How to Find the Volume of a Cylinder
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Definition
A cylinder has two congruent circular bases connected by a curved surface. The radius \(r\) is the distance from the center of a base to its edge, and the height \(h\) is the perpendicular distance between the bases.
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A Cylinder's Volume
The base is a circle with area \(\pi r^2\). Multiplying by height gives:
\(V = \pi r^2h\)
Volume is measured in cubic units. Unless an approximation is requested, leave answers exactly in terms of \(\pi\).
Example
If \(r = 4\) cm and \(h = 7\) cm, then \(V = \pi(4^2)(7) = 112\pi\).
Volume of a Cylinder
Think of this lesson as more than a rule to memorize. Volume 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 Volume of a Cylinder
1) A cylinder has radius \(3\) cm and height \(5\) cm. Find its volume.
2) A cylinder has diameter \(8\) in. and height \(7\) in. Find its volume.
3) A cylinder has radius \(5\) ft and height \(10\) ft. Find its volume.
4) A cylinder has radius \(2\) units and height \(9\) units. Find its volume.
5) A cylinder has radius \(4\) m and height \(6\) m. Find its volume.
6) A cylinder has volume \(144\pi\) and radius \(6\). Find the height.
7) A cylinder has volume \(200\pi\) and height \(8\). Find the radius.
8) A cylinder has diameter \(10\) cm and height \(12\) cm. Find the volume.
9) A cylinder has radius \(3\) and height \(4\). If height is doubled, find the new volume.
10) A cylinder has radius \(2\) in and height \(6\) in. Find its volume.
11) Use \(\pi \approx 3.14\) to approximate the volume of a cylinder with radius \(4\) and height \(9\).
12) A cylinder has height \(10\) and volume \(490\pi\). Find the radius.
13) A cylinder has radius \(6\). Its height equals its diameter. Find the volume.
14) A semicylindrical trough has radius \(4\) ft and length \(10\) ft. Find half the full-cylinder volume.
15) Two cylinders have height \(8\) and radii \(5\) and \(3\). How much greater is the larger volume?
16) The base circumference is \(18\pi\) and height is \(4\). Find the volume.
17) The base area is \(64\pi\) and height is \(11\). Find the volume.
18) A cylinder has volume \(100\pi\) and height \(25\). Find its diameter.
19) A cylinder's radius increases from \(4\) to \(6\) while height stays \(10\). By how much does volume increase?
20) Water from a full cylinder with radius \(6\) and height \(10\) is poured into a cylinder with radius \(4\). What height will it reach?
1) Use \(V = \pi r^2h\).
Substitute: \(V = \pi(3^2)(5)\).
Compute: \(45\pi\).
Answer: \(\color{red}{45\pi\ \text{cm}^3}\)
2) Radius is \(4\).
Volume: \(\pi(4^2)(7)=112\pi\).
Answer: \(\color{red}{112\pi\ \text{in}^3}\)
3) Use \(V = \pi r^2h\).
Substitute: \(V = \pi(5^2)(10)\).
Compute: \(250\pi\).
Answer: \(\color{red}{250\pi\ \text{ft}^3}\)
4) Use \(V = \pi r^2h\).
Substitute: \(V = \pi(2^2)(9)\).
Compute: \(36\pi\).
Answer: \(\color{red}{36\pi\ \text{units}^3}\)
5) Use \(V = \pi r^2h\).
Substitute: \(V = \pi(4^2)(6)\).
Compute: \(96\pi\).
Answer: \(\color{red}{96\pi\ \text{m}^3}\)
6) Set \(144\pi = \pi(6^2)h\).
Cancel \(\pi\): \(144=36h\).
Solve \(h=4\).
Answer: \(\color{red}{4\ \text{units}}\)
7) Set \(200\pi = \pi r^2(8)\).
Then \(r^2=25\).
So \(r=5\).
Answer: \(\color{red}{5\ \text{units}}\)
8) Radius is \(5\) cm.
Volume: \(\pi(5^2)(12)=300\pi\).
Answer: \(\color{red}{300\pi\ \text{cm}^3}\)
9) New height is \(8\).
Volume: \(\pi(3^2)(8)=72\pi\).
Answer: \(\color{red}{72\pi\ \text{cubic units}}\)
10) Use \(V = \pi r^2h\).
Substitute: \(V = \pi(2^2)(6)\).
Compute: \(24\pi\).
Answer: \(\color{red}{24\pi\ \text{in}^3}\)
11) Exact volume: \(144\pi\).
Approximate: \(144\cdot3.14 = 452.16\).
Answer: \(\color{red}{452.16\ \text{cubic units}}\)
12) Set \(490\pi = \pi r^2(10)\).
Then \(r^2 = 49\).
So \(r = 7\).
Answer: \(\color{red}{7\ \text{units}}\)
13) Height is \(12\).
Volume: \(\pi(6^2)(12)=432\pi\).
Answer: \(\color{red}{432\pi\ \text{cubic units}}\)
14) Full volume: \(\pi(4^2)(10)=160\pi\).
Half is \(80\pi\).
Answer: \(\color{red}{80\pi\ \text{ft}^3}\)
15) Larger: \(\pi(5^2)(8)=200\pi\).
Smaller: \(\pi(3^2)(8)=72\pi\).
Difference: \(128\pi\).
Answer: \(\color{red}{128\pi\ \text{cubic units}}\)
16) From \(2\pi r=18\pi\), \(r=9\).
Volume: \(\pi(9^2)(4)=324\pi\).
Answer: \(\color{red}{324\pi\ \text{cubic units}}\)
17) Volume equals base area times height.
Compute \(64\pi\cdot11=704\pi\).
Answer: \(\color{red}{704\pi\ \text{cubic units}}\)
18) Set \(100\pi = \pi r^2(25)\).
Then \(r^2=4\), so \(r=2\).
Diameter is \(4\).
Answer: \(\color{red}{4\ \text{units}}\)
19) Old volume: \(160\pi\).
New volume: \(360\pi\).
Increase: \(200\pi\).
Answer: \(\color{red}{200\pi\ \text{cubic units}}\)
20) Original volume: \(\pi(6^2)(10)=360\pi\).
Set \(360\pi = \pi(4^2)h\).
Solve \(h=22.5\).
Answer: \(\color{red}{22.5\ \text{units}}\)
Volume of a Cylinder Practice Quiz
