When it comes to geometry, two terms that are often used interchangeably are frustum and truncated solid. While both may refer to a shape with a unique geometry, they are not the same thing. In this article, we will attempt to explain the difference between the frustum and a truncated solid.

Frustum

A frustum is a geometric shape that is formed by slicing a solid shape into two parallel planes. The resulting shape that remains is called a frustum. A frustum has a unique geometry, and its properties can be derived by analyzing the original shape that it was derived from.

The most common type of frustum is a cone frustum. A cone frustum is derived by slicing a cone with a plane parallel to its base, and the resulting shape is a truncated cone. A frustum can also be derived from a pyramid, where the top portion of the pyramid is sliced off with a plane parallel to its base.

A frustum can be described by its height, base area, and the area of its top surface. These properties can be used to calculate various parameters of the frustum, such as its volume and surface area.

Truncated Solid

A truncated solid, on the other hand, refers to a shape that is derived by slicing a solid with a plane that is not parallel to its base. The resulting shape is a solid that has its top portion missing, but still retains its original geometry to some extent.

A truncated solid can be derived from a cone, pyramid, or any other solid shape. The resulting shape can vary depending on the angle of the slice and the shape of the solid that was used to derive it.

One of the most common types of truncated solids is a truncated pyramid. A truncated pyramid is derived by slicing a pyramid at an angle that is not parallel to its base. The resulting shape is a pyramid that has its top portion missing, but its base remains intact.

A truncated solid can be described by its height, base area, top area, and the angle of the slice. These properties can be used to calculate various parameters of the truncated solid, such as its volume and surface area.

Difference Between Frustum And Truncated Solid

Now that we have a basic understanding of what a frustum and a truncated solid are, let us examine the key differences between the two shapes.

1. Parallel vs. Non-Parallel Planes

The most significant difference between a frustum and a truncated solid is how they are derived. A frustum is formed by slicing a shape with two parallel planes, whereas a truncated solid is derived by slicing a shape with a non-parallel plane.

2. Retains Original Geometry vs. Modified Geometry

When a frustum is derived, the resulting shape retains the original geometry of the solid that it was derived from. The only difference is that the top portion of the shape is missing. On the other hand, a truncated solid may have a modified geometry due to the angled slice that was used to derive it.

3. Description by Base and Top Surface vs. Angle of the Slice

A frustum is described by its height, base area, and top surface area. A truncated solid is described by its height, base area, top area, and the angle of the slice that was used to derive it.

4. Unique Geometry vs. Multiple Shapes

A frustum has a unique geometry that can be derived from a cone or a pyramid. On the other hand, a truncated solid can be derived from any solid shape, including cones, pyramids, cylinders, and spheres.

Conclusion

While a frustum and a truncated solid may seem similar at first glance, they are two distinct shapes with unique properties. A frustum is derived by slicing a solid with two parallel planes, whereas a truncated solid is derived by slicing a solid with a plane that is not parallel to its base. A frustum retains the original geometry of the solid that it was derived from, while a truncated solid may have a modified geometry. Additionally, a frustum is described by its height, base area, and top surface area, whereas a truncated solid is described by its height, base area, top area, and the angle of the slice that was used to derive it. By understanding these key differences, we can appreciate the unique properties of each shape and their applications in various fields, including mathematics, engineering, and architecture.