3, not a general rule for hemispheres.
Hemisphere Volumes by Radius
Volume and total surface area for common radii.
| Radius | Volume | Total surface area |
|---|---|---|
| 1 | 2.09 | 9.42 |
| 2 | 16.76 | 37.70 |
| 3 | 56.55 | 84.82 |
| 5 | 261.80 | 235.62 |
| 6 | 452.39 | 339.29 |
| 10 | 2094.40 | 942.48 |
Values are rounded to two decimal places for display. Because volume scales with the cube of the radius, jumps between rows grow quickly: radius 10 holds roughly 1,000 times the volume of radius 1, not just 10 times.
Hemisphere vs. sphere: what actually changes
A hemisphere keeps the same radius as the sphere it was cut from, which is why its volume is always exactly half the sphere's (4/3) pi r cubed. Surface area does not follow that same simple half rule. A full sphere's entire skin is curved, 4 pi r squared, but slicing it in half removes some of that curved skin and replaces it with a flat circular disk, pi r squared. That disk is smaller than the curved area it replaced, which is why a hemisphere's total surface area, 3 pi r squared, is more than half of the sphere's 4 pi r squared, not exactly half.
This distinction matters for real objects. A dome-shaped roof only needs the curved surface figure for its shingles or panels, since nobody covers the flat base where the walls attach. A cast concrete footing, by contrast, needs the total surface area when calculating the surface exposed to forming material on every side, including the flat bottom. Picking the wrong one of the two surface numbers is one of the most common errors people make with this shape.
Volume behaves more predictably. Because it scales with the cube of the radius, small radius changes produce large volume swings. A hemisphere with a 10 cm radius does not hold twice as much as one with a 5 cm radius, it holds eight times as much, since 2 cubed is 8. That cube relationship is worth remembering any time you are scaling up a mold, bowl, or storage vessel design and estimating how much more material or capacity a size increase will need.
Common mistakes
- Halving the sphere's surface area for the total. Cutting a sphere exposes a new circular face, so the total is 3 pi r squared, not 2 pi r squared.
- Using the diameter as the radius. A dome spanning 12 meters has a radius of 6, and cubing the wrong one is an 8x volume error.
- Forgetting the cube. Volume grows with r cubed, so doubling the radius gives 8 times the volume, not 2 times.
- Mixing up curved surface area with total surface area when ordering material. Painting or glazing a dome only needs the curved figure, 2 pi r squared, while a mold or casting needs the total, 3 pi r squared.
- Applying the hemisphere formula to a bowl that is not a true half-sphere. Shallow or deep bowls are spherical caps, which use a different formula that includes the cap's height, not just its radius.
- Skipping unit conversion before comparing results. A volume in cubic centimeters cannot be compared directly with one in cubic inches without converting first, even though both describe the same hemisphere.
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