Find The Height Of A Cylinder When Given The Volume And Radius

🔑 Key Concepts

• The volume of a cylinder is calculated using \(V=\pi r^2h\).
• When volume and radius are known, substitute those values and solve the equation for \(h\).
• The radius must be squared before multiplying by \(\pi\).
• Use \(3.14\) for \(\pi\) when the problem specifically directs you to do so.
• Dividing the volume by the area of the circular base gives the cylinder’s height.

✏️ Worked Example

Restate the Problem

A cylinder has a volume of \(602.88\) cubic centimeters and a radius of \(4\) centimeters. Find its height.

Use \(3.14\) for \(\pi\).

Step 1: Identify the Known Values

Begin with the cylinder volume formula:

\(V=\pi r^2h\)

The problem gives the volume and radius. The height is unknown:

\(V=602.88\text{ cm}^3\)
\(r=4\text{ cm}\)
\(\pi\approx3.14\)
\(h=?\)

Substitute these known values into the formula before solving for \(h\).

Step 2: Substitute and Simplify

\(602.88=(3.14)(4)^2h\)

Square the radius:

\(4^2=16\)

\(602.88=(3.14)(16)h\)

Multiply \(3.14\) by \(16\):

\(602.88=50.24h\)

The value \(50.24\) is the area of the circular base:

\(\pi r^2\approx50.24\text{ cm}^2\)

Step 3: Solve for the Height

The equation is:

\(602.88=50.24h\)

Divide both sides by \(50.24\):

\(\displaystyle \frac{602.88}{\color{red}{50.24}} = \frac{50.24h}{\color{red}{50.24}} \)

\(12=h\)

\(h=12\)

Height of the cylinder:
\(\boxed{h=12\text{ cm}}\)

Step 4: Check the Solution

Substitute \(r=4\) and \(h=12\) into the original formula:

\(V=\pi r^2h\)

\(V=(3.14)(4)^2(12)\)

\(V=(3.14)(16)(12)\)

\(V=(50.24)(12)\)

\(V=602.88\text{ cm}^3\)

The calculated volume matches the volume given in the problem, so the height is correct.

Why Does Dividing by the Base Area Find the Height?

A cylinder’s volume can be written more generally as:

\(V=Bh\)

The value \(B\) represents the area of the circular base:

\(B=\pi r^2=50.24\text{ cm}^2\)

Since the cylinder has \(602.88\) cubic centimeters of volume, dividing that volume by \(50.24\) square centimeters tells us how many one-centimeter layers of that base area are stacked vertically:

\(\displaystyle h=\frac{V}{B}=\frac{602.88}{50.24}=12\)

Therefore, the cylinder is \(12\) centimeters tall.

🧠 Math Vocabulary

• Cylinder: A three-dimensional solid with two congruent, parallel circular bases.
• Radius: The distance from the center of a circular base to its outer edge.
• Height: The perpendicular distance between the two circular bases of a cylinder.
• Base area: The area of one circular base, calculated using \(B=\pi r^2\).
• Volume: The amount of three-dimensional space contained inside a solid.
• Cubic units: Units used to measure volume, such as cubic centimeters or \(\text{cm}^3\).
• Substitution: Replacing a variable in a formula with a known value.
• Inverse operation: An operation used to undo another operation, such as division undoing multiplication.
• Unknown dimension: A measurement that must be determined from the other information in a problem.

💡 Main Idea

When a cylinder’s volume and radius are known, substitute those values into \(V=\pi r^2h\), simplify the area of the circular base, and divide the volume by that base area. In this problem, the base area is \(50.24\) square centimeters, so \(602.88\div50.24=12\). The cylinder’s height is therefore \(12\) centimeters.

📚 What You Should Already Know

Students should know how to calculate the area of a circle, evaluate a squared value, substitute known measurements into a formula, multiply decimals, and solve a one-step equation by dividing both sides by the same number.

🚀 What Comes Next

Students can extend this reasoning by finding a missing radius, solving for unknown dimensions of cones, working with exact volumes in terms of \(\pi\), and applying volume formulas to real-world containers and composite solids.

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