Conical Frustum Calculator

What is a Conical Frustum?

A conical frustum is the portion of a cone that lies between two parallel planes cutting the cone. It's essentially a cone with the top cut off, resulting in a shape with circular bases of different sizes. Conical frustums are common in everyday objects like lampshades, buckets, and certain types of cups.

How to Calculate Conical Frustum Properties

To fully understand a conical frustum, we need to calculate several key properties: its volume, surface area, slant height, and lateral surface area. Each of these properties provides unique information about the frustum's size and shape.

Formulas

Here are the essential formulas for a conical frustum:

1. Volume (V):

\[ V = \frac{1}{3}\pi h(R^2 + r^2 + Rr) \]

2. Surface Area (SA):

\[ SA = \pi(R^2 + r^2) + \pi(R + r)s \]

3. Slant Height (s):

\[ s = \sqrt{h^2 + (R - r)^2} \]

4. Lateral Surface Area (LSA):

\[ LSA = \pi(R + r)s \]

Where:

• \(R\) is the radius of the larger base
• \(r\) is the radius of the smaller base
• \(h\) is the height of the frustum
• \(s\) is the slant height
• \(\pi\) (pi) is approximately 3.14159

Calculation Steps

1. Determine the radii of both bases (R and r) and the height (h) of the frustum
2. Calculate the slant height using \(s = \sqrt{h^2 + (R - r)^2}\)
3. Calculate the volume using \(V = \frac{1}{3}\pi h(R^2 + r^2 + Rr)\)
4. Calculate the surface area using \(SA = \pi(R^2 + r^2) + \pi(R + r)s\)
5. Calculate the lateral surface area using \(LSA = \pi(R + r)s\)

Example and Visual Representation

Let's calculate the properties of a conical frustum with the following dimensions:

• Radius of larger base (R) = 5 units
• Radius of smaller base (r) = 3 units
• Height (h) = 4 units

1. Slant height: \(s = \sqrt{4^2 + (5 - 3)^2} = \sqrt{16 + 4} = \sqrt{20} \approx 4.47\) units
2. Volume: \(V = \frac{1}{3}\pi \cdot 4(5^2 + 3^2 + 5 \cdot 3) = \frac{1}{3}\pi \cdot 4(25 + 9 + 15) = \frac{196\pi}{3} \approx 205.26\) cubic units
3. Surface Area: \(SA = \pi(5^2 + 3^2) + \pi(5 + 3)4.47 \approx 201.06\) square units
4. Lateral Surface Area: \(LSA = \pi(5 + 3)4.47 \approx 112.34\) square units

Here's a visual representation of this conical frustum:

In this diagram, you can see the conical frustum with its key dimensions labeled. The red lines represent the radii of the bases, the green line represents the height, and the purple dashed line represents the slant height. The blue outline shows the shape of the frustum, including its circular bases.