Finding The Volume Of Cones And Cylinders

🔑 Key Concepts

• The volume of a cylinder is calculated using \(V=\pi r^2h\).
• The volume of a cone is calculated using \(\displaystyle V=\frac{1}{3}\pi r^2h\).
• A cone has exactly one-third the volume of a cylinder when both solids have the same radius and height.
• The cylinder and cone share the same circular base area, \(\pi r^2\), and the same perpendicular height.
• Three congruent cones with the same radius and height would fill the corresponding cylinder.

✏️ Worked Example

Same Radius and Height

The cone fits inside the cylinder, and both solids have:

\(r=6\text{ cm}\)
\(h=12\text{ cm}\)

Because the radius and height match, the cone’s volume will be exactly one-third of the cylinder’s volume.

Volume of the Cylinder

Use the cylinder volume formula:

\(V_{\text{cylinder}}=\pi r^2h\)

Substitute \(r=6\) and \(h=12\):

\(V_{\text{cylinder}}=\pi(6)^2(12)\)

\(V_{\text{cylinder}}=\pi(36)(12)\)

\(V_{\text{cylinder}}=432\pi\)

Cylinder volume:
\(\boxed{432\pi\text{ cm}^3}\)

Volume of the Cone

Use the cone volume formula:

\(\displaystyle V_{\text{cone}}=\frac{1}{3}\pi r^2h\)

Substitute the same radius and height:

\(\displaystyle V_{\text{cone}}=\frac{1}{3}\pi(6)^2(12)\)

\(\displaystyle V_{\text{cone}}=\frac{1}{3}\pi(36)(12)\)

\(\displaystyle V_{\text{cone}}=\frac{432\pi}{3}\)

\(V_{\text{cone}}=144\pi\)

Cone volume:
\(\boxed{144\pi\text{ cm}^3}\)

Comparing the Formulas

The cylinder and cone use the same radius and height, so the expression \(\pi r^2h\) is identical in both calculations.

\(V_{\text{cylinder}}=\pi r^2h\)

\(\displaystyle V_{\text{cone}}=\frac{1}{3}\pi r^2h\)

The cone formula multiplies the entire cylinder expression by \(\displaystyle \frac{1}{3}\):

\(\displaystyle V_{\text{cone}}=\frac{1}{3}V_{\text{cylinder}}\)

\(\displaystyle V_{\text{cone}}=\frac{1}{3}(432\pi)\)

\(\boxed{V_{\text{cone}}=144\pi\text{ cm}^3}\)

The factor \(\displaystyle \frac{1}{3}\) is a permanent part of the cone volume formula because a cone occupies one-third of the corresponding cylinder.

Verify the One-Third Relationship

Compare the cone’s volume to the cylinder’s volume:

\(\displaystyle \frac{V_{\text{cone}}}{V_{\text{cylinder}}} = \frac{144\pi}{432\pi}\)

\(\displaystyle \frac{V_{\text{cone}}}{V_{\text{cylinder}}} = \frac{144}{432}\)

\(\displaystyle \frac{V_{\text{cone}}}{V_{\text{cylinder}}} = \frac{1}{3}\)

\(\boxed{V_{\text{cone}}=\frac{1}{3}V_{\text{cylinder}}}\)

The ratio confirms that the cone contains exactly one-third as much volume as the cylinder.

Three Cones Fill the Cylinder

Since one cone contains \(144\pi\) cubic centimeters, three identical cones contain:

\(3(144\pi)=432\pi\)

\(3V_{\text{cone}}=V_{\text{cylinder}}\)

This matches the cylinder’s calculated volume:

\(V_{\text{cylinder}}=432\pi\text{ cm}^3\)

Therefore, three cones with the same radius and height would fill one cylinder with those dimensions.

🧠 Math Vocabulary

• Cylinder: A three-dimensional solid with two congruent, parallel circular bases.
• Cone: A three-dimensional solid with one circular base that narrows to a single vertex.
• Radius: The distance from the center of a circular base to its outer edge.
• Height: The perpendicular distance between a cylinder’s bases or from a cone’s base to its vertex.
• Base area: The area of the circular base, calculated using \(B=\pi r^2\).
• Volume: The amount of three-dimensional space contained inside a solid.
• Corresponding solids: Solids compared using matching measurements, such as the same radius and height.
• One-third relationship: The relationship showing that a cone occupies one-third the volume of a cylinder with the same base and height.
• Cubic units: Units used to measure volume, such as cubic centimeters or \(\text{cm}^3\).

💡 Main Idea

A cone and cylinder with the same radius and height share the same circular base area and perpendicular height. The cylinder’s volume is \(\pi r^2h\), while the cone’s volume is one-third of that amount. For a radius of \(6\) centimeters and height of \(12\) centimeters, the cylinder has a volume of \(432\pi\) cubic centimeters and the cone has a volume of \(144\pi\) cubic centimeters.

📚 What You Should Already Know

Students should know how to calculate the area of a circle, evaluate a squared radius, substitute values into formulas, multiply and divide expressions containing \(\pi\), and interpret fractions as multiplicative comparisons.

🚀 What Comes Next

Students can apply this relationship to find a missing cone or cylinder volume without completing two separate calculations, solve for missing dimensions, compare capacities, and explain why the cone formula includes the factor \(\displaystyle \frac{1}{3}\).

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