Octagonal Prism

Octagonal Prism is a 3D shape with two octagon-shaped ends connected by eight rectangular sides. Thus an octagonal has 10 faces out of which 8 are rectangular and 2 are octagonal. In this article, we’ll talk about what an octagon prism is, its surface area and volume formulas, and real-life examples where we see octagonal objects.

Prism Meaning

Prism are three dimemensional geometrical shape whose two ends are similar. The two identical surfaces are generally on the opposite side of each other and separated by a height. Based on the shape of these two identical faces, prism are classified into various types such as Triangular Prism, Rectangular Prism etc.

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One of such prism is Octagonal Prism which is discussed below:

What is an Octagonal Prism?

An octagonal prism is a 3D shape with two octagonal bases and eight rectangular sides. Think of it like a can of soda, but instead of a circle, the top and bottom are octagons, and the sides are rectangles. It has 24 edges, which are the lines where its faces meet. There are also 16 vertices, which are the corner points. Altogether, the octagonal prism has 10 faces, combining the surfaces of its octagonal bases and rectangular sides.

Cross-Section of Octagonal Prism

A cross-section of a prism is like a “slice” or a view you get when you cut through the prism. Imagine if you had a loaf of bread, and you cut it to see what the inside looks like, that slice you see is a cross-section. In the case of a prism, it’s the shape you get when you cut straight through it. So, if you were to slice through an octagonal prism, the cross-section would be an octagon, just like one of its bases. It helps understand the prism’s shape and how it looks from different angles.

Octagonal Prism Faces, Edges And Vertices

An octagonal prism possesses 10 faces, consisting of 8 rectangular lateral faces and 2 octagonal bases. It also has a total of 18 edges and 10 vertices. The vertices of the octagonal prism coincide with the vertices of the two octagonal bases, connected by lines that form rectangles along the prism’s sides. The edges of the octagonal prism encompass the 16 edges formed by the two octagonal bases (8 + 8), along with the additional 2 edges created by the sides connecting the bases. This description captures the essential geometric features of an octagonal prism in terms of its faces, edges, and vertices.

Property	Numbers
Faces	10
Edges	18
Vertices	10

Net For Octagonal Prism

The net of an octagonal prism is like a plan that shows the unfolded surface of the prism. Picture opening it up, flattening it, and laying out the surface to see all its sides clearly. The resulting shape is a flat, two-dimensional figure known as the net. If you fold this net, you can recreate the original octagonal prism. The net displays that the prism has octagonal bases and rectangular lateral faces. Simply put, it’s a visual guide that illustrates how to put together the prism from a flat, folded shape.

Real-Life Examples of Octagon Prism

Real-life examples of an octagonal prism can be found in various places:

• Building Construction: Some architectural structures may have columns or pillars in the shape of octagonal prisms, providing both aesthetic appeal and structural support.
• Lamp Posts: Street lamp posts are often designed with an octagonal prism shape, contributing to the overall design while maintaining stability.
• Decorative Columns: Decorative columns in buildings, especially in classical architecture, may take on the form of octagonal prisms, adding a touch of elegance.
• Candle Holders: Certain candle holders or decorative items can feature an octagonal prism shape, combining functionality with an appealing design.

Surface Area of an Octagonal Prism

The Surface Area of Octagonal prism has two parts:

• Lateral Surface Area
• Total Surface Area

Lateral Surface Area (LSA) Formula

• The LSA is the combined area of all the sides (excluding the top and bottom faces) of the octagonal prism.
• To find it, you multiply the perimeter of the base (the total length of all the sides of the octagon) by the height of the prism.

LSA = Perimeter of Base × Height

Total Surface Area (TSA) Formula

• The TSA includes both the lateral surface area and the area of the top and bottom faces.
• To calculate it, you add the LSA to the area of one of the octagonal bases.

TSA = LSA + Base Area

• The Base Area is half of the product of the perimeter of the base and the apothem (the distance from the center of the octagon to the midpoint of one of its sides). Formula for Base Area:

Base Area = 1/2 × Perimeter of Base × Apothem

Volume of an Octagonal Prism

Volume of Octagon Prism means calculating how much space the octagonal prism occupies by considering the area of its base and its height. The apothem is the distance from the center of the octagon to the midpoint of one of its sides.

For an octagonal prism, you find its volume by multiplying the area of the base (octagon) by the height of the prism.

Volume = Base Area × Height

The Base Area for an octagon is found using the formula:

Base Area = Perimeter of Base × Apothem

Therefore, the complete formula for the volume of an octagonal prism is:

Volume=(1/2 × Perimeter of Base × Apothem) × Height

Solved Examples of Octagon Prism

Example 1: Calculate the volume of an octagonal prism with a height of 8 units, a base perimeter of 32 units, and an apothem length of 5 units.

Solution:

The formula for the volume of an octagonal prism is given by:

Volume = 1/2 × Perimeter of Base × Apothem × Height

Substitute the given values:

Volume = 1/2 × 32 units × 5 units × 8 units

Volume = 640 cubic units

the volume of the octagonal prism is 640 cubic units

Example 2: Determine the total surface area of an octagonal prism with a height of 10 meters and a base perimeter of 40 meters. The apothem length is 7 meters.

Solution:

The total surface area (TSA) of an octagonal prism is given by the sum of the lateral surface area (LSA) and the base area. The formulas are:

LSA = Perimeter of Base × Height

Base Area= 1/2 × Perimeter of Base × Apothem

Substitute the given values:

LSA = 40m × 10 m = 400m²

Base Area = 1/2 × 40m × 7m = 140m²

Now,

TSA = LSA + Base Area = 400m² + 140m²= 540m²

the total surface area of the octagonal prism is 540 square meters.

Practice Questions of Octagon Prism

Q1. What is the lateral surface area of an octagonal prism with a base perimeter of 40 units and a height of 8 units?

Q2. Determine the volume of an octagonal prism with an apothem of 6 units, a base perimeter of 32 units, and a height of 10 units.

Q3. A decorative column has an octagonal prism shape with a base perimeter of 24 meters and a height of 15 meters. Calculate its total surface area.

Q4. How many edges does an octagonal prism have if each of its octagonal bases has 8 edges, and there are 8 rectangular sides?

Octagonal Prism – FAQs

1. What is an Octagonal Prism?

An octagonal prism is a three-dimensional geometric shape with two octagonal bases connected by eight rectangular sides. It resembles a can with octagonal ends and rectangular sides.

2. How many Edges does an Octagonal Prism have?

An octagonal prism has 24 edges. These edges are the lines where the faces of the prism, including the octagonal bases and rectangular sides, meet.

3. What is the Cross-Section of an Octagonal Prism?

The cross-section of an octagonal prism is an octagon. When you cut through the prism, the shape you see is similar to one of its bases, which is an octagon.

4. What are the Vertices of an Octagonal Prism?

An octagonal prism has 16 vertices. These vertices represent the corner points where the edges of the prism meet.

5. How do you Calculate the Surface Area of an Octagonal Prism?

The surface area is calculated by finding the sum of the lateral surface area (LSA) and the base area. LSA is the perimeter of the base multiplied by the height, and the base area is half the product of the perimeter of the base and the apothem.

6. What are the Properties of an Octagonal Prism?

An octagonal prism has 24 edges, 16 vertices (corner points), and 10 faces. Its faces include the two octagonal bases and eight rectangular sides.

7. How do you Calculate the Volume of an Octagonal Prism?

The volume of an octagonal prism is found by multiplying the area of its octagonal base by its height. The formula is Volume = (1/2) × Perimeter of Base × Apothem × Height.