Pentagonal Prism

The Pentagonal Prism (named for the five-sided base) is a polygon with five sides (called edges) and five faces (called triangular sides). It has two diagonals, as well as two other interior diagonals, that meet at a center.

The edges of the Pentagonal Prism are all isosceles right triangles. The diagonals are equidistant. Therefore, the sides of the Pentagonal Prism are in a right-angle triangle with an obtuse angle at the apex of the Prism, and two acute angles, which make a smaller right angle.

A Pentagonal Prism is one of the Convex regular polygons and the only one with diagonals that are right isosceles triangles. The Pentagonal Prism can be split into two isosceles triangles. The smallest of the two isosceles triangles is the so-called Pentagonal Prism, while the biggest is the regular polygon.

The Pentagonal Prism is a regular solid. If one side of the Pentagonal Prism is divided into ten equal parts, it results in a regular pentagon. If two sides of the Pentagonal Prism are divided into five equal parts each, it results in an isosceles trapezoid.

The regular pentagon is the only regular solid with an even number of edges. The regular pentagon has 5 faces (the Pentagonal Prism). The regular pentagon can also be made by joining the midpoints of three of the sides of a regular triangle.

Convex

This solid is Convex; that is, any point inside the solid can be connected to a point on the boundary of the solid by a straight line that does not go through the center of the solid.

Projections

The Pentagonal Prism may be considered a three-dimensional analogue of the Prism and pyramid. The three-dimensional projections of the Pentagonal Prism are the pyramid, the triangular Prism, and the Pentagonal pyramid. The base of the pyramids, the triangular Prisms, and the Pentagonal Prisms, and their corresponding solid figures, are analogous to the base, the sides, and the apex of the Prism and pyramid.

A Pentagonal Prism is an asymmetrical Convex polyhedron with five faces meeting at fivefold axes and with a triangular base. In other words, a Pentagonal Prism is a Convex polygon with five sides, six faces, and five corners. It is a member of the Kepler–Poinsot polyhedra, the Johnson solids, and the Kepler–Poinsot polyhedra family.

The Pentagonal Prism is named in honor of William Sturgeon (or Sturgeon's Prisms). It was one of the first Convex polyhedra to be named after it was introduced to the community in 1735 by William Sturgeon of London in his work A Treatise on Plane Convexity. The term "sturgeon's Prisms" was eventually shortened to "sturgeons' Prisms" or "sturgeons". Sturgeon is also given credit as the creator of the Pentagonal Prism; however, a different source claims the Prism originated much earlier in China and was first described by the mathematician Chen Zai in the 9th century.

The Pentagonal Prism is the only regular polyhedron with only four faces, the only regular polyhedron with only three regular faces, and the only regular polyhedron with just one non-regular face (its fifth face is regular). The Prism is a Petrie polyhedron, having Petrie duals of one and five faces, and can be formed as the vertex arrangement of two truncated regular dodecahedra sharing their Pentagonal faces.

Geometrical properties

It has the Schläfli symbol {5,5} and Coxeter plane symmetry D5. It is a Johnson solid with a Coxeter diagram. It is also a member of the Kepler–Poinsot polyhedra family and its Convex hull in space is a Kepler–Poinsot polyhedra.

The Pentagonal Prism has three dihedral angles of 65.5°, as well as three triangular faces that are congruent to one another: {60°, 60°, 30°, 90°} and {60°, 60°, 60°, 60°} – both with the same ratio of.

The Pentagonal Prism can be used to construct two other polyhedra. With two Pentagonal Prisms placed end to end, the Prism has a Rectangular pyramidal base. It is a truncated Cube and can be regarded as having an extra "square" base face. Two Pentagonal Prisms can be edge-to-edge to form a rectangle, and a rectangle and a triangular face can be connected in a Pentagonal pyramid, thereby forming a rhombicosidodecahedron.

Forms

A Pentagonal Prism can be constructed by truncating the faces of a Pentagonal pyramid, a regular star polyhedron, or by a combination of both methods.

Geometrically, a regular Pentagonal Prism is formed by truncating the faces of a regular pentagon. Truncating the outermost 5 faces of the pentagon would form the Pentagonal Prism, leaving a pyramid with 5 triangular faces and a pyramid with 5 pyramidal (square) faces. This is a Pentagonal pyramid. The pyramid with the Pentagonal faces has 12 edges of the pentagon.

Since there are 5 faces and 12 edges, the Prism has 60 edges of the pentagon, and so a total of 20 faces. One vertex is the center of the pyramid, another is its apex, and a Pentagonal Prism can be regarded as being the meeting point of the pyramid and the pentagon.

Truncating the innermost 5 faces of the pentagon (with 5 pyramidal faces and 5 triangular faces) would create a Pentagonal pyramid, a star polyhedron. The truncation creates a vertex which is a flat plane Surface. In the diagram below, it is indicated by a pentagon and a Pentagonal Prism, respectively.

Other possible methods of building Pentagonal Prisms are the truncation of faces of a regular decagon, and the truncation of the pyramids of a regular decagon. The first method creates a decagonal Prism, while the second creates an icosidodecahedron. The first method can also be regarded as a truncation of the decagon face.

What is a Pentagonal Prism?

• A Prism that consists of two Pentagonal bases the top and the bottom and has five Rectangular sides is known as a Pentagonal Prism.

• A Pentagonal Prism is a type of heptahedron with 15 edges, 10 vertices, and 7 faces.

• A Pentagonal Prism has Pentagonal bases which give five sides of the Prism

• The other name for a Pentagonal Prism is a five-sided polygon Prism.

What is a Prism?

In mathematics, a Prism is generally a three-dimensional box that is a solid figure with a uniform cross-section and it has two common bases.

• Face: Faces of the Pentagonal Prism can be defined as the flat side of a 3-dimensional object.

• Base: The base is defined as one of two congruent and parallel sides of an object.

• Edge: An edge is a line of intersection of two faces on a solid object.

• Vertex: Vertex is the joining point where two edges of a Prism meet.

Properties of a Pentagonal Prism:

Pentagonal Prism faces, edges, vertices have been listed below in a tabular form.

Here’s a diagram showing Pentagonal Prism faces, edges, vertices-

Types of Pentagonal Prisms-

The two types of Pentagonal Prisms are –

• Regular Pentagonal Prisms

• Right Pentagonal Prisms

What is a Regular Pentagonal Prism?

• In a regular Pentagonal Prism, all the sides of the Pentagonal Prism are of exactly equal length.

• All the Rectangular faces of Pentagonal Prisms are congruent.

• The Rectangular faces in a regular Pentagonal Prism are said to be lateral when the Pentagonal faces of the regular Pentagonal Prism are bases.

• The lateral side faces of Pentagonal Prisms are known as lateral edges.

What is a Rectangular Pentagonal Prism?

• If a Prism has two congruent and parallel Pentagonal faces and five Rectangular faces which are perpendicular to the triangular ones of the Pentagonal Prism are known as Rectangular Pentagonal Prisms.

Surface Area of a Pentagonal Prism-

The Surface Area of a Prism can be defined as the sum of the Areas of the lateral faces of a Prism and its bases.

Length(a) = Apothem of the Pentagonal Prism

Length(b) = Base of the Pentagonal Prism

Length(h) = Height of the Pentagonal Prism

Let’s find the Surface Area of the Pentagonal Prism using the given information,

Volume of a Pentagonal Prism-

The Volume of a Pentagonal Prism can be defined as the amount of space inside the Pentagonal Prism.

In the diagram given above,

Length(a) = Apothem of the Pentagonal Prism

Length(b) = Base of the Pentagonal Prism

Length(h) = Height of the Pentagonal Prism

Let’s find the Volume of the Pentagonal Prism,

Questions on Pentagonal Prism-

1. What are Prism faces and define Pentagonal Prism faces?

The faces of a Pentagonal Prism can be defined as a flat side of a 3-dimensional object. There are seven Prism faces.

2. Calculate the Surface Area and the Volume of the Pentagonal Prism with the given measurements as apothem length equal to 4 cm, base length equal to 8 cm, and height equal to 10 cm?

Let’s list down the given information,

Apothem Length(a) of the Pentagonal Prism, a = 4 cm

Base length (b) of the Pentagonal Prism, b = 8 cm

Height (h)of the Pentagonal Prism, h = 10 cm

We know the formula for the Volume of a Pentagonal Prism,

Volume of the Pentagonal Prism = (5/2) ×a ×b ×h cubic units

= 5/2 × (4×8×10)

= 5/2 × (320)

= 5 × (160)

= 800 cubic units

Therefore, the Volume of the given Pentagonal Prism is 800 cm3.

We know the formula for the Surface Area of a Pentagonal Prism,

Surface Area of the Pentagonal Prism = 5×a ×b +5×b×h square units

= 5 (4×8) + 5 (8×10)

= 5(32) + 5(80)

= 160 + 400

= 560

Therefore, the Surface Area of the given Pentagonal Prism is 560 cm2.

3. Calculate the Volume of a Pentagonal Prism with the given measurements as apothem length equal to 10 cm, base length equal to 12 cm, and height equal to 16 cm?

Let’s list down the given information,

Apothem Length(a) of the Pentagonal Prism, a = 10cm

Base length (b) of the Pentagonal Prism, b = 12 cm

Height (h)of the Pentagonal Prism, h = 16 cm

We know the formula for the Volume of a Pentagonal Prism,

Volume of the Pentagonal Prism = (5/2) ×a ×b ×h cubic units

= 5/2 × (10×12×16)

= 5/2 × (1920)

= 5 × (960)

= 4,800 cubic units

Therefore, the Volume of the given Pentagonal Prism is 4,800 cm3 .