10
units
31.4159
units
78.5398
sq units
157.0796
sq units
235.6194
sq units
261.7994
cu units
0.5
10
units
31.4159
units
78.5398
sq units
157.0796
sq units
235.6194
sq units
261.7994
cu units
0.5
The Hemisphere Calculator computes the volume, total surface area, curved surface area, and base area of a hemisphere -- exactly half of a sphere. A hemisphere is formed when a plane passes through the center of a sphere, dividing it into two equal halves.
The fundamental hemisphere formulas are:
$$V = \frac{2}{3}\pi r^3$$
$$SA_{\text{total}} = 3\pi r^2$$
$$SA_{\text{curved}} = 2\pi r^2$$
where $$r$$ is the radius. The total surface area includes both the curved dome surface ($$2\pi r^2$$) and the flat circular base ($$\pi r^2$$). The volume is exactly half the sphere volume $$\frac{4}{3}\pi r^3$$.
Hemispheres appear extensively in architecture and structural engineering. Domed structures from the Roman Pantheon to modern geodesic domes use hemispherical geometry. The dome shape efficiently distributes loads through compression, allowing large spans without internal supports. Computing the surface area of a hemispherical dome determines the amount of roofing material, paint, or waterproofing membrane required.
In pressure vessel design, hemispherical end caps (heads) are the strongest and most efficient shape for containing internal pressure. The stress in a hemispherical head is exactly half that in the cylindrical shell of the same thickness, making hemispheres the preferred choice for high-pressure applications in chemical and aerospace engineering.
Geophysics and meteorology use hemisphere concepts extensively. The Northern and Southern Hemispheres of Earth each have a surface area of approximately $$2\pi R^2_{\text{Earth}} \approx 2.55 \times 10^8$$ km2 (curved surface only). Weather patterns, ocean currents, and the Coriolis effect are all analyzed within hemispherical frameworks.
In optics and antenna design, hemispherical lenses and reflectors are common. Hemispherical microphones capture sound from all directions above a surface (2-pi steradian solid angle). The curved surface area determines the collection aperture and sensitivity pattern.
In cooking and food science, hemispherical molds are popular for desserts, chocolate bonbons, and ice cream scoops. Knowing the volume helps determine portion sizes and ingredient quantities. A hemispherical scoop of ice cream with radius 3 cm holds about $$\frac{2}{3}\pi(27) \approx 56.5$$ cm3, or roughly 56.5 milliliters.
This calculator provides all four key measurements for any hemisphere, making it useful for engineering, architecture, science, and everyday applications.
The calculator applies standard hemisphere formulas. Volume is $$(2/3)\pi r^3$$ (half the sphere volume). The curved surface area is $$2\pi r^2$$ (half the sphere's surface). The base area is $$\pi r^2$$ (a flat circle). Total surface area is the sum: $$2\pi r^2 + \pi r^2 = 3\pi r^2$$.
The Volume is the space inside the hemisphere. The Total Surface Area includes both the curved dome and the flat base. The Curved Surface Area is just the dome portion (useful for coating or roofing calculations). The Base Area is the flat circular face. For a solid hemisphere resting on a table, the base area is the contact region.
Inputs
Results
V = (2/3)*pi*125 = (250/3)*pi ~ 261.80. Total SA = 3*pi*25 = 75*pi ~ 235.62. Curved = 50*pi ~ 157.08. Base = 25*pi ~ 78.54.
Inputs
Results
A 15m radius hemispherical dome has ~2121 m2 total surface area and encloses ~7069 m3 of space.
A hemisphere is exactly half a sphere, so its volume is exactly half: $$V_{\text{hemisphere}} = \frac{1}{2} \times \frac{4}{3}\pi r^3 = \frac{2}{3}\pi r^3$$. If you know the sphere volume, simply divide by 2 to get the hemisphere volume.
The curved dome surface alone is $$2\pi r^2$$ (half the sphere's $$4\pi r^2$$). However, a hemisphere also has a flat circular base with area $$\pi r^2$$. The total surface area is the sum: $$2\pi r^2 + \pi r^2 = 3\pi r^2$$. If the hemisphere is open on the bottom (like a bowl), use only the curved area.
A hemispherical bowl uses the same volume formula: $$V = \frac{2}{3}\pi r^3$$. If the bowl has a specific wall thickness $$t$$, the interior volume uses the inner radius: $$V = \frac{2}{3}\pi (r - t)^3$$.
The surface-area-to-volume ratio for a hemisphere is $$\frac{3\pi r^2}{(2/3)\pi r^3} = \frac{9}{2r}$$. This ratio decreases with increasing radius, meaning larger hemispheres are more volume-efficient relative to their surface area.
Yes. Rearrange the formula: $$r = \left(\frac{3V}{2\pi}\right)^{1/3}$$. For example, if the volume is 500 cubic units, then $$r = (750/\pi)^{1/3} \approx 6.20$$ units.
Hemispherical domes distribute applied loads (gravity, wind, snow) through membrane compression rather than bending. Every point on the dome surface is in a state of uniform compressive stress, which masonry and concrete handle exceptionally well. This allows domes to span large distances without internal columns.
Roboculator Team
The Roboculator Team explains calculations, planning tools, and practical formulas in clear language for real-life situations.
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