Sphere and Hemisphere

A sphere is the most symmetric 3D shape: the set of all points equidistant from a central point. Examples include balls, planets, and bubbles. A hemisphere is half a sphere , like a bowl or a dome. This lesson gives all the surface-area and volume formulas.

Definitions

A sphere of radius $r$ is the set of all points in 3D space at distance $r$ from a fixed point (the centre). It has no edges or vertices , just one continuous curved surface.

A hemisphere of radius $r$ is half a sphere, bounded by a flat circular base of radius $r$ and a curved surface.

Formulas

For a sphere of radius $r$:

• Surface area: $4 \pi r^2$.
• Volume: $\tfrac{4}{3} \pi r^3$.

For a hemisphere of radius $r$:

• Curved surface area: $2 \pi r^2$ (half of sphere).
• Total surface area: $2 \pi r^2 + \pi r^2 = 3 \pi r^2$ (curved + flat base).
• Volume: $\tfrac{2}{3} \pi r^3$ (half of sphere).

Why the formulas are right

The surface area $4\pi r^2$ and volume $\tfrac{4}{3} \pi r^3$ of a sphere are derived using calculus or by careful integration. We accept them here.

A neat way to remember the volume: it is exactly twice the volume of a cone of the same radius and height as the radius , namely $V_\text{cone} = \tfrac{1}{3}\pi r^2 \cdot r = \tfrac{1}{3}\pi r^3$, so the sphere's volume is $\tfrac{4}{3}\pi r^3 = 4 \cdot \tfrac{1}{3}\pi r^3 = 4 V_\text{cone}$. Equivalently: a sphere fits inside a cylinder of the same radius and height $2r$, with the sphere's volume being $\tfrac{2}{3}$ of the cylinder's. Archimedes is said to have been so pleased with this discovery that he asked for the cylinder-with-sphere figure to be engraved on his tomb.

A useful relation: the Archimedes ratio

For a sphere fitting exactly inside a cylinder of the same radius (and height $= 2r$):

• Cylinder volume $= \pi r^2 \cdot 2r = 2\pi r^3$.
• Sphere volume $= \tfrac{4}{3}\pi r^3$.
• Ratio: sphere : cylinder $= 2 : 3$.

So the sphere occupies exactly two-thirds of its enclosing cylinder. Beautiful.

Worked examples

Example 1. A sphere has radius $7$ cm. Find surface area and volume. (Use $\pi = \tfrac{22}{7}$.)

$\text{SA} = 4\pi r^2 = 4 \cdot \tfrac{22}{7} \cdot 49 = 616$ cm$^2$. $V = \tfrac{4}{3} \pi r^3 = \tfrac{4}{3} \cdot \tfrac{22}{7} \cdot 343 = \tfrac{30184}{21} \approx 1436.95$ cm$^3$.

Example 2. A hemisphere has radius $7$ cm. Find curved SA, total SA, and volume.

CSA $= 2\pi r^2 = 308$ cm$^2$. TSA $= 3\pi r^2 = 462$ cm$^2$. $V = \tfrac{2}{3}\pi r^3 \approx 718.47$ cm$^3$.

Example 3. A spherical ball has volume $4851$ cm$^3$. Find its radius.

$\tfrac{4}{3}\pi r^3 = 4851 \Rightarrow r^3 = \tfrac{3 \cdot 4851}{4 \pi} = \tfrac{3 \cdot 4851}{4 \cdot 22/7} = \tfrac{3 \cdot 4851 \cdot 7}{88} = \tfrac{101871}{88} \approx 1157.6$. So $r \approx \sqrt[3]{1157.6} \approx 10.5$. Cleaner: assume the question's numbers give $r = \tfrac{21}{2}$ exactly.

Example 4. A spherical balloon expands to twice its original radius. By what factor does the volume increase?

Volume scales with $r^3$. Doubling $r$ multiplies $V$ by $8$.

Example 5. A hemispherical bowl has internal radius $14$ cm. How much water can it hold?

Volume $= \tfrac{2}{3} \pi r^3 = \tfrac{2}{3} \cdot \tfrac{22}{7} \cdot 2744 = \tfrac{120736}{21} \approx 5749$ cm$^3$ $\approx 5.75$ litres.

Try it yourself

1. Sphere of radius $5$. Find surface area and volume.
2. Hemisphere of radius $10$. Find CSA, TSA, V.
3. A spherical ball has surface area $314$ cm$^2$ ($\pi \approx 3.14$). Find the radius.
4. A spherical ball has volume $\tfrac{4}{3}\pi \cdot 125$. Find the radius.
5. Compare the volumes of two spheres of radii $1$ and $2$.
6. A sphere of radius $r$ exactly fits inside a cylinder of the same radius and height $2r$. Find the ratio of volumes.
7. A hemispherical dome has radius $7$. Find the area of the curved surface.
8. A spherical balloon expands so its radius triples. By what factor does its volume increase?
9. Find the volume of a sphere of radius $\tfrac{1}{2}$.
10. A hemispherical bowl holds $1450$ cm$^3$ of water. Find its radius (approximately).

Pitfalls / Insight

• Sphere has only one surface formula. $4\pi r^2$. There is no "curved" vs "total" distinction.
• Hemisphere has two surface formulas. Curved ($2\pi r^2$) and total ($3\pi r^2$, including the base).
• Don't forget the $\tfrac{1}{3}$ in the sphere volume formula. $V = \tfrac{4}{3}\pi r^3$, not $\tfrac{4}{2}\pi r^3$.

Insight. The sphere is the shape Nature loves: water drops, planets, soap bubbles all approximate spheres because the sphere is the shape with the smallest surface area for a given volume. The formulas $4\pi r^2$ and $\tfrac{4}{3}\pi r^3$ capture this perfect symmetry.