Volume and Surface Area of a Frustum of a Cone: An Analytical Approach

A regular cone with a top radius of 8 cm is cut at a certain height to create a frustum with a top radius of 3 cm and a height of 10 cm. This article delves into the process of calculating both the volume and the total surface area of such a frustum. We will use analytical geometry along with the Pythagorean theorem to derive the necessary dimensions and perform the calculations.

Introduction

In this section, we introduce the concept of a frustum of a cone and the specific dimensions provided for the calculation. A full cone with a base radius of 8 cm is cut at a certain height, resulting in a frustum with a top radius of 3 cm and a height of 10 cm.

Dimensions

Top radius (r2): 3 cm Bottom radius (r1): 8 cm Height of the frustum (H): 10 cm

Calculation of the Volume of the Frustum

The volume of a frustum of a cone can be calculated using the following formula:

V (1/3) * π * H * (r12 r22 r1 * r2)

Substituting the provided dimensions:

V (1/3) * π * 10 * (82 32 8 * 3)

V (1/3) * π * 10 * (64 9 24)

V (1/3) * π * 10 * 97

V (100/3) * π * 9.7

V ≈ 1015.3 cm3

Calculation of the Total Surface Area of the Frustum

The total surface area of a frustum of a cone can be calculated using the formula:

S π * (r12 r22) π * (r1 r2) * L

Where L is the slant height of the frustum, which can be found using the Pythagorean theorem:

L sqrt((r1 - r2)2 H2)

L sqrt((8 - 3)2 102)

L sqrt(52 100)

L sqrt(125) ≈ 11.18 cm

Now, inserting the values into the surface area formula:

S π * (82 32) π * (8 3) * 11.18

S π * (64 9) π * 11 * 11.18

S π * 73 π * 123.08

S π * (73 123.08) ≈ 31856/7 cm2

Geometry of the Frustum

Given the dimensions of the full cone from which the frustum was derived, we can further analyze the geometry of the frustum:

Let's denote the height of the full cone as h1 and the height of the cut cone as h2. We know that h2 10 cm and h1 - h2 6 cm (since the top radius is 3 cm, we can use similar triangles to find h2):

3/h2 8/h1

h1 80/3 ≈ 26.67 cm

Using the Pythagorean theorem for the full cone:

L sqrt(h12 r12)

L sqrt(26.672 82)

L sqrt(711.49 64) ≈ 26.39 cm

And for the smaller cone:

L' sqrt(h22 r22)

L' sqrt(102 32)

L' sqrt(100 9) ≈ 10.49 cm

Conclusion

The volume of the given frustum is approximately 1015.3 cm3, and the total surface area is approximately 31856/7 cm2. This comprehensive analysis covers the geometric properties of the frustum, including the detailed calculations for bottom and top radii, height, and slant heights, leading to the final volume and surface area.

Additional Resources

For further reading and more detailed calculations, you may refer to:

Cone and Frustum Calculators Math Is Fun - Cone Geometry