Pentagonal Pyramid
A pentagonal pyramid is a type of pyramid where the base of the pyramid is in the shape of a pentagon and has its five lateral sides or faces shaped like triangles. In geometry, a pyramid is a 3D shape with a polygon-shaped base, and the sides in the shape of a triangle all combined by the apex. Let us learn the various properties of a pentagonal pyramid.
Definition of Pentagonal Pyramid
A pentagonal pyramid is a 3D-shaped object which has the base as a pentagon upon which are five triangle-shaped lateral sides that meet at the apex. This apex is what forms the pentagonal pyramid together as it combines the triangular sides and the pentagonal base together. A regular pentagonal pyramid's base is a regular pentagon with the lateral faces shaped like equilateral triangles. The image below shows the shape of a pentagonal pyramid.
Properties of a Pentagonal Pyramid
A pentagonal pyramid has the general properties of a pyramid and has some special properties with respect to its pentagonal base. Hence, the properties of a pentagonal pyramid are:
- A regular pentagonal pyramid has 6 faces(5 equilateral triangles as the lateral sides and a regular pentagon base).
- There are 6 vertices in a pentagonal pyramid along with 10 edges.
- There are 6 faces in total in any pentagonal pyramid.
- A pentagonal pyramid can also have isosceles triangles as its lateral sides.
Volume of a Pentagonal Pyramid
The volume of a pentagonal pyramid is calculated to measure the number of cubic units occupied by the pyramid. As we already know the area of a pentagon is (5/2) × side × apothem of the pentagon, we get the formula of the volume of a pentagonal pyramid as:
V= (1/3) Area of the pentagonal base × height
= (1/3) × [(5/2) × side of the base × apothem] × height
Volume of a pentagonal pyramid = (5/6) abh
Where,
- a is the apothem length of the pentagonal pyramid.
- b is the base length.
- h is the height of the pentagonal pyramid.
Surface Area of a Pentagonal Pyramid
The surface area of a pentagonal pyramid is the measure of the area of all the faces of the pyramid. Every pyramid has a different formula depending on the base of the pyramid i.e. the shape of the polygon. In the case of the pentagonal pyramid, the formula for surface area is:
Surface Area of a Pentagonal Pyramid = 5/2 × b(a + s)
Where,
- a is the apothem length of the pentagonal pyramid
- b is the base length of the pentagonal pyramid
- s is the slant height of the pyramid
Pentagonal Pyramid Net
The net of the pentagonal pyramid is the base of the pyramid-shaped as a pentagon while the sides or faces of the pyramid are shaped like triangles. The method to draw the net of a pentagonal pyramid is first let us begin by drawing the base of the pyramid i.e. a pentagon. Next, along the 5 sides of the pentagon, we draw 5 triangles as the sides or faces or lateral faces of the pentagonal pyramid. When we get all the triangles together and connect them at the vertex, we get the apex or the top of the pyramid that forms the entire pentagonal pyramid. Below is an image for your reference.
Related Topics
Listed below are a few interesting topics related to the concept of pentagonal pyramids:
Examples on Pentagonal Pyramid
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Example 2: From the following image, identify the pentagonal pyramid.
Solution: All four images are the different types of pyramids that we see in math. The different types are - square pyramid, triangular pyramid, pentagonal pyramid, and hexagonal pyramid. In the above image, we can see all four types of pyramids. Image c is a Pentagonal pyramid because it has a pentagonal base and triangular lateral sides all meeting at the common apex. While a is a square, b is a triangular, and d is a hexagonal pyramid.
FAQs on Pentagonal Pyramid
What is a Pentagonal Pyramid?
A pentagonal pyramid is a 3D shape with the base as the shape of a pentagon along with the faces shaped like a triangle. The apex or the top of the pentagonal pyramid forms the pyramid by joining all the 5 triangular faces along with the base to the vertex of the triangle which becomes the apex. A pentagonal pyramid has 6 vertices, 6 faces, and 10 edges.
What is the Formula to Find the Volume of a Pentagonal Pyramid?
The formula to find the volume of a pentagonal pyramid is:
Volume of a pentagonal pyramid = (5/6)abh
Where a is the apothem length of the pentagonal pyramid, b is the base length, and h is the height of the pentagonal pyramid
What is the Formula to Find the Surface Area of a Pentagonal Pyramid?
The formula to find the surface area of a pentagonal pyramid is:
Surface Area of a Pentagonal Pyramid = 5/2 × b(a + s)
Where a is the apothem length of the pentagonal pyramid, b is the base length of the pentagonal pyramid, and s is the slant height of the pyramid
What is the Net of a Pentagonal Pyramid?
The net of the pentagonal pyramid is the base is the shape of a pentagon and along the base, there are 5 triangles that become the faces of the pyramid once it is all combined by the vertex of the triangle which becomes the apex or top of the pyramid. When the pentagonal pyramid is flattened, the shapes are seen clearly.
What Type of Triangles Makes a Pentagonal Pyramid?
A regular pentagonal triangle usually is made up of equilateral triangles that are the faces or the lateral area of the pyramid. But, the triangles can be different as well such as it could be an isosceles triangle or a scalene triangle, in an irregular pentagonal pyramid.
What are the Properties of a Pentagonal Pyramid?
Every pyramid has its own properties depending on its polygon base. The properties of a pentagonal pyramid are:
- It has 5 triangles as the faces and one pentagon base
- It has 6 vertices along with 10 edges
- It has 6 faces in total
- A pentagonal pyramid can also have isosceles triangles as its faces or sides.
How Many Triangular Faces are there in a Pentagonal Pyramid?
There are five triangular faces in a pentagonal pyramid that are shaped as equilateral triangles.