How to Find the Surface Area of a Cylinder
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Definition
A cylinder has two congruent circular bases and one curved side. Surface area measures all outside area and is written in square units.
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Surface Area Formula
The two circular bases have area \(2\pi r^2\). The curved side, or lateral area, unwraps into a rectangle with area \(2\pi rh\). Therefore:
\(SA = 2\pi r^2 + 2\pi rh\)
For a label or side covering only, use \(LA = 2\pi rh\). Leave answers in terms of \(\pi\) unless an approximation is requested.
Example
If \(r = 3\) and \(h = 5\), then \(SA = 2\pi(3^2) + 2\pi(3)(5) = 48\pi\).
Surface Area of a Cylinder
Think of this lesson as more than a rule to memorize. Surface Area of a Cylinder is about three-dimensional measurement, volume, and surface area. A strong student does not rush to the first formula on the page; they pause, identify the structure of the problem, and then choose the tool that matches that structure. That pause is what prevents most mistakes.
Geometry formulas work because they measure a specific feature: length around, space inside, or space enclosed by a solid. Match the question to the measurement first.
Here is the teacher way to approach the topic. First, read the problem slowly and underline the information that is actually given. Next, name the target: are you finding a value, simplifying an expression, comparing two quantities, solving for a variable, or interpreting a graph? Once the target is clear, the calculation becomes much less mysterious because every step has a job.
- Sketch or label the shape.
- Decide whether the question asks for length, area, volume, or surface area.
- Substitute values into the matching formula.
- Keep units squared for area and cubed for volume.
A helpful habit is to say what each number represents before using it. For example, if a number is a denominator, a radius, a slope, a common difference, or a coefficient, it should not be treated like an ordinary loose number. Its role tells you where it belongs in the formula. This is especially important on ACT-style questions because many wrong answer choices come from using the right numbers in the wrong places.
Another good habit is to keep the work organized vertically. Write one clean line for substitution, one line for simplifying, and one line for the final answer. If the problem has units, keep the units attached. If the problem has signs, exponents, or parentheses, copy them carefully from one line to the next. Most errors in this topic are not caused by a hard idea; they are caused by dropping a negative sign, combining unlike terms, using the wrong denominator, or skipping a check.
When you finish, ask a quick reasonableness question. Should the answer be positive or negative? Should it be larger or smaller than the original number? Does it fit the graph, table, shape, or equation? Can you plug it back into the original problem? This final check turns the lesson from a procedure into understanding.
On a test, the goal is not to write the longest solution. The goal is to write enough clear work that you can see the structure, avoid traps, and recover quickly if one line goes wrong. Practice the examples below with that mindset: identify the type of problem, choose the matching rule, show the substitution, simplify carefully, and check the answer before moving on.
Free printable Worksheets
Exercises for Surface Area of a Cylinder
1) A closed cylinder has radius \(3\) units and height \(5\) units. Find its surface area.
2) A cylinder has diameter \(8\) and height \(7\). Find the total surface area.
3) A closed cylinder has radius \(5\) units and height \(10\) units. Find its surface area.
4) Find the lateral area of a cylinder with radius \(6\) and height \(4\).
5) A closed cylinder has radius \(7\) units and height \(3\) units. Find its surface area.
6) A label covers only the side of a cylinder with radius \(4\) and height \(12\). Find the label area.
7) An open-top cylinder has radius \(5\) and height \(9\). Include the bottom but not the top.
8) A closed cylinder has surface area \(160\pi\) and radius \(5\). Find the height.
9) A cylinder has lateral area \(84\pi\) and height \(7\). Find the radius.
10) A closed cylinder has radius \(5\) units and height \(8\) units. Find its surface area.
11) Two identical closed cylinders each have radius \(3\) and height \(10\). Find their combined surface area.
12) The base circumference is \(12\pi\) and height is \(5\). Find surface area.
13) A cylinder has radius \(4\) and height twice its diameter. Find surface area.
14) For a closed cylinder with radius \(8\), how much greater is total surface area than lateral area?
15) A closed cylinder has surface area \(288\pi\) and diameter \(12\). Find the height.
16) A cylinder is open on the bottom but closed on top. Radius is \(7\) and height is \(11\). Find the outside area.
