What Are the Surface Area and Volume of a Pentagonal Pyramid?
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 Pyramid: Properties, Formulas & Uses
1. What is a pentagonal pyramid?
A pentagonal pyramid is a three-dimensional shape that has a pentagon as its base and five triangular faces that rise to meet at a single point called the apex. It is a type of polyhedron, specifically a pyramid with a five-sided base.
2. How many faces, vertices, and edges does a pentagonal pyramid have?
A pentagonal pyramid has a specific number of faces, vertices, and edges. The count is as follows:
- Faces: It has 6 faces in total – one pentagonal base and five triangular side faces.
- Vertices: It has 6 vertices (corners) – five vertices form the base, and one is the apex where the triangular faces meet.
- Edges: It has 10 edges – five edges form the base, and five lateral edges connect the base vertices to the apex.
3. What is the difference between a regular and an irregular pentagonal pyramid?
The main difference lies in the properties of their base and faces. A regular pentagonal pyramid has a regular pentagon as its base (all sides and angles are equal) and its lateral faces are congruent isosceles triangles. In contrast, an irregular pentagonal pyramid has an irregular pentagon as its base, meaning the sides and angles of the base are not all equal, which can result in triangular faces of different shapes and sizes.
4. What is the formula for calculating the volume of a pentagonal pyramid?
The volume of a pentagonal pyramid is calculated by finding one-third of the product of its base area and its height. The formula is: Volume (V) = (1/3) × Base Area × Height (h). Here, 'Base Area' is the area of the pentagonal base, and 'h' is the perpendicular distance from the base to the apex.
5. How is the total surface area of a pentagonal pyramid calculated?
The total surface area (TSA) of a pentagonal pyramid is the sum of the area of its base and the area of all its five triangular faces. The calculation involves two parts:
- Base Area: The area of the pentagon at the bottom.
- Lateral Surface Area: The combined area of the five triangular faces.
Therefore, the formula is: TSA = Area of Pentagonal Base + (5 × Area of one Triangular Face). For a regular pyramid, this simplifies to TSA = Base Area + (1/2) × Perimeter of Base × Slant Height.
6. What does the 'net' of a pentagonal pyramid represent and why is it useful?
The net of a pentagonal pyramid is a two-dimensional pattern that can be folded to create the three-dimensional pyramid. It consists of one pentagon (the base) with five triangles attached to its sides. The net is extremely useful for understanding the shape's construction and for calculating its surface area, as it lays out all the faces flat, making it easier to measure and sum their individual areas.
7. How does a pentagonal pyramid differ from a pentagonal prism?
While both are 3D shapes with a pentagonal component, they are fundamentally different. The key differences are:
- Bases: A pyramid has one pentagonal base and an apex, whereas a prism has two parallel, congruent pentagonal bases.
- Lateral Faces: A pyramid's lateral faces are triangles that meet at the apex. A prism's lateral faces are rectangles that connect the corresponding sides of the two bases.
- Vertices: A pentagonal pyramid has 6 vertices, while a pentagonal prism has 10 vertices.
8. Can we find examples of pentagonal pyramids in real life?
Yes, although less common than square-based pyramids, examples can be found. Some modern architectural designs use pentagonal pyramid shapes for roofs, skylights, or decorative structures. They can also be seen in the design of certain crystals, monuments, or even in smaller objects like dice for games (d10s can be formed by joining two pentagonal pyramids at their bases) and decorative paperweights.
