Triangular Prism

Triangular Prism is a three-dimensional geometric shape with two identical triangular faces connected by three rectangular faces. It is one of the classifications of prism. It is named a triangular prism because it has a triangle across its cross-section.

This article covers the meaning of prism and triangular prism, the properties of the prism, the formula of a triangular prism, and the net of a triangular prism. We will also see the types of triangular prism on the basis of uniformity and alignment and verify Euler’s rule for triangular prism.

What is Prism?

A prism is a specific type of polyhedron that features identical polygons at both its top and bottom. The remaining faces of a prism are referred to as lateral faces, and these faces share the same shape throughout their length.

They are often named according to the shape of their cross-sections. For instance, a triangular prism has a triangle cross-section a hexagonal prism resembles a metallic nut, and a rectangular prism has a similarity to a fish tank.

What is a Triangular Prism?

A triangular prism is a three-dimensional shape characterized by two identical triangular faces connected by three rectangular faces. These rectangular faces are called lateral faces, and the triangular faces are known as bases, which can also be referred to as the top and bottom faces of the prism.

To describe its dimensions, we use parameters such as the length of the prism l, the height of the triangular base h, and the length of the bottom edge of the triangular base b.

Triangular Prism

Charecterstics of Triangular Prism

Some of the key characteristics of triangular prism are:

• Faces: A triangular prism has five faces in total. Two of these faces are triangular, forming the bases of the prism, while the other three are rectangular or parallelogram-shaped, connecting the corresponding sides of the two triangles.

• Edges: A triangular prism has six edges. Each triangular base has three edges, and there are three additional edges connecting the corresponding vertices of the two bases.

• Vertices: There are six vertices (corner points) in a triangular prism. Each triangular base has three vertices, and there are three additional vertices where the edges meet.

• Symmetry: A triangular prism exhibits symmetry along its axis perpendicular to the triangular bases. This means that if you rotate the prism around this axis by 180 degrees, it will look the same.

• Cross-sectional Shapes: If you were to slice a triangular prism perpendicular to its length, the cross-sectional shape would always be a triangle.

Examples of Triangular Prism

Some examples of triangular prism include:

• Tents
• Cheese
• Glass Triangular Prism for practical

Types of Triangular Prism

The types of triangular prism are divided on the two basis;

• On the basis of uniformity (regular prism and irregular prism) and
• On the basis of alignment (right prism and oblique prism).

Let’s discuss these classification in detail.

Triangular Prism on the Basis of Uniformity

On the basis of uniformity, the triangular prism is divided into two:

• Regular triangular Prism
• Irregular triangular Prism

Regular Triangular Prism

A regular triangular prism is a three-dimensional shape where both triangular bases are regular triangles.

A regular triangle is a type of triangle where all sides are equal, and the angles between these sides measure 60°. Also, the lateral faces, or the sides, of the regular triangular prism take the form of rectangles.

Irregular Triangular Prism

An irregular triangular prism is a three-dimensional figure that deviates from this regularity.

In an irregular triangular prism, at least one of the triangular bases is not an equilateral triangle. This means that the sides of the base triangle in an irregular triangular prism may have different lengths, and the angles between these sides are not fixed at 60°.

Triangular Prism on the Basis of Alignment

On the basis of alignment, the triangular prism is divided into two:

• Right triangular Prism
• Oblique triangular Prism

Right Triangular Prism

A right triangular prism is a specific type of prism where the angle formed between the edges of the triangular bases and the edges of the rectangular faces is precisely 90°.

This means that the triangular bases meet the rectangular faces at right angles. All other properties of triangular prisms, such as the number of faces, edges, and vertices, remain the same for a right triangular prism.

Oblique Triangular Prism

An oblique triangular prism differs in that its lateral faces are not perpendicular to its bases. In this type of prism, each lateral face takes the shape of a parallelogram. This implies that the angles between the lateral faces and the bases are not necessarily 90 degrees.

In essence, an oblique triangular prism doesn’t have the strict right-angle alignment between its triangular ends and its rectangular sides. Instead, the lateral faces form parallelograms, allowing for more flexibility in the geometric configuration of the prism.

