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 and formulas of the triangular prism, and the net of a triangular prism. We will also see different types of triangular prism based on uniformity and alignment.
What is Prism?
A prism is a specific type of polyhedron that features identical polygons at both its top and bottom connected with rectangular faces. The remaining faces on the side 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 PrismCharecterstics 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 experiments
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.
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 equilateral triangles.
A equilateral 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
Properties of Triangular Prism
Parts of a Triangular Prism | Numbers |
|---|
Face of a Triangular Prism | 5 |
|---|
Edge of a Triangular Prism | 9 |
|---|
Vertex of a Triangular Prism | 6 |
|---|
A triangular prism is easily identifiable by its key characteristics. Here are the important properties explained in neutral language:
- 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 NetSurface 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.
Also Read:
Solved Examples on Triangular Prism
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
Question 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.
Question 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.
Question 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.
Question 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.
Conclusion
Triangular prism is a three-dimensional shape with two triangular bases and three rectangular sides. Triangular prisms come in different types, such as right, oblique, and regular, each with unique characteristics and properties. Triangular prisms have various practical uses in both everyday life and specialized fields:
- Architecture: Triangular prisms are used in building designs, particularly for roofs, bridges, and support structures, providing stability and a unique shape.
- Optics: Glass triangular prisms are used to split or bend light in optics and science experiments, such as in spectrometers and lasers.
- Tents and Shelters: Many tents are shaped like triangular prisms because of their simple and stable structure.
- Aquariums: Triangular prism-shaped tanks can be found in certain types of aquariums for unique visual effects.
- Mathematics and Education: Triangular prisms are used in geometry lessons to help students understand 3D shapes and their properties, such as volume and surface area.
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