Find the volume of a sphere.
The radius is cubed and multiplied by 4/3 times π.
Volume: (4/3) × π × radius³
The sphere volume calculator finds how much space a perfectly round ball takes up. You enter the radius, the distance from the centre to the surface, and it returns (4/3)πr³.
Spheres turn up in cricket and football balls, ball bearings, marbles, laddoos, spherical storage tanks, planets and raindrops. The volume tells you how much material a solid ball contains, how much gas or liquid a spherical tank stores, or how many small spheres can be made by melting a big one, which is a classic Class 10 question. Because the radius is cubed, the answer is very sensitive to it, so measure carefully and remember that the input is the radius, not the diameter.
1. Find the radius r. If you measured the diameter, halve it; if you measured the circumference C, use r = C ÷ (2π).
2. Cube the radius: r × r × r.
3. Multiply by π.
4. Multiply by 4 and divide by 3: V = (4/3)πr³.
5. Write the answer in cubic units, and convert cm³ to litres by dividing by 1,000 if needed.
6. For a hemisphere, halve the result: (2/3)πr³.
Archimedes showed that a sphere fits inside a cylinder of the same radius and height 2r, and that its volume is exactly two-thirds of that cylinder's. The cylinder's volume is πr² × 2r = 2πr³, and two-thirds of that is (4/3)πr³. He was so proud of this result that he asked for a sphere inside a cylinder to be carved on his tomb. The same two-thirds ratio holds for their surface areas.
Slice the sphere into thin horizontal discs. A disc at height y from the centre has radius √(r² − y²), so its area is π(r² − y²). Adding up, or integrating, these areas from y = −r to y = r gives π[r²y − y³/3] evaluated over that range, which equals π(2r³ − 2r³/3) = (4/3)πr³. This is where the 4/3 comes from. Differentiating the volume with respect to r gives 4πr², the sphere's surface area.
Because volume depends on r³, doubling the radius makes the volume eight times as large. A ball with radius 10% bigger holds about 33% more. This is why melting one large sphere into many small ones produces surprisingly many: a sphere of radius 6 cm yields 27 spheres of radius 2 cm. Real objects are rarely perfect spheres. The Earth is slightly flattened, and fruits and laddoos are irregular, so treat the result as a good estimate.
A sweet shop in Madurai makes large festive laddoos about 3.5 centimetres in radius, and the owner wants the volume of one to estimate the mixture needed.
Answer: Volume 179.5944
Entering the diameter instead of the radius, which makes the volume eight times too large.
Squaring the radius instead of cubing it, which mixes up volume with surface area.
Forgetting the 4/3 factor or using 3/4 instead.
Using the sphere formula for a hemisphere without halving.
Rounding the radius heavily before cubing, which magnifies the error in the volume.
Estimating material in balls, bearings, marbles and spherical sweets.
Finding the capacity of spherical gas or water storage tanks.
Solving melting and recasting problems, such as how many small spheres come from a large one.
Estimating volumes of planets, droplets and bubbles in science.
Class 9 and 10 surface area and volume exercises.
Where does the 4/3 come from?
It comes from the calculus used to work out this formula, by adding up thin circular slices through the sphere. It is a fixed number — you do not need to work it out yourself.