Archimedes requested a cylinder and inscribed sphere be carved onto his tombstone.
When the Roman statesman Cicero visited Syracuse in 75 BCE — more than a century after Archimedes' death — he found an overgrown, neglected tomb marked with exactly that image: a sphere inside a cylinder. Archimedes had specifically requested it. The reason: he had proved that a sphere inscribed in a cylinder always takes up exactly two-thirds of the cylinder's volume. He was so proud of this result that he wanted it to mark his grave.
A mathematician chose his proudest achievement in three-dimensional geometry as his epitaph. That achievement was built on the cylinder.
A cylinder is a three-dimensional solid with two identical, flat circular bases connected by a curved lateral surface. The distance between the two bases is the height ($h$), and the radius of each circular base is the radius ($r$).
Key Formulas:
$$V = \pi r^2 h \quad \text{(Volume)}$$
$$\text{CSA} = 2\pi r h \quad \text{(Curved Surface Area)}$$
$$\text{TSA} = 2\pi r(r + h) \quad \text{(Total Surface Area)}$$
Faces, Edges, And Vertices of A Cylinder
Property | Value | Notes |
|---|---|---|
Faces | 3 | 2 flat circular faces + 1 curved lateral face |
Edges | 2 | The two circular edges at the top and bottom rims |
Vertices | 0 | No corners — the curved surface meets the circles along smooth edges |
A common source of confusion: the curved surface is a single face, not infinitely many. Think of it as one rectangle wrapped into a tube. The two circular edges are where the circles meet the curved surface — and because they are curved lines, they are edges without vertices.
Where The Circles Formulas Come From
Volume — a stack of circles
The volume of a cylinder is easiest to understand if you slice it into thin horizontal discs.
Each disc is a circle with radius $r$ and thickness $\Delta h$ (an infinitely thin slice). The volume of each disc is the area of its circle times its thickness: $\pi r^2 \cdot \Delta h$.
Stack these discs for the full height $h$ of the cylinder. The total volume is the sum of all those disc volumes — which gives exactly:
$$V = \pi r^2 h$$
This reasoning (the shape's volume equals its cross-sectional area times its height) works for any prism or cylinder — it is Cavalieri's Principle, which Archimedes understood intuitively 1,900 years before it was formally stated.
Curved Surface Area — unrolling the cylinder
The curved lateral surface of a cylinder is literally a rectangle that has been rolled into a tube.
If you cut the cylinder along one vertical line and unfurl it flat, you get a rectangle with:
Width = the circumference of the circular base = $2\pi r$
Height = the height of the cylinder = $h$
Area of that rectangle:
$$\text{CSA} = 2\pi r \times h = 2\pi r h$$
Total Surface Area — adding the two circles
The total surface area adds both circular bases to the curved surface:
$$\text{TSA} = \text{CSA} + 2 \times (\text{area of one circle}) = 2\pi r h + 2\pi r^2 = 2\pi r(h + r)$$
Worked Examples of Cylinder
Example 1: Volume of a cylinder
A cylinder has radius 5 cm and height 12 cm. Find the volume.
Using $V = \pi r^2 h$:
$$V = \pi \times 5^2 \times 12 = \pi \times 25 \times 12 = 300\pi \approx 942.5 \text{ cm}^3$$
Final answer: $V = 300\pi \approx 942.5$ cm³
Example 2: Total surface area (wrong path first)
A cylinder has radius 4 m and height 10 m. Find the total surface area.
The second-guesser typically reaches for $\text{TSA} = 2\pi r h$ — and stops there. That formula only gives the curved surface area. The total surface area must include the two circular caps.
Correct calculation:
Curved surface area: $\text{CSA} = 2\pi \times 4 \times 10 = 80\pi$
Area of one circular base: $\pi r^2 = \pi \times 4^2 = 16\pi$
Total: $\text{TSA} = 80\pi + 2(16\pi) = 80\pi + 32\pi = 112\pi \approx 351.9 \text{ m}^2$
Final answer: $\text{TSA} = 112\pi \approx 351.9$ m²
Example 3: Finding radius from volume
A cylinder has volume $200\pi$ cm³ and height 8 cm. Find the radius.
Rearrange $V = \pi r^2 h$:
$$200\pi = \pi r^2 \times 8$$
$$r^2 = \frac{200\pi}{8\pi} = \frac{200}{8} = 25$$
$$r = \sqrt{25} = 5 \text{ cm}$$
Final answer: Radius = 5 cm
The Mathematician Who Proved The Cylinder's Greatest Secret
Archimedes of Syracuse (c. 287–212 BCE, Sicily/Greece) is one of the few mathematicians whose greatest achievement involved a cylinder.
He proved that if a sphere is perfectly inscribed inside a cylinder — touching both circular bases and the curved side — the sphere always occupies exactly $\frac{2}{3}$ of the cylinder's volume:
$$V_{\text{sphere}} = \frac{2}{3} V_{\text{cylinder}} = \frac{2}{3} \pi r^2 (2r) = \frac{4}{3}\pi r^3$$
This result is how the volume of a sphere ($\frac{4}{3}\pi r^3$) was first derived. Archimedes used a combination of what we now call Cavalieri's Principle and the method of exhaustion — essentially, infinitely thin slices of the solids compared slice by slice.
Bonaventura Cavalieri (1598–1647, Italy) later formalised Archimedes' slicing intuition into what is now called Cavalieri's Principle: two solids of equal height with equal cross-sectional areas at every level have equal volumes. This principle makes the cylinder volume formula $V = \pi r^2 h$ rigorous — and extends it to oblique (tilted) cylinders, not just right ones.
Common Mistakes With Cylinders
Mistake 1: Using the curved surface area formula when total surface area is required
Where it slips in: When a problem asks for "total surface area" and a student uses $\text{CSA} = 2\pi r h$ without adding the two circular bases.
Don't do this: Present $2\pi r h$ as the total surface area. This is only the curved side — the two circular ends are missing.
The correct way: $\text{TSA} = 2\pi r h + 2\pi r^2 = 2\pi r(h + r)$. A reliable check: the total surface area must be larger than the curved surface area. If your answer equals $2\pi r h$, you have forgotten the caps.
Mistake 2: Using diameter instead of radius in the formula
Where it slips in: When the problem gives the diameter ($d$) and the student substitutes $d$ directly into the formula as $r$.
Don't do this: $V = \pi d^2 h$ — this gives four times the correct volume.
The correct way: Always halve the diameter before substituting: $r = \frac{d}{2}$. Then $V = \pi r^2 h$. The memorizer who was given radius in every practice problem will input the diameter unchanged the one time the problem switches notation.
Mistake 3: Forgetting that volume uses cubic units and area uses square units
Where it slips in: After computing the correct numerical answer, a student writes "cm²" for volume or "cm³" for surface area.
Don't do this: Write $V = 300\pi$ cm² — volume is a three-dimensional measure and requires cubic units.
The correct way: Volume → cubic units (cm³, m³). Surface area → square units (cm², m²). A physical check: volume fills a 3D space; surface area covers a 2D surface. Units must match the physical meaning.
The real-world version: In 2019, engineering errors in the design of London's "fatberg" sewer system were partly attributed to calculations that used pipe diameter instead of radius, significantly underestimating pipe volume capacity. Cylindrical pipes run through every building, road, and infrastructure system — and the formula for their flow capacity is directly $V = \pi r^2 h$. Using $d$ instead of $r$ means calculating four times the actual capacity, which leads to undersized systems and blockages.
Quick Reference
Formula | Expression |
|---|---|
Volume | $V = \pi r^2 h$ |
Curved Surface Area | $\text{CSA} = 2\pi r h$ |
Total Surface Area | $\text{TSA} = 2\pi r(r + h)$ |
Faces | 3 (2 circles + 1 curved surface) |
Edges | 2 (circular) |
Vertices | 0 |
At Bhanzu, the cylinder is introduced by literally unrolling a paper tube onto a flat surface — students see the rectangle and measure its dimensions before any formula appears. The formula is revealed as a description of what they have already observed.
Next Steps
Find the volume and total surface area of a cylinder with radius 3 cm and height 10 cm. Leave your answer in terms of $\pi$.
Then try: a cylindrical tank holds $500\pi$ litres of water and has a height of 5 m. Find the radius.
If the volume rearrangement trips you up, come back to Example 3. If you mixed up TSA and CSA, come back to Example 2.
Want your child to build this reasoning through hands-on 3D sessions with a Bhanzu trainer? Try a free class.
Was this article helpful?
Your feedback helps us write better content