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Cylinder

Cylinder

A three-dimensional geometric figure with parallel congruent bases. The bases can be shaped like any closed plane figure (not necessarily a circle) and must be oriented identically.

Note: The word cylinder often refers to a right circular cylinder.

 

Oblique and right cylinders. Oblique: Volume=Bh. Right: Volume=Bh, Lateral Area=hC, Total Surface Area=2B+hC, where h=height,...

 

 

See also

Right cylinder, circular cylinder, oblique cylinder, height of a cylinder, volume, lateral surface, lateral surface area, surface area

Key Formula

V=πr2hA=2πr2+2πrhV = \pi r^2 h \qquad\qquad A = 2\pi r^2 + 2\pi r h
Where:
  • VV = Volume of a right circular cylinder
  • AA = Total surface area of a right circular cylinder
  • rr = Radius of the circular base
  • hh = Height (altitude) — the perpendicular distance between the two bases
  • π\pi = Pi, approximately 3.14159

Worked Example

Problem: Find the volume and total surface area of a right circular cylinder with radius 5 cm and height 12 cm.
Step 1: Write down the volume formula and substitute the given values.
V=πr2h=π(5)2(12)V = \pi r^2 h = \pi (5)^2 (12)
Step 2: Compute the base area first, then multiply by the height.
V=π⋅25⋅12=300π≈942.5 cm3V = \pi \cdot 25 \cdot 12 = 300\pi \approx 942.5 \text{ cm}^3
Step 3: Now write the surface area formula and substitute.
A=2πr2+2πrh=2π(5)2+2π(5)(12)A = 2\pi r^2 + 2\pi r h = 2\pi(5)^2 + 2\pi(5)(12)
Step 4: Evaluate each part: the two circular bases and the lateral (side) surface.
A=2π(25)+2π(60)=50π+120π=170π≈534.1 cm2A = 2\pi(25) + 2\pi(60) = 50\pi + 120\pi = 170\pi \approx 534.1 \text{ cm}^2
Answer: The volume is 300π≈942.5300\pi \approx 942.5 cm³ and the total surface area is 170π≈534.1170\pi \approx 534.1 cm².

Another Example

This example works backwards — you are given the volume and must solve for a missing dimension, which is a common real-world application.

Problem: A cylindrical water tank holds exactly 2,000 litres. Its interior height is 1.5 m. Find the radius of the tank in metres, to two decimal places. (1 m³ = 1,000 litres.)
Step 1: Convert litres to cubic metres so the units match.
2,000 L=2,0001,000=2 m32{,}000 \text{ L} = \frac{2{,}000}{1{,}000} = 2 \text{ m}^3
Step 2: Start from the volume formula and solve for the radius.
V=πr2h  ⟹  r2=VπhV = \pi r^2 h \;\Longrightarrow\; r^2 = \frac{V}{\pi h}
Step 3: Substitute the known values.
r2=2π(1.5)=21.5π=43πr^2 = \frac{2}{\pi(1.5)} = \frac{2}{1.5\pi} = \frac{4}{3\pi}
Step 4: Take the square root to find the radius.
r=43π=23π≈23.0699≈0.65 mr = \sqrt{\frac{4}{3\pi}} = \frac{2}{\sqrt{3\pi}} \approx \frac{2}{3.0699} \approx 0.65 \text{ m}
Answer: The radius of the tank is approximately 0.650.65 m (65 cm).

Frequently Asked Questions

What is the difference between a right cylinder and an oblique cylinder?
In a right cylinder, the line segment connecting the centres of the two bases is perpendicular to those bases, so the sides stand straight up. In an oblique cylinder, this segment is tilted, making the figure lean to one side. Both types use the same volume formula V=BhV = Bh (where BB is the base area and hh is the perpendicular height), but the surface area calculation for an oblique cylinder is more complex.
Does a cylinder have to have circular bases?
Strictly speaking, no. The general mathematical definition of a cylinder requires only two parallel, congruent bases that are closed plane figures — they could be ellipses, rectangles, or other shapes. However, in most school courses, 'cylinder' means a right circular cylinder unless stated otherwise.
How do you find the lateral surface area of a cylinder?
For a right circular cylinder, imagine 'unrolling' the curved side into a flat rectangle. Its width equals the circumference of the base (2πr2\pi r) and its height equals hh. So the lateral surface area is 2πrh2\pi r h. The total surface area then adds the areas of the two circular bases: 2πr22\pi r^2.

Cylinder vs. Cone

CylinderCone
Number of basesTwo parallel, congruent basesOne base and one apex (point)
Volume formulaV=πr2hV = \pi r^2 hV=13πr2hV = \tfrac{1}{3}\pi r^2 h
Lateral surface area2πrh2\pi r hπrl\pi r l (where ll is slant height)
Cross-section (parallel to base)Same size circle at every heightCircle that shrinks as you move toward the apex
Relationship—A cone's volume is exactly one-third of a cylinder's with the same base and height

Why It Matters

Cylinders appear constantly in science and daily life — beverage cans, pipes, pistons, and storage tanks are all cylindrical. You will need cylinder formulas in geometry, physics (pressure in a pipe, moment of inertia), and chemistry (graduated cylinders). Mastering the volume and surface area formulas also builds the foundation for understanding more complex solids of revolution in calculus.

Common Mistakes

Mistake: Using the diameter instead of the radius in the formulas.
Correction: Always check whether a problem gives the radius or the diameter. If you are given the diameter dd, divide by 2 first: r=d/2r = d/2. Plugging the diameter directly into πr2h\pi r^2 h will give a volume four times too large.
Mistake: Confusing height with slant height for an oblique cylinder.
Correction: The height hh in the volume formula is always the perpendicular distance between the two bases, not the length measured along the slanted side. For a right cylinder these are the same, but for an oblique cylinder they differ.

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