What Is a Pentagonal Pyramid Definition Properties Surface Area and Volume Formula
A pentagonal pyramid is a pyramid named after its pentagonal base upon which are set up five triangular faces that meet at a point known as the vertex. The regular pentagonal pyramid is considered to be one of the Johnson solid.
It has lateral faces that are of equilateral triangles and a base that is a regular pentagon. It implies that all the sides of the base of the pentagonal are equal, as are the angles between the sides.
A pentagonal pyramid can be observed as a lid of the “regular icosahedron (a convex polyhedron with 20 faces 30 edges and 12 vertices) and the remaining portion forms the gyroelongated pentagonal pyramid, J₁₁.
Pentagonal Pyramid Faces
The total number of pentagonal pyramid faces is 6. The 5 side faces of a pentagonal prism are triangles and the base of the pentagonal pyramid is pentagonal.
Pentagonal Prism Edges and Vertices
The total number of pentagonal prism edges and vertices (corner points) are 6 and 10 respectively.
Volume of Pentagonal Based Pyramid
The volume of the pyramid refers to the total space enclosed between its faces.
Consider a pentagonal pyramid
We know the base of this pyramid is pentagonal.
Let the edge length of the pentagonal pyramid be’ s’, the height of the pyramid be ‘h’, and the apothem length of the pyramid be ‘a’.
Thus, the volume of the pentagonal pyramid is given as:
Surface Area of Pentagonal Pyramid
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The surface area of a regular pentagonal pyramid can be calculated by using the equation with three different variables:
There are three different distance marks in the diagram.
The distance from the centre to the midpoint of the side is marked with ‘a’
The length of a side of the pentagonal base is marked with ‘b’
The slant height is marked with ‘s’
Accordingly, the surface area of a regular pentagonal pyramid is given as:
Pentagonal Prism
A pentagonal prism is a three-dimensional object that has two bases: top and bottom.
The 5 sides of a pentagonal prism are rectangular.
Try to sketch a pentagonal prism on a piece of paper using straight lines. Then imagine that it is stretching from a sheet of paper. The three-dimensional shape is formed known as a pentagonal prism.
Pentagonal Prism Faces
The total number of faces of the pentagonal prism is 7.
Pentagonal Prism Edges and Vertices
The total number of pentagonal prism edges and vertices are 15 and 10 respectively.
Types of Pentagonal Prism
Pentagonal prisms are of three types:
Regular Pentagonal Prism
Right Pentagonal Prism
Oblique Pentagonal Prism
Let's discuss each of them:
Regular Pentagonal Prism - A regular pentagonal prism is a prism whose all 5 sides are equal in length.
Right Pentagonal Prism - If all the pentagonal prism faces are congruent and parallel and rectangular faces are perpendicular to the pentagonal faces, it is known as a right pentagonal prism.
Oblique Pentagonal Prism - If the pentagonal prism faces are not exactly on top of each other i.e. when the rectangular faces are not perpendicular to the pentagonal faces, it is known as an oblique pentagonal prism.
Solved Example
1. Find the Volume of a Pentagonal Pyramid Given the Side Length 5 cm, Apothem Length of 2 cm, and a Height of 9 cm.
Given,
s = 5 cm
a = 2 cm
h = 9 cm
Using the formula:
V = 5/6 x a x s x h
= 5/6 x 2 x 5 x 9
= 5/6 x 90
= 5 x 15
= 75 cm
Therefore, the volume of a pentagonal pyramid is 75 cm.
2. Find the Total Surface Area of a Pentagonal Pyramid Given the Side Length 9 cm, Apothem Length of 6 cm, and a Slant Height of 12 cm.
Solution:
Given,
s = 9 cm
a = 6 cm
l = 12 cm
The perimeter of the pentagonal pyramid is the sum of all its sides.
p = 5(9) = 45 cm
Lateral Surface Area of Pentagonal Pyramid = 1/2 x pl
= 1/2 x 45 x 12
= 270 cm²
Pentagonal Pyramid Base Area = 1/2 x perimeter x apothem
= 1/2 x 45 x 6
= 135 cm²
The formula for the total surface area of pentagonal pyramid = Lateral Surface Area of Pyramid + Base Area
TSA = Lateral Surface Area + Base Area
= 270 + 135
= 405 cm² .
FAQs on Pentagonal Pyramids Explained with Formulas and Properties
1. What is a pentagonal pyramid?
A pentagonal pyramid is a three-dimensional solid with a pentagon as its base and five triangular faces that meet at a single point called the apex. It is a type of pyramid in solid geometry.
- Base: 1 pentagon
- Faces: 5 triangular faces + 1 pentagonal base = 6 faces
- Edges: 10 edges
- Vertices: 6 vertices
2. How many faces, edges, and vertices does a pentagonal pyramid have?
A pentagonal pyramid has 6 faces, 10 edges, and 6 vertices.
- Faces: 5 triangular faces + 1 pentagonal base
- Edges: 5 base edges + 5 lateral edges = 10
- Vertices: 5 base vertices + 1 apex = 6
3. What is the formula for the volume of a pentagonal pyramid?
The volume of a pentagonal pyramid is given by the formula V = (1/3) × B × h, where B is the area of the base and h is the height.
- B = area of the pentagonal base
- h = perpendicular height from apex to base
4. How do you find the surface area of a pentagonal pyramid?
The surface area of a pentagonal pyramid is the sum of the base area and the areas of its five triangular faces.
- Total Surface Area = Base Area + Lateral Area
- Lateral Area = (1/2) × Perimeter of base × Slant height
5. What is a regular pentagonal pyramid?
A regular pentagonal pyramid is a pyramid with a regular pentagon as its base and equal triangular faces. In this solid:
- All base sides are equal.
- All base angles are equal.
- All lateral edges are equal in length.
6. How do you calculate the base area of a regular pentagonal pyramid?
The base area of a regular pentagonal pyramid is found using the formula A = (1/4) √(5(5 + 2√5)) s², where s is the side length.
- s = side length of the pentagon
7. What is the difference between a pentagonal prism and a pentagonal pyramid?
The main difference is that a pentagonal prism has two parallel pentagonal bases, while a pentagonal pyramid has only one base and an apex.
- Pentagonal prism: 2 pentagonal bases, rectangular faces
- Pentagonal pyramid: 1 pentagonal base, 5 triangular faces
- Volume formula (pyramid): (1/3)Bh
- Volume formula (prism): Bh
8. How do you find the slant height of a regular pentagonal pyramid?
The slant height of a regular pentagonal pyramid is found using the Pythagorean theorem.
- l² = h² + a²
- h = vertical height
- a = apothem of the base
9. Can you give an example of finding the volume of a pentagonal pyramid?
Yes, the volume is calculated using V = (1/3)Bh.
- Suppose base area B = 60 cm²
- Height h = 12 cm
10. What are the real-life applications of a pentagonal pyramid?
A pentagonal pyramid appears in architecture, design, and geometric modeling.
- Architectural roofs and towers
- Decorative structures and monuments
- 3D geometry problems in mathematics
