Sphere: Volume and Surface Area

Insert the radius into V = 4/3·π·r³: multiply the radius by itself three times, times π, times 4/3. Example: r = 3 cm gives V = 4/3·π·27 = 36π ≈ 113.1 cm³. If the diameter is given, halve it first: r = d/2. Watch the order: only r is cubed, π and 4/3 remain factors. The unit is always a cubic unit like cm³ or m³, because three lengths are multiplied. For litres use 1000 cm³ = 1 l. A quick sanity check: the sphere must hold less than the enclosing cube with edge 2r, here 216 cm³.

Rearrange the formula: r = ∛(3V/(4π)). So multiply the volume by 3, divide by 4π and take the cube root. Example: V = 500 cm³ gives 3·500/(4π) = 1500/12.566 ≈ 119.4 and thus r = ∛119.4 ≈ 4.92 cm. Check: 4/3·π·4.92³ ≈ 499 cm³ ✓. The most common mistake is taking the square root instead of the cube root; but the volume depends on r³, so the cube root is needed. On the calculator use the ∛ key or raise to the power 1/3. Analogously the surface gives r = √(O/(4π)), there with a square root.

The volume V = 4/3·πr³ measures the capacity, i.e. how much fits inside the sphere; the surface O = 4πr² measures the skin of the sphere, i.e. how much material is needed to wrap or paint it. You can tell them apart by exponent and unit: volume has r³ and cm³, surface has r² and cm². For radius r = 3 cm both numerical values happen to be 36π ≈ 113.1, but with different units; that is a peculiarity of r = 3. Also note the scaling: doubling the radius quadruples the surface and multiplies the volume by eight. That is why small bodies cool faster: they have relatively much surface per volume.

The classical route is due to Archimedes: he compared the sphere with the circumscribed cylinder (radius r, height 2r) and proved that the sphere fills exactly 2/3 of its volume. The cylinder holds πr²·2r = 2πr³, two thirds of that is 4/3·πr³. In upper school you can verify this with the volume of revolution: the semicircle f(x) = √(r² − x²) rotates over [−r; r] around the x-axis, and V = π·∫(r² − x²) dx gives exactly 4/3·πr³. The surface formula is tied to it: O = 4πr² is the derivative of the volume with respect to r, because a sphere grows in shells like an onion.

The volume is simply half: V = 2/3·πr³. With the surface you must be careful, because the circular cut face πr² is added to the half shell 2πr²: O = 2πr² + πr² = 3πr². Example with r = 3 cm: V = 2/3·π·27 = 18π ≈ 56.5 cm³ and O = 3π·9 = 27π ≈ 84.8 cm². The typical mistake is to give the surface of the hemisphere as half the sphere surface 2πr² and forget the cut face; that is only correct if the open shell is explicitly meant (e.g. a bowl without a lid). So read carefully whether the solid is closed.