Proving the Volume Ratios of a Cone, Hemisphere, and Cylinder with Equal Bases and Heights

In geometry, the relationship between the volumes of a cone, a hemisphere, and a cylinder with equal bases and heights can be explored using basic volume formulas. This article demonstrates how to show that these volumes maintain a specific ratio of 1:2:3.

Introduction to the Volumes of Geometric Shapes

The volume of geometric shapes is a fundamental concept in both mathematics and physics. For this problem, we will focus on three shapes: a cone, a hemisphere, and a cylinder. All these shapes have the same base and the same height. By using the formulas for their volumes, we can determine the ratio of these volumes.

Volume Formulas

Volume of a Cone

The volume of a cone (Vc) is given by the formula:

V_c frac{1}{3} pi r^2 h

Volume of a Hemisphere

The volume of a hemisphere (Vh) is given by the formula:

V_h frac{2}{3} pi r^3

Since the height of the hemisphere is equal to its radius, we can set h r:

V_h frac{2}{3} pi r^3 frac{2}{3} pi r^2 h quad text{since } h r

Volume of a Cylinder

The volume of a cylinder (Vcy) is given by the formula:

V_{cy} pi r^2 h

Expressing the Volumes in Terms of h

Since the problem states that all the shapes have the same height, we can express the volumes in terms of h using h r:

Cone:

V_c frac{1}{3} pi r^2 h

Hemisphere (using h r):

V_h frac{2}{3} pi r^2 h frac{2}{3} pi r^2 r frac{2}{3} pi r^2 r

Cylinder:

V_{cy} pi r^2 h pi r^2 r

Expressing the Volumes in the Ratio 1:2:3

Now we can express the volumes in a ratio:

Vc: Vh: Vcy frac{1}{3} pi r^2 h : frac{2}{3} pi r^2 h : pi r^2 h

By eliminating the fractions, multiplying each term by 3:

1: 2: 3

Conclusion

In conclusion, the volumes of a cone, hemisphere, and cylinder with equal bases and heights are in the ratio 1:2:3. This result is demonstrated using the volume formulas and expressing the volumes in terms of a common height h.

By following these steps, we have proved that the volumes of these three shapes maintain a specific ratio, which is a fundamental concept in solid geometry.

Keywords: cone volume, hemisphere volume, cylinder volume, volume ratio