Triangular Prism 

What do tents, rooftops, and Toblerone bars have in common? They all share the unique 3D structure — a triangular prism! From understanding its shape and types to mastering the triangular prism volume formula and surface area of a right triangular prism, this guide brings this interesting figure of 3D geometry to life. So, let’s unfold this prism one face at a time!

1.0Introduction to Triangular Prism

In geometry, a triangular prism is a three-dimensional solid shape that has two triangular bases and three rectangular lateral faces. It is one of the basic types of prisms and is frequently seen in everyday objects and architecture. Understanding the shape, structure, and formulas related to a triangular prism helps in solving problems related to volume and surface area in mathematics.

2.0Shape of a Triangular Prism

A triangular prism looks like a stretched triangle. It has two congruent triangles (which are the bases) and three rectangular faces that connect the sides of the two triangles. These faces are known as the lateral faces.

• Faces: 5 (2 triangular bases and 3 rectangular lateral faces)
• Edges: 9
• Vertices: 6

The triangular prism is similar to a tent. It's a prism because the cross-section along its length remains the same, and it's called triangular because the base is a triangle.

3.0Types of Triangular Prism

Triangular prisms can be classified into two main types:

Right Triangular Prism

• A right triangular prism has its side faces (rectangles) perpendicular to the triangular bases.
• All the lateral faces are rectangles, and the height of the prism is the distance between the two triangular bases.
• This is the most commonly studied type of triangular prism in school mathematics.

Oblique Triangular Prism:

• In this prism, the rectangular faces are not perpendicular to the triangular bases.
• It is slanted, and the height is not directly equal to the length of the side faces.
• Less common in basic geometry but useful in advanced applications.

4.0Properties of Triangular Prism

Let’s go over the main properties of a triangular prism:

• It has two triangular bases and three rectangular lateral faces.
• It has 6 vertices, 9 edges, and 5 faces.
• The volume depends on the area of the base triangle and the height (length) of the prism.
• The surface area includes the area of the two triangular bases plus the area of the three rectangles.

These properties make triangular prisms important in geometry, architecture, and engineering.

5.0Formula Related to Triangular Prism

Triangular Prism Volume Formula 

The triangular prism volume formula helps in calculating the amount of space inside the prism. The volume depends on the area of the triangular base and the height (length) of the prism. 

$Volume=BaseArea\times Height(orlength)$

If the base triangle has a base b and height h, and the length of the prism is l, then:

$Volume=\frac{1}{2}\times b\times h\times l$

This is the triangular prism formula used to calculate how much space the prism occupies.

Surface Area of Right Triangular Prism

Total Surface Area of Right Triangular Prism: To calculate the total surface area of a right triangular prism, we add the areas of all five faces — two triangles and three rectangles. 

$SurfaceArea=BaseArea(bothtriangles)+LateralSurfaceArea(3rectangles)$

$\text{Surface area of right triangular Prism}=2\times (\frac{1}{2}\times b\times h)+(a+b+c)\times l$

Lateral Surface Area of Right Triangular Prism: The lateral surface area of a right triangular prism is calculated by adding the areas of all the rectangular faces of the prism. Like this: 

$\text{Lateral Surface Area of Prism}=(a+b+c)\times l$

In both the formulas: 

• 𝑎, 𝑏, 𝑐 are the sides of the triangle
• 𝑙 is the length (or height) of the prism

6.0Pyramid vs Triangular Prism

It is common to confuse a pyramid and a triangular prism, but they are different shapes. Let’s understand the basic difference between these two similar yet different shapes of Mathematics: 

7.0Triangular Prism Examples: Numericals

Problem 1: A right triangular prism has a triangular base with sides 5 cm, 12 cm, and 13 cm. The length of the prism is 7 cm. Find the surface area of the prism.

Solution: First, calculate the area of the triangle base:

This is a right triangle (5-12-13 is a Pythagorean triplet).

$\text{Base area}=\frac{1}{2}\times 5\times 12=30{\text{cm}}^2$

Now use the surface area formula:

$SurfaceArea=2\times \text{Base Area}+(\text{Perimeter of triangle})\times \text{Length}$

Perimeter of Triangle = 5+12+13 = 30 cm

$\text{Surface Area}=2\times 30+30\times 7=60+210=270{\text{cm}}^2$

Problem 2: A water tank is in the shape of a triangular prism. The cross-section is a triangle with a base of 8 m and a height of 5 m. The length of the tank is 15 m. How much water (in litres) can it hold when full?

Solution: First, find the volume in cubic meters:

$\text{volume}=\frac{1}{2}\times 8\times 5\times 15=300{\text{m}}^3$

Since 1 m³ = 1000 liters:

Water it can hold = 300 × 1000 = 3,00,000 liters

Problem 3: A triangular prism has a volume of 180 cm³. The triangular base has a base of 6 cm and a height of 5 cm. Find the length of the prism.

Solution: We know:

$\text{Volume}=\frac{1}{2}\times \text{base}\times \text{height}\times \text{length}$

Substitute values:

$180=\frac{1}{2}\times 6\times 5\times l$

$180=15l\Rightarrow l=\frac{180}{15}=12\text{cm}$