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Volume of Right Circular Cone: Definition and Examples | EDU.COMEDU.COMResourcesBlogGuidePodcastPlanBackHomesvg]:size-3.5">Math Glossarysvg]:size-3.5">Volume of Right Circular ConeVolume of Right Circular Cone: Definition and ExamplesTable of ContentsVolume of a Right Circular Cone Definition of Volume of a Right Circular Cone

A right circular cone is a three-dimensional shape with a circular base and a single vertex. The volume of a right circular cone measures the total space occupied by the cone. This volume equals one-third the product of the base area and height, expressed by the formula V=13πr2hV = \frac{1}{3} \pi r^{2}hV=31​πr2h, where rrr is the radius of the base circle and hhh is the height of the cone.

A right circular cone forms when a right triangle revolves around one of its legs, creating an axis that connects the vertex (apex) to the center of the circular base. When comparing cones and cylinders with the same radius and height, the volume of a cone is exactly one-third the volume of the cylinder. This means three identical cones could fill up a cylinder with matching dimensions.

Examples of Volume of a Right Circular Cone Example 1: Comparing Cylinder and Cone Volumes Problem:

A cylinder and a cone have the same radius and same height. If the volume of a cylinder is 1,038 cubic inches, what is the volume of the cone?

Step-by-step solution:

Step 1, Remember that the volume of a cone equals one-third of the volume of a cylinder with the same radius and height.

Step 2, Use the relationship to find the cone's volume: Volume of cone = 13×\frac{1}{3} \times31​× Volume of cylinder

Step 3, Substitute the given cylinder volume: Volume of cone = 13×1,038\frac{1}{3} \times 1,03831​×1,038

Step 4, Calculate the final volume: Volume of cone = 346 cubic inches.

Example 2: Finding Volume with Known Radius and Height Problem:

Find the volume of the right circular cone if the radius is 7 units and height is 9 units.

Step-by-step solution:

Step 1, Identify what we know: Radius of the cone (r) = 7 units and Height of the cone (h) = 9 units

Step 2, Apply the volume formula: Volume = 13πr2h\frac{1}{3} \pi r^{2} h31​πr2h

Step 3, Substitute the values into the formula: Volume = 13×227×7×7×9\frac{1}{3} \times \frac{22}{7} \times 7 \times 7 \times 931​×722​×7×7×9

Step 4, Calculate step by step:

First find r2r^2r2 = 7×77 \times 77×7 = 49 Multiply by height: 49×949 \times 949×9 = 441 Multiply by π\piπ (227\frac{22}{7}722​): 441×227441 \times \frac{22}{7}441×722​ = 1,386 Multiply by 13\frac{1}{3}31​: 1,386×131,386 \times \frac{1}{3}1,386×31​ = 462

Step 5, Write the final answer: Volume = 462 cubic units.

Example 3: Finding Radius from Volume and Height Problem:

If the volume of the right circular cone is 24.5π cubic inches and height is 6 inches, then what will be the radius of the cone?

Step-by-step solution:

Step 1, Write down what we know:

Volume of the cone = 24.5π cubic inches Height of the cone (h) = 6 inches

Step 2, Use the volume formula and substitute the known values: 13πr2h=24.5π\frac{1}{3} \pi r^{2} h = 24.5\pi31​πr2h=24.5π

Step 3, Simplify by dividing both sides by π: 13r2×6=24.5\frac{1}{3} r^{2} \times 6 = 24.531​r2×6=24.5

Step 4, Solve for r²:

Multiply both sides to remove the fraction: 2r2=24.52r^{2} = 24.52r2=24.5 Divide both sides by 2: r2=24.52=12.25r^{2} = \frac{24.5}{2} = 12.25r2=224.5​=12.25

Step 5, Find the radius by taking the square root: r=12.25=3.5r = \sqrt{12.25} = 3.5r=12.25​=3.5 inches

Comments(2)SSwimmerEvanNovember 5, 2025I've used this glossary page to teach my students. The formula explanation and examples really helped them grasp the volume of a right circular cone!

MCMs. CarterSeptember 17, 2025This explanation really helped my kids understand the volume of a cone! The examples made it easy to compare shapes, and we even tried measuring cones at home. Great resource for hands-on learning!

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