What is a Triangular Prism?
A triangular prism is a three-dimensional (3D) shape with two identical triangle-shaped faces joined by three rectangular faces. The bases are the triangular faces, and the lateral faces are the rectangular faces. The top and bottom (faces) of the prism are other names for the bases.
The bases and sides of the triangular prism are either congruent, or oblique. The prism’s edges connect the corresponding sides. The edges of the two equilateral triangles that make up the prism’s bases are parallel to one another. To understand the structure of a triangular prism we can refer to the net of the prism as shown in the image.
Properties
- A triangular prism has 5 faces, 9 edges, and 6 vertices.
- It has two triangle-shaped faces and three rectangular faces that make a polyhedron.
- The triangular prism is said to be semiregular if the triangular bases are equilateral and the other faces are squares rather than rectangles.
- A triangle is the shape of any cross-section of a triangular prism.
Right Triangular Prism
In a right triangular prism the triangular faces are perpendicular to the lateral rectangular faces as the angle formed by the intersection of the triangular and rectangular faces is 90 degrees. A right triangular prism has six vertices, nine edges, and five faces.
What do we understand by the Volume of a Triangular Prism?
The volume of a triangular prism V with a base length b, height of the triangular base h, and the distance between the triangular bases or the length of the prism l, is calculated using the formula:
V = Base area \(\times\) length of the prism = \(\frac{1}{2}~\times~(b~\times~h)~\times~l\)
What do we understand by the Surface Area of a Triangular Prism?
The surface area of a triangular prism is the space that it takes up based on all its faces. It represents the total area of all the faces of the prism. Thus, the following formula is used to determine surface area:
Surface Area = Area of bottom + Area of front + Area of back + Area of a side + Area of a side
Triangular Prism Net
The net of a triangular prism is a pattern that appears when the surface of the prism is opened, made flat, and arranged so that each face can be seen clearly. The net of a three dimensional shape has only two dimensions. A triangular prism can be created by folding this net. It demonstrates the rectangular-shaped lateral faces and the bases being triangle-shaped. The triangles and rectangles can be seen clearly in the net of a triangular prism in the image:
Interesting Facts
- Volume = Base Area \(\times\) Prism Height
- Number of faces = 5
- Number of edges = 9
- Number of vertices = 6
- Base shape of the prism is triangular.
- Surface Area = (Area of bottom + Area of front + Area of back + Area of a side + Area of a side)
- Rectangles or squares are the shapes of sides.
Solved Triangular Prism Examples
Example 1: Find the volume of the triangular prism with base, 6 cm, height, 5 cm and length, 10 cm.
Solution:
Volume of a triangular prism = \(\frac{1}{2}~\times~(b~\times~h)~\times~l\) [Formula for the volume of a triangular prism]
V = \(\frac{1}{2}~\times~(6~\times~5)~\times~10\) [Substitute 6 for b, 5 for h, and 10 for l]
Volume (V) = \(150~cm^3\) [Evaluate]
Hence, the volume of the triangular prism is \(150~cm^3\).
Example 2: The triangular end of a triangular prism has a base of 5 m and height 6 m. If the length of each side is 10 m and the width of each side is 5 m then determine the surface area of the prism.
Solutions: We have,
Base length of triangular end, b = 5 m
Height of the base, h = 6 m
Length of each side, l = 10 m
Width of each side, w = 5 m
Surface Area = (Area of bottom + Area of front + Area of back + Area of a side + Area of a side) [Surface area of triangular prism formula]
Area of front = Area of back = Area of triangular end
= \(\frac{1}{2}~\times~b~\times~h\) [Area of triangular end formula]
= \(\frac{1}{2}~\times~5~\times~6\) [Substitute 5 for b, and 6 for h]
= \(5~\times~3\) [Simplify]
= \(15~m^2\) [Multiply]
Area of each side = Area of bottom
= \(l~\times~w\) [Area of rectangular side formula]
= \(10~\times~5\) [Substitute 10 for l and 5 for w]
= \(50~m^2\) [Multiply]
Surface Area = (Area of bottom + Area of front + Area of back + Area of a side + Area of a side)
= \(50~m^2\) + \(15~m^2\) + \(15~m^2\) + \(50~m^2\) + \(50~m^2\)
= \(180~m^2\)
Hence, the surface area of the triangular prism is \(180~m^2\).
Example 3: A glass triangular prism used in the laboratories has a base area of 20 square units, its length is 10 units. Find the volume of the prism.
Solution:
We have,
Base area = 20 square units
Length = 10 units
Volume of triangular prism V = Base area \(\times~\) length of the prism
= \(20~\times~10\) [Substitute 20 for base area and 10 for length of the prism]
Volume of triangular prism = \(200~\text{units}^3\)
Therefore, the volume of the given prism is \(200~\text{units}^3\).
A rectangular prism has bases that are rectangle-shaped, while a triangular prism has bases that are triangle-shaped.
A triangular prism has 9 vertices, 6 edges, 3 rectangular faces, and 2 triangular faces.
A triangular prism has five faces, of which two form the top and bottom bases of the prism and three are lateral rectangular faces.
Volume = base area \(\times\) prism length is the formula for determining the volume of a triangular prism. As a result, the volume of a triangular prism equals its length times its base area.
Volume = base area \(\times\) prism length. The volume of a triangular prism does not depend on the angle of the triangular base, as can be seen from the relationship above.