17) A cylinder with radius \(2\) and height \(5\) is enlarged by scale factor \(3\). Find the new surface area.
18) Radius increases from \(2\) to \(5\) while height stays \(6\). By how much does surface area increase?
19) A cylinder has volume \(125\pi\) and radius \(5\). Find total surface area.
20) Which closed cylinder has greater surface area: radius \(3\), height \(18\); or radius \(6\), height \(6\)? By how much?
1) Use \(SA = 2\pi r^2 + 2\pi rh\).
Compute: \(2\pi(3^2) + 2\pi(3)(5) = 48\pi\).
Answer: \(\color{red}{48\pi\ \text{units}^2}\)
2) Radius is \(4\).
Surface area: \(2\pi(4^2)+2\pi(4)(7)=88\pi\).
Answer: \(\color{red}{88\pi\ \text{square units}}\)
3) Use \(SA = 2\pi r^2 + 2\pi rh\).
Compute: \(2\pi(5^2) + 2\pi(5)(10) = 150\pi\).
Answer: \(\color{red}{150\pi\ \text{units}^2}\)
4) Use \(LA = 2\pi rh\).
Compute \(2\pi(6)(4)=48\pi\).
Answer: \(\color{red}{48\pi\ \text{square units}}\)
5) Use \(SA = 2\pi r^2 + 2\pi rh\).
Compute: \(2\pi(7^2) + 2\pi(7)(3) = 140\pi\).
Answer: \(\color{red}{140\pi\ \text{units}^2}\)
6) Use lateral area only.
Compute \(2\pi(4)(12)=96\pi\).
Answer: \(\color{red}{96\pi\ \text{square units}}\)
7) Use one base plus lateral area.
Compute \(\pi(5^2)+2\pi(5)(9)=115\pi\).
Answer: \(\color{red}{115\pi\ \text{square units}}\)
8) Set \(160\pi = 2\pi(5^2)+2\pi(5)h\).
Simplify \(160 = 50 + 10h\).
Solve \(h=11\).
Answer: \(\color{red}{11\ \text{units}}\)
9) Set \(84\pi = 2\pi r(7)\).
Solve \(84=14r\), so \(r=6\).
Answer: \(\color{red}{6\ \text{units}}\)
10) Use \(SA = 2\pi r^2 + 2\pi rh\).
Compute: \(2\pi(5^2) + 2\pi(5)(8) = 130\pi\).
Answer: \(\color{red}{130\pi\ \text{units}^2}\)
11) One cylinder: \(78\pi\).
Two cylinders: \(156\pi\).
Answer: \(\color{red}{156\pi\ \text{square units}}\)
12) From \(2\pi r=12\pi\), \(r=6\).
Surface area: \(72\pi+60\pi=132\pi\).
Answer: \(\color{red}{132\pi\ \text{square units}}\)
13) Height is \(16\).
Surface area: \(2\pi(4^2)+2\pi(4)(16)=160\pi\).
Answer: \(\color{red}{160\pi\ \text{square units}}\)
14) The difference is the two bases.
Compute \(2\pi(8^2)=128\pi\).
Answer: \(\color{red}{128\pi\ \text{square units}}\)
15) Radius is \(6\).
Set \(288 = 72 + 12h\).
Solve \(h=18\).
Answer: \(\color{red}{18\ \text{units}}\)
16) Use one top plus lateral area.
Compute \(\pi(7^2)+2\pi(7)(11)=203\pi\).
Answer: \(\color{red}{203\pi\ \text{square units}}\)
17) Original surface area: \(28\pi\).
Area scale factor: \(3^2=9\).
New surface area: \(252\pi\).
Answer: \(\color{red}{252\pi\ \text{square units}}\)
18) Old area: \(32\pi\).
New area: \(110\pi\).
Increase: \(78\pi\).
Answer: \(\color{red}{78\pi\ \text{square units}}\)
19) Find height: \(125\pi=\pi(5^2)h\), so \(h=5\).
Surface area: \(100\pi\).
Answer: \(\color{red}{100\pi\ \text{square units}}\)
20) First: \(126\pi\).
Second: \(144\pi\).
Difference: \(18\pi\).
Answer: \(\color{red}{\text{second cylinder by }18\pi}\)
Surface Area of a Cylinder Practice Quiz