Other Types of Prism

• Rectangular Prism
• Pentagonal Prism
• Square Prism
• Hexagonal Prism

Triangular Prism Faces Edges Vertices

Parts of a Triangular Prism	Numbers
Face of a Triangular Prism	5
Edge of a Triangular Prism	9
Vertex of a Triangular Prism	6

Properties of Triangular Prism

A triangular prism is easily identifiable by its key characteristics. Here are the important properties explained in neutral language:

• Number of Faces: A triangular prism has five faces. These include two triangular bases and three rectangular sides.
• Number of Edges: It possesses nine edges, that representing the lines where the faces meet.
• Number of Vertices: A triangular prism has six vertices.
• Polyhedral Nature: It falls under the category of Polyhedra, that specifically characterized by two triangular bases and three rectangular sides.
• Base Shape: The base of a triangular prism is in the shape of a triangle.
• Side Shape: The sides are in the shape of rectangles, providing a consistent structure along the length of the prism.
• Equilateral Triangular Bases: The two triangular bases are equilateral triangles, meaning all sides of these triangles are of equal length.
• Cross-Section Shape: Any cross-section of triangular prism results in the shape of a triangle.
• Congruent Bases: The two triangular bases are identical to each other that implies their congruence

Triangular Prism Net

The net of a triangular prism is like a blueprint that unfolds the surface of the prism. By folding this net, you can recreate the original triangular prism.

The net illustrates that the prism has triangular bases and rectangular lateral faces. In simpler terms, it’s a visual guide that shows how the prism can be assembled from a flat, folded shape.

Triangular Prism Net

Surface Area of a Triangular Prism

The Surface Area of a Triangular Prism is divided into two parts Lateral Surface Area and Total Surface Area

Lateral Surface Area (LSA) of a Triangular Prism:

The lateral surface area (LSA) of a triangular prism is the total area of all its sides excluding the top and bottom faces. The formula to calculate the lateral surface area is given by:

Lateral Surface Area (LSA) = (s₁ + s₂ + h)L

Here, s₁, s₂, and s₃ are the lengths of the edges of the base triangle, and L is the length of the prism.

For a right triangular prism, the formula is:

Lateral Surface Area = (s₁ + s₂ + h)L

OR

Lateral Surface Area = Perimeter × Length

Here, (h) represents the height of the base triangle, (L) is the length of the prism, and s₁ and s₂ are the two edges of the base triangle.

Total Surface Area (TSA) of a Triangular Prism

The total surface area (TSA) of a triangular prism is found by adding the area of its lateral surface (the sides) and twice the area of one of its triangular bases. For a right triangular prism, where one of the bases is a right-angled triangle, the formula for the total surface area is given by:

Total Surface Area (TSA) = (b × h) + (s₁ + s₂ + s₃) L

Here, s₁, s₂, and s₃ are the edges of the triangular base, (h) is the height of the base triangle, (l) is the length of the prism, and (b) is the bottom edge of the base triangle.

For a right triangular prism specifically, the formula simplifies to:

Total Surface Area = (s₁ + s₂ + h) L + b × h

Where,

• b is the bottom edge of the base triangle.
• h is the height of the base triangle.
• L is the length of the prism.
• s₁ and s₂ represent the two edges of the base triangle.
• bh represents the combined area of the two triangular faces.
• (s₁ + s₂ + h) L represents the combined area of the three rectangular side faces.

This formula essentially accounts for the areas of all the faces (rectangular and triangular) of the prism, providing a comprehensive measure of its total surface area.

Volume of Triangular Prism

The volume of triangular prism refers to the amount of space it occupies in the three-dimensional space. The formula to compute the volume of triangular prism is expressed as:

Volume (V) = 1/2 × base edge × height of the triangle × length of the prism

Where,

• base edge b: This is the length of one of the edges forming the base triangle.
• height of the triangle h: It represents the perpendicular distance from the base to the opposite vertex, forming the triangle.
• length of the prism l: This indicates the overall length of the prism along its axis.

By using these values in the formula, one can calculate the volume of the triangular prism.

Euler’s Formula for Triangular Prism

Euler’s formula states that in any polyhedron, the sum of the number of faces (F) and vertices (V) is equal to two more than the number of edges (E).

Consider a triangular prism. Euler’s formula, which relates the number of faces F, vertices V, and edges E of a polyhedron, is given by:

F + V = E + 2

Now, for the triangular prism:

• The number of faces F is 5.
• The number of vertices V is 6.
• The number of edges E is 9.

Substituting these values into Euler’s formula:

5 + 6 = 9 + 2

This simplifies to:

11 = 11

The result confirms that Euler’s formula is true for the given triangular prism, validating the relationship between the number of faces, vertices, and edges.

Triangular Prism – Solved Examples

Example 1. Consider a triangular prism with a base edge of 4 cm, a height of the triangular base as 6 cm, and an overall length of the prism as 10 cm. Find the volume of triangular prism.

