Cylinder — Shape, Formula, Examples
Cylinder — Shape, Formula, Examples
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 (hhh), and the radius of each circular base is the radius (rrr).
Key Formulas:
V=πr²h (Volume)
CSA=2πrh (Curved Surface Area)
TSA=2πr(r+h) (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 Δh (an infinitely thin slice). The volume of each disc is the area of its circle times its thickness: πr²⋅Δh.
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=πr²h.
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 = 2πr
- Height = the height of the cylinder = h
Area of that rectangle: CSA=2πr×h=2πrh.
Total Surface Area — adding the two circles
The total surface area adds both circular bases to the curved surface:
TSA=CSA+2×(area of one circle)=2πrh+2πr²=2πr(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=πr²h:
V=π×5²×12=π×25×12=300π≈942.5 cm³.
Final answer: V=300π≈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 TSA=2πrh — 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: CSA=2π×4×10=80π
- Area of one circular base: πr²=π×4²=16π
- Total: TSA=80π+2(16π)=80π+32π=112π≈351.9 m².
Final answer: TSA=112π≈351.9 m²
Example 3: Finding radius from volume
A cylinder has volume 200π cm³ and height 8 cm. Find the radius.
Rearrange V=πr²h:
200π=πr²×8.
r²=200π/8π=200/8=25.
r=√25=5 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 2/3 of the cylinder's volume:
Vsphere=2/3Vcylinder=2/3πr²(2r)=4/3πr³.
This result is how the volume of a sphere 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 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=πr²h 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 CSA=2πrh without adding the two circular bases.
Don't do this: Present 2πrh as the total surface area. This is only the curved side — the two circular ends are missing.
The correct way: TSA=2πrh+2πr²=2πr(h+r).
Mistake 2: Using diameter instead of radius in the formula
Where it slips in: When the problem gives the diameter and the student substitutes directly into the formula as r.
Don't do this: V=πd²h — this gives four times the correct volume.
The correct way: Always halve the diameter before substituting: r=d/2. Then V=πr²h.
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=300π 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.
Quick Reference
| Formula | Expression |
|---|---|
| Volume | V=πr²h |
| Curved Surface Area | CSA=2πrh |
| Total Surface Area | TSA=2πr(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.
Frequently Asked Questions
Does a cylinder have faces, edges, and vertices?
A cylinder has 3 faces, 2 edges, and 0 vertices. The 3 faces are the two flat circular bases and the one curved lateral surface. The 2 edges are the circular boundaries where the curved surface meets each base. There are no vertices because there are no corners where edges meet at a point.
What is the difference between curved surface area and total surface area of a cylinder?
Curved surface area (CSA) is only the lateral surface — the rectangle that wraps around the side. Total surface area (TSA) adds both circular bases. Use CSA when you need to know how much material wraps the side and TSA when you need the total outer surface.
Is an oblique cylinder's volume the same formula?
Yes. Cavalieri's Principle guarantees this: even if the cylinder is tilted (oblique), as long as every horizontal cross-section is the same circle with radius r and the perpendicular height is h, the volume is still πr²h.
How does the cylinder volume compare to the sphere that fits inside it?
The sphere inscribed in a cylinder has radius equal to the cylinder's radius and height equal to the diameter: the cylinder volume is πr²(2r). The sphere volume is \frac{4}{3}πr³. The ratio is \frac{4}{3} / 2 = \frac{2}{3} — the result Archimedes requested on his tomb.
What are real-world uses of cylinder geometry?
Engineering uses the volume formula to size pipes, tanks, and pressure vessels. Medicine uses it to calculate drug doses in cylindrical syringes. Architecture uses it in columns, towers, and vaults. Astronomy uses it to model the shapes of stars and planetary bodies.
✍️ Written By
BT
Content Creator and Editor