The Hemisphere Calculator converts radius into volume and surface area for a solid half-ball. A hemisphere is half of a sphere cut through a great circle, so the solid has a curved dome and a flat circular base. The page redraws a labelled figure with the radius marked to the entered value.
Enter radius or diameter. The engine returns volume, curved surface area, base area and total surface area so the base is never forgotten on exam sheets that ask for total area. Related pages cover the full Sphere Calculator when both halves are present, and the Capsule Calculator when two hemispheres close a cylinder.
Calculate the volume of a hemisphere
Volume of a hemisphere is half the volume of a full sphere with the same radius. The Hemisphere Calculator uses the two-thirds coefficient that comes from splitting (4/3) * pi * r^3 in half. Radius must be positive. Diameter input divides by two before volume runs.
V = (2/3) * pi * r^3The cubic dependence on radius still applies: doubling radius multiplies hemisphere volume by eight. When a worksheet gives volume instead of radius, recover r from the cube root of (3 * V) / (2 * pi), then refresh surface outputs from that same radius.
For the reference radius r = 6:
V = (2/3) * pi * 216 = 144 * pi approx 452.39That value is exactly half of the sphere volume 288 * pi at r = 6. Exact form 144 * pi is preferred on papers that keep pi factored. Decimal mode shows approx 452.39 using pi approx 3.14159 for display only.
A common mistake is to treat hemisphere volume as (4/3) * pi * r^3 without the half. Another is to confuse the hemisphere with a hemicylinder or a bowl that is not a true half-sphere. This page assumes a planar diametral cut through the center.
Calculate the total surface area
Total surface area of a solid hemisphere includes the curved dome and the flat circular base. The Hemisphere Calculator reports both parts and their sum so coating or material questions that need the closed solid stay correct. Curved area alone is half of 4 * pi * r^2.
Curved SA = 2 * pi * r^2
Base area = pi * r^2
Total SA = 3 * pi * r^2Forgetting the base under-counts total surface area by one-third relative to the correct total. Exam language matters: "curved surface area" means 2 * pi * r^2; "total surface area" means 3 * pi * r^2. The result panel labels each quantity so the wording on the question maps to the right line.
For r = 6:
Curved SA = 2 * pi * 36 = 72 * pi approx 226.19
Base area = pi * 36 = 36 * pi approx 113.10
Total SA = 3 * pi * 36 = 108 * pi approx 339.29Half of a full sphere's surface area 144 * pi would be 72 * pi, which matches the curved part only. Total hemisphere surface area is larger because of the disk.
Solve a hemisphere with radius 6
Use r = 6 as the shared fixture with the sphere page so students can compare half-ball and full-ball numbers side by side. Diameter is 12. Volume and total surface area use the formulas above with clean pi multiples. **1.
Diameter**
d = 2 * 6 = 122. Volume
V = (2/3) * pi * 216 = 144 * pi approx 452.393. Curved surface area
Curved SA = 2 * pi * 36 = 72 * pi approx 226.194. Total surface area
Total SA = 3 * pi * 36 = 108 * pi approx 339.29Starting from volume 144 * pi recovers r = 6. Starting from total surface area 108 * pi recovers the same radius via r = sqrt(TSA / (3 * pi)). Those seeds keep the hemisphere engine aligned with the sphere r = 6 fixture.
When a problem describes a dome that is less than a half-sphere, the solid is a spherical cap, not a hemisphere. Cap formulas need height as well as radius and are outside this page. When two hemispheres sit on the ends of a cylinder, volume is cylinder volume plus sphere volume; use the Capsule Calculator for that compound solid.
Bowls and domes in architecture often approximate hemispheres. Confirm the cut is through the center: if the flat face radius equals the sphere radius, the solid is a true hemisphere. If the flat face is a smaller parallel circle, the solid is a smaller spherical cap and volume is less than (2/3)*pi*r^3 for that sphere radius.
Coating estimates must match the question wording. Paint on the outside of a hemispherical dome sitting on a floor may omit the base (use curved SA only). A free solid hemisphere includes the base in total SA. The Hemisphere Calculator prints both so the choice stays visible.
Recover radius from a known measure
When a worksheet gives volume instead of radius, solve V = (2/3) * pi * r^3 for r. Rearrange to r^3 = (3 * V) / (2 * pi), then take the positive cube root. Example: if V = 144 * pi, then r^3 = (3 * 144 * pi) / (2 * pi) = 216, so r = 6.
Refresh curved and total surface areas from that same radius.
When total surface area is known, use TSA = 3 * pi * r^2 so r = sqrt(TSA / (3 * pi)). For TSA = 108 * pi, r^2 = 36 and r = 6. Curved-only problems use CSA = 2 * pi * r^2 instead, which recovers a different relation: r = sqrt(CSA / (2 * pi)). Matching the wording on the question to the correct formula avoids a one-third surface error.
Unit note: radius and diameter share one length unit. Volume uses cube units of that length; every surface measure uses square units. Mixing centimetres of radius with metres of a related full-sphere diameter breaks the half-ball comparison.
Frequently asked questions
What is hemisphere volume?
Hemisphere volume is V = (2/3) * pi * r^3. For r = 6, V = 144 * pi approx 452.39.
What is total surface area of a hemisphere?
Total surface area is 3 * pi * r^2, which adds the curved dome 2 * pi * r^2 to the flat base pi * r^2. For r = 6, total SA = 108 * pi approx 339.29.
Does the Hemisphere Calculator include the flat base?
Yes. The Hemisphere Calculator reports curved area, base area and total surface area separately so the base is visible.
How does hemisphere volume relate to a sphere?
Hemisphere volume is exactly half of sphere volume at the same radius. At r = 6, sphere volume 288 * pi halves to 144 * pi.
Is curved surface area half of a sphere?
Yes. Curved SA = 2 * pi * r^2 is half of 4 * pi * r^2. Total SA for a solid hemisphere is not half of the sphere surface because of the added base.
Can I start from volume?
Yes. Recover r = cube root of (3 * V) / (2 * pi), then compute surface areas from that radius.
What if the problem says "curved surface only"?
Use 2 * pi * r^2 and ignore the base. Total surface area questions need 3 * pi * r^2 instead.
What units does the Hemisphere Calculator use?
Length units on radius carry through to cube units for volume and square units for every surface measure.
Summary
Hemisphere Calculator returns volume, curved surface area, base area and total surface area for a solid half-ball from radius or diameter, with substituted arithmetic and a labelled figure. Worked numbers match r = 6: V = 144 * pi and total SA = 108 * pi.
Use the Sphere Calculator for the full ball and the Capsule Calculator when hemispheres close a cylindrical body.