A hemisphere is half of a sphere, like a bowl or a dome. Enter the radius to find its volume and both surface areas.
Enter the radius.
Read the volume and surface areas.
Volume: V = ⅔ π r³
CSA: 2 π r²
TSA: 3 π r²
A hemisphere is exactly half of a sphere, cut through its centre. Bowls, domes of temples and mosques, the ends of gas cylinders and storage tanks, and the bottom of a kadhai are all close to hemispheres. This calculator takes only the radius and returns the volume, the curved surface area and the total surface area.
The two surface areas matter because real objects differ. An open bowl has only its curved surface, while a solid hemisphere, such as a paperweight or a dome built on a flat floor, also has a flat circular face. Class 9 and Class 10 mensuration questions deliberately switch between the two, and combined-solid problems often stack a hemisphere on a cylinder or a cone.
1. Measure the radius r. If you know the diameter, halve it.
2. Volume: V = ⅔ π r³. Cube the radius first, then multiply by π and by ⅔.
3. Curved surface area: CSA = 2 π r², half the surface of a full sphere.
4. Total surface area of a solid hemisphere: add the flat circle πr², so TSA = 3 π r².
5. For capacity, convert cm³ to litres by dividing by 1000.
6. If the textbook uses π = 22/7, expect answers that differ slightly from the full-precision value.
Archimedes showed that a sphere fits snugly inside a cylinder of the same radius and height 2r, and that the sphere's volume is exactly two thirds of that cylinder's. The cylinder's volume is πr² × 2r = 2πr³, so the sphere's is ⁴⁄₃πr³. Half of that is ⅔πr³ for the hemisphere. A related fact: a hemisphere holds twice as much as a cone with the same base radius and a height equal to r, since the cone's volume is ⅓πr³. This pair often appears in Class 10 combined-solid problems.
Archimedes also found that the surface of a sphere equals the curved surface of that same snug cylinder, 2πr × 2r = 4πr². Remarkably, any band of the sphere between two parallel cuts has the same area as the matching band of the cylinder: the sphere's band is narrower but more tilted, and the two effects cancel. Half the sphere therefore has a curved surface of 2πr². A solid hemisphere adds its flat circular base, πr², giving the total 3πr². An open bowl, having no base, uses only 2πr².
Volume grows with the cube of the radius and surface area with its square. A 10% error in measuring the radius becomes roughly a 21% error in surface area and a 33% error in volume. So measure the radius carefully, ideally from the diameter across the rim. For a thick bowl, the inner radius gives the capacity and the outer radius gives the outside paint area. The volume of the material itself is ⅔π(R³ − r³), a common Class 10 question about hemispherical shells.
A steel utensil shop in Moradabad makes solid hemispherical brass paperweights with a radius of 10.5 cm and wants to know the brass volume and the area to be polished, including the flat base.
Volume = ⅔ π r³: = ⅔ × 3.14159265 × 10.5³ = 2,424.5241
Curved surface area = 2 π r²: = 692.7212
Total surface area = 3 π r²: = 1,039.0818
Answer: Volume 2,424.5241; Curved surface area 692.7212; Total surface area 1,039.0818
Using the total surface area 3πr² for an open bowl, which has no flat face.
Entering the diameter as the radius, which makes the volume eight times too large.
Using the outer radius to find how much a thick bowl holds; capacity needs the inner radius.
Squaring the radius instead of cubing it in the volume formula.
Treating a shallow bowl or dome as a full hemisphere when its depth is less than its radius.
Finding the capacity of hemispherical bowls, tanks and vessels in litres.
Estimating paint or plaster for domes of buildings, temples and water tanks.
Calculating metal needed for solid or hollow hemispherical castings.
Solving Class 10 combined-solid problems, such as a toy made of a cone on a hemisphere.
Working out the end volumes of capsule-shaped tanks and gas cylinders.
Which value of π is used?
The full-precision value 3.14159…; textbook answers using 22/7 may differ slightly.
Is a bowl's surface the total surface area?
An open bowl has only the curved surface, 2πr².