Solution:

Given:

• Base edge (b) = 4 cm
• Height of the triangular base (h) = 6 cm
• Length of the prism (l) = 10 cm

The formula for the volume (V) of a triangular prism is:

V= 1/2 × b × h × l

Substitute the given values into the formula:

V= 1/2 × 4cm × 6cm × 10cm

V=120cm³

∴ the volume of the triangular prism is 120cm³

Example 2. A triangular prism has a triangular base with sides measuring 8 cm, 15 cm, and 17 cm. The height of the triangular base is 10 cm, and the overall length of the prism is 12 cm. Calculate the surface area of triangular prism.

Solution:

Given:

• Sides of the triangular base (a, b, c) = 8 cm, 15 cm, 17 cm (this is a right-angled triangle)
• Height of the triangular base (h) = 10 cm
• Length of the prism (l) = 12 cm

The formula for the surface area (A) of a triangular prism is:

A=2 × area of base triangle + perimeter of base × height of prism

First, calculate the area of the base triangle using Heron’s formula:

s= (a + b + c)/2

Area= √[s × (s – a) × (s – b) × (s – c)]

s = (8 + 15 + 17) / 2 = 20

Area= √20 × (20 – 8) × (20 – 15) × (20 – 17)

Area= √20 × 12 × 5 × 3

= √3600

= 60cm²

Now, substitute the values into the surface area formula:

A= 2 × 60cm² +(8+15+17)cm × 10cm

A= 120cm² + 40cm × 10cm

A= 520cm²

∴ the surface area of the triangular prism is 520 cm²

Example 3. Consider a triangular prism with a base edge of 9 cm, a height of the triangular base as 16 cm, and an overall length of the prism as 20 cm. Find the volume of triangular prism.

Solution:

Given:

• Base edge (b) = 9 cm
• Height of the triangular base (h) = 16 cm
• Length of the prism (l) = 20 cm

The formula for the volume (V) of a triangular prism is:

V= 1/2 × b × h × l

Substitute the given values into the formula:

V= 1/2 × 9 cm × 16 cm × 20cm

V= 1440 cm³

∴ the volume of the triangular prism is 1440 cm³

Triangular Prism – Practice Questions

1. A triangular prism has a triangular base with sides measuring 10 cm, 18 cm, and 25 cm. The height of the triangular base is 12 cm, and the overall length of the prism is 15 cm. Calculate the total surface area of triangular prism.

2. Consider a triangular prism with a base edge of 8 cm, a height of the triangular base as 10 cm, and an overall length of the prism as 16 cm. Find the volume of triangular prism.

3. A triangular prism has a triangular base with sides measuring 5 cm, 9 cm, and 13 cm. The height of the triangular base is 15 cm, and the overall length of the prism is 25 cm. Calculate the lateral surface area of triangular prism.

4. Consider a triangular prism with a base edge of 9 cm, a height of the triangular base as 17 cm, and an overall length of the prism as 30 cm. Find the volume of triangular prism.

Triangular Prism – FAQs

What is a Triangular Prism?

A triangular prism is a 3D geometric shape that consists of two triangular bases and three rectangular sides connecting to the corresponding vertices of the triangles. It has a total of five faces, nine edges, and six vertices.

What is the Volume of a Triangular Prism?

The volume (V) of a triangular prism is calculated using the formula:

V= 1/2 × base edge × height of the triangle × length of the prism

What is the Surface Area of a Triangular Prism?

The surface area (A) of a triangular prism involves finding the areas of the two triangular bases and three rectangular sides, then adding them:

A= 2 × area of base triangle + perimeter of base × height of prism

How many vertex does a Triangular Prism have?

A triangular prism has six vertices (corner points).

What is a Right Triangular Prism?

A right triangular prism is a specific type of triangular prism where the triangular bases are right-angled triangles, meaning one of the angles is 90 degrees.

What is the Difference between a Triangular Prism and a Rectangular Prism?

The key difference lies in the shape of their bases. A triangular prism has triangular bases, while a rectangular prism has rectangular bases. Triangular prisms have fewer edges and vertices compared to rectangular prisms with the same number of faces.

How many Edge does a Triangular Prism have?

A triangular prism has nine edges (line segments where the faces meet).

How many Face does a Triangular Prism have?

A triangular prism has five faces.

How to find the Area of a Triangular Prism?

We can find the area of a triangular prism using formula:

Total Surface Area (TSA) = (b × h) + (s₁ + s₂ + s₃) L