S=2\pi r^2+2\pi rhVariables
- S: total surface area
- r: radius
- h: height
How to use this formula
Adds the areas of two circular bases and the curved lateral surface.
Important notes
- For lateral area only, omit the 2πr² term.
- Use consistent length units.
Quick example
For r=2 and h=5, S=28π.
Applicability, worked calculation, and verification
Domain and applicability
Apply Total Surface Area of a Cylinder only when the listed variables are defined, denominators are nonzero, and the problem satisfies the assumptions stated on this page.
Assumptions and domain checks
- For lateral area only, omit the 2πr² term.
- Use one consistent unit system and confirm whether lengths, areas, volumes, and angles use the dimensions implied by the formula.
Do not use this formula when
- Do not use Total Surface Area of a Cylinder when the stated lengths, angles, or geometric conditions do not match the figure or use inconsistent units.
Boundary and special cases
- For Total Surface Area of a Cylinder, check zero, negative, and extreme input values before relying on the result.
- When using Total Surface Area of a Cylinder, confirm that denominators, radicals, logarithms, and domain restrictions remain valid for the chosen values.
Equivalent and alternative forms
- Keep the canonical LaTeX form S=2\pi r^2+2\pi rh for copying; rearrange only after preserving equivalence and domain restrictions.
Worked example
For r=2 and h=5, S=2π(2²)+2π(2)(5)=8π+20π=28π≈87.965.
- Compute the two circular ends: 2πr²=8π.
- Compute the lateral area: 2πrh=20π.
- Add the components to obtain 28π.
Independent verification
The net consists of two circles and a rectangle of width 2πr and height h, giving the same total.
Common mistakes
- Before substituting values into Total Surface Area of a Cylinder, map each variable to its definition and preserve every sign, exponent, subscript, and grouping mark.
- For the Total Surface Area of a Cylinder, do not report a length, area, or volume with the wrong unit dimension, and keep intermediate precision until the final rounding step.
Continue the workflow
Use Total Surface Area of a Cylinder in your own work
- Check the domainMatch the variables and assumptions to the problem before substituting values.
- Copy the exact notationPreserve grouping, signs, and exponents in
S=2\pi r^2+2\pi rh. - Edit or convertOpen the expression in the LaTeX editor, then export it for your document or web page.
Review and verification
Last reviewed: 2026-07-26
Review method: Reviewed the notation, variable definitions, applicability conditions, worked workflow, and verification method for Total Surface Area of a Cylinder.
Verified against: OpenStax Algebra and Trigonometry
Automated quality check: Passed the core formula indexing gate.
Formula references
- Algebra and Trigonometry 2eOpenStax, Rice University — Reviewed algebra, functions, sequences, trigonometry, and analytic geometry definitions and examples.
Frequently asked questions
What is the Total Surface Area of a Cylinder used for?
Adds the areas of two circular bases and the curved lateral surface.
Can I copy this formula as LaTeX?
Yes. Copy S=2\pi r^2+2\pi rh or open it in the LaTeX editor.
What should I check before using it?
Confirm that each variable, unit, domain restriction, and assumption matches the problem.