Pentagonal Pyramid

Pentagonal pyramid is a type of pyramid with a pentagon base. It is a 3D geometric structure that has 5 triangular sides that meet at a single closing point. In this article, you will learn about the basic meaning of pyramid and pentagonal pyramid, and various properties of pentagonal pyramid.

In this article, we will learn about, Pentagonal Pyramid Definition, Formulas of Pentagonal Pyramids, Examples, and others in detail.

What is a Pyramid?

A pyramid is a three-dimensional geometric shape that is characterized by a polygonal base connected to a single point, known as the apex or vertex. The base can be any polygon, such as a square, rectangle, or in the case of a pentagonal pyramid, a pentagon. The sides or faces of the pyramid are triangles, connecting each vertex of the base to the apex. Pyramids are defined by their base shape, and the height is the distance from the base to the apex.

Pentagonal Pyramid Definition

A pentagonal pyramid is a 3-dimensional object with a pentagon-shaped base and five triangular sides meeting at the top, forming an apex. The apex brings together the triangular sides and the pentagonal base. In a regular pentagonal pyramid, the base is a regular pentagon, and the triangular sides are equilateral triangles. The image added below shows the pentagonal pyramid as,

Properties of Pentagonal Pyramid

A pentagonal pyramid have all the properties general pyramid with some specific features because,

Faces: A regular pentagonal pyramid possesses 6 faces and 5 of them are equilateral triangles forming the lateral sides, and the base is a regular pentagon.
Vertices: The pentagonal pyramid has 6 vertices (corners) where the triangular sides and the base meet.
Edges: With 5 edges connecting the apex to each vertex of the base and 5 more connecting the vertices around the base, there are 10 edges in total.
While a regular pentagonal pyramid has equilateral triangles as lateral sides, it can also exhibit isosceles triangles, adding flexibility to its geometric characteristics.

Pentagonal Pyramid Faces Edges Vertices

A pentagonal pyramid has 6 Faces, 6 Vertices and 5 Edges

Types of Pentagonal Pyramid

There are various types of pentagonal pyramids, each having its own distinct characteristics that are,

Regular Pentagonal Pyramid
Irregular Pentagonal Pyramid
Right Pentagonal Pyramid
Oblique Pentagonal Pyramid

Regular Pentagonal Pyramid

All faces of this type are identical, and all edges have the same length.
It features a regular pentagon as its base and equilateral triangles as its lateral faces.
Known for its exceptional symmetry and uniformity.

Irregular Pentagonal Pyramid

This type deviates from regularity seen in the previous type.
It has a pentagonal base with sides of different lengths, resulting in non-uniform lateral faces.
Can exhibit a distinct and unique appearance.

Right Pentagonal Pyramid

In a right pentagonal pyramid, the apex is directly above the center of the pentagonal base.
It stands upright, creating a symmetrical shape.
A pyramid with a perfectly centered point.

Oblique Pentagonal Pyramid

In contrast to the right pentagonal pyramid, an oblique pentagonal pyramid has an apex positioned off-center.
This results in an inclined or tilted appearance.
A pyramid leaning to one side.

Pentagonal Pyramid Formula

Various pentagonal pyramid formulas are,

Volume of Pentagonal Pyramid
Surface Area of Pentagonal Pyramid

Volume of a Pentagonal Pyramid

The volume of a pentagonal pyramid tells us how much space it takes up. To find it, you use the formula,

Volume of a Pentagonal Pyramid (V) = (5/6) × Base Length × Apothem Length × Height

The base length (b) and the apothem length (a) are measured from the center to the midpoint of a side in the pentagon forming the base. The height (h) is the distance from the base to the apex (top) of the pyramid.

Surface Area of a Pentagonal Pyramid

Surface area of pentagonal pyramid is the sum of all the faces of the pentagonal pyramid and the formula for the same is,

Surface Area of a Pentagonal Pyramid = (5/2) × Base Length × (Apothem Length + Slant Height)

The slant height (s) is the distance from the apex to the midpoint of any side of the base. Both formulas help understand and calculate the size of a pentagonal pyramid.

Net of Pentagonal Pyramid

To create the net of a pentagonal pyramid, start by drawing a pentagon as the base. Extend triangles from each side of the pentagon to represent the faces of the pyramid. These triangles come together at the top vertex, forming the apex and completing the pentagonal pyramid. The pentagon serves as the base, while the triangles act as the sides or lateral faces. When you connect all the triangles, you form the entire pyramid. This visual representation helps understand how the flat, unfolded shape relates to the three-dimensional pyramid. The Pentagonal Pyramid Net is shown in the image added below,

Applications of Pentagonal Pyramid

Pentagonal pyramids, with their distinctive shape and characteristics, have practical uses in various fields. Some examples are given below

Architecture

Architects draw inspiration from pentagonal pyramids when designing iconic structures.
They are utilized in grand entranceways, modern skyscrapers, or as decorative elements in buildings to enhance architectural designs aesthetically.

Education

Pentagonal pyramids serve as effective teaching tools for geometry lessons.
They assist students in comprehending the concept of polyhedra, exploring the relationship between faces, edges, and vertices, and providing a tangible representation of three-dimensional geometry.

Art and Design

Artists and designers often find creative inspiration in the geometric elegance of pentagonal pyramids.
The shape is incorporated into sculptures, jewelry, and various creative works, adding sophistication and intrigue to artistic endeavors.

Research

Pentagonal pyramids, along with other polyhedra, serve as subjects of mathematical research.
Mathematicians study their properties, understand mathematical relationships, and gain new insights into the realm of geometry.

Related Resources

Examples of Pentagonal Pyramid

Example 1: In a pentagonal pyramid, if the base length is 9 cm, the slant height is 12 cm, and the apothem length is 8 cm, what is the volume of the pyramid?

Solution:

Given,

Base Length = 9 cm

Slant Height = 12 cm

Apothem Length = 8 cm

V = ⅓ × Base Area × Height

V = ⅓ × (5/4 × Base Length × Apothem Length) × Slant Height

V = ⅓ × (5/4 × 9 × 8) × 12

V = ⅓ (5/4 × 72) × 12

V = ⅓ × (360/4) × 12

V = ⅓ × 90 × 12

V = ⅓ × 1080 = 360 cm³

Example 2: If the base length of a pentagonal pyramid is 9 cm, slant height is 12 cm, and apothem length is 8 cm, what is its total surface area?

Solution:

Given,

Base Length = 9 cm

Slant Height = 12 cm

Apothem Length = 8 cm

A = Base Area + Lateral Area

A = 5/4 × Base Length × Apothem Length + 5/2 × Base Length × Slant Height

A = 5/4 × [ 9 × 8] + 5/2 × [9 × 12]

A = 5/4 × 72 + 5/2 × 108

A = 360/4 + 540/2

A = 90 + 270 = 360 cm²

Example 3: In a pentagonal pyramid, if the base length is 7 cm, slant height is 10 cm, and apothem length is 6 cm, what is the volume of the pyramid?

Solution:

Given,

Base Length = 7 cm

Slant Height = 10 cm

Apothem Length = 6 cm

V = ⅓ × Base Area × Height

V = ⅓ × (5/4 × Base Length × Apothem Length) × Slant Height

V = ⅓ × (5/4 × 7 × 6) × 10

V = ⅓ × (5/4 × 42) × 10

V = ⅓ × 105 × 10

V = 1050/3 = 350 cm³

Example 4: If the base length of a pentagonal pyramid is 6 cm, slant height is 9 cm, and apothem length is 5 cm, what is its total surface area?

Solution:

Given,

Base Length = 6 cm

Slant Height = 9 cm

Apothem Length = 5 cm

A = Base Area + Lateral Area

A = 5/4 × Base Length × Apothem Length + 5/2 × Base Length × Slant Height

A = 5/4 × (6 × 5) + 5/2 (6 × 9)

A = 5/4 × 30 + 5/2 × 54

A = 150/4 + 270/2

A = 37.5 + 135 = 172.5 cm²

Example 5: Consider a pentagonal pyramid with a base length of 8 units, a slant height of 13 units, and an apothem length of 7 units. Calculate both the volume and the total surface area of the pyramid.

Solution:

Given,

Base Length = 8 units

Slant Height = 13 units

Apothem Length = 8 units

V = 1/3×Base Area×Height

V = 1/3×( 5/4×Base Length×Apothem Length)×Slant Height

V = 1/3×( 5/4×8×7)×13

V = 1/3×( 5/2 ×56)×13

V = 1/3×140×13

V = 3640/3 = 1213.33 units³

A = Base Area + Lateral Area

A = 5/4×Base Length×Apothem Length + 5/2×Base Length×Slant Height

A= 5/4 ×(8×7) + 5/2×(8×13) = 5/4×56+ 5/2 ×104 = 280/4 + 520/2 = 70+260

A = 330 units²

Practice Problem on Pentagonal Pyramid

Problem 1: In a pentagonal pyramid, the base length is 6 cm, the slant height is 8 cm, and the apothem length is 4 cm. Calculate the volume of the pyramid.

Problem 2: If the base length of a pentagonal pyramid is 10 cm, the slant height is 15 cm, and the apothem length is 7 cm, find the total surface area of the pyramid.

Problem 3: Consider a pentagonal pyramid with a base length of 12 units, a slant height of 18 units, and an apothem length of 9 units. Calculate both the volume and the total surface area of the pyramid.

Problem 4: In a hexagonal pyramid, the base length is 5 cm, the slant height is 9 cm, and the apothem length is 3 cm. Determine the volume of the pyramid.

Problem 5: If the base length of a hexagonal pyramid is 12 cm, the slant height is 14 cm, and the apothem length is 8 cm, find the total surface area of the pyramid.

Pentagonal Pyramid-FAQs

1. What is a Pentagonal Pyramid?

A pentagonal pyramid is a three-dimensional geometric shape with a pentagon as its base and five triangular sides that meet at a single point, forming the apex or vertex.

2. How is the Volume of a Pentagonal Pyramid Calculated?

The volume of a pentagonal pyramid is determined using the formula

V = (5/6) × Base Length × Apothem Length × Height

3. What is the Formula for Pentagonal Pyramid?

The Formula for Pentagonal Pyramid are,

Volume of Pentagonal Pyramid (V) = (1/3) × Area of Pentagonal Base × Height

Surface Area of a Pentagonal Pyramid (S) = (5/2) × b(a+s)