# Triangular Prism

A triangular prism is a three-dimensional solid consisting of two identical triangular bases joined together by three rectangular faces. The lateral faces are rectangular. A triangular prism has 5 faces (3 rectangular lateral faces and 2 triangular bases), 9 edges, and 6 vertices.

A common example of triangular prisms is prisms used in the physics lab for refracting white light. Other real-life examples are camping tents or chocolate candy bars.

A triangular prism can be regular or irregular based on the uniformity of its cross-section. It can also be right or oblique, depending on the alignment of its bases.

## Regular and Irregular Triangular Prisms

1. Regular Triangular Prism â€“ Its 2 bases are equilateral triangles.
2. Irregular