Frustum

Pentagonal frustum

Usech kvadrat piramid

Square frustum

Triangulated monorecied tetrahedron

Frustum with symbols

Frustum is a geometric shape that results from slicing a pyramid or a cone with a plane parallel to its base, leaving two portions: a smaller pyramid or cone at the top (which is removed) and the remaining bottom part, which is the frustum. The term "frustum" comes from the Latin word frustum, meaning "piece" or "crumb". The frustum is a common shape in everyday objects, such as drinking glasses, lampshades, and buckets.

Definition

A frustum is defined by three main components: the top and bottom bases, which are parallel and similar polygons, and the lateral surface connecting the edges of the two bases. In the case of a conical frustum, the bases are circles. The height (h) of a frustum is the perpendicular distance between the bases. The volume (V) of a frustum can be calculated using the formula:

• For a pyramidal frustum:

\[ V = \frac{h}{3} \left( A_1 + A_2 + \sqrt{A_1A_2} \right) \] where \(A_1\) and \(A_2\) are the areas of the top and bottom bases, respectively.

• For a conical frustum:

\[ V = \frac{\pi h}{3} \left( r_1^2 + r_2^2 + r_1r_2 \right) \] where \(r_1\) and \(r_2\) are the radii of the top and bottom circles, respectively.

The surface area of a frustum includes the areas of the top and bottom bases plus the lateral surface area. The lateral surface area (A_lateral) of a conical frustum can be found with the formula: \[ A_{lateral} = \pi (r_1 + r_2) s \] where \(s\) is the slant height of the frustum.

Applications

Frustums are encountered in various fields, including architecture, engineering, and manufacturing. They are used in the design of structures, such as domes and towers, to provide stability and aesthetic appeal. In engineering, frustums are part of the design of machine components, such as gears and bearings, to ensure proper function and efficiency. In everyday life, frustums are seen in objects like paper cups, flower pots, and certain types of packaging.

Mathematical Properties

The mathematical study of frustums involves exploring their properties, such as volume, surface area, and the relationships between their dimensions. This study is crucial in fields like geometry and calculus, where understanding the properties of shapes is fundamental to solving complex problems.

See Also

• Pyramid
• Cone
• Geometry
• Volume
• Surface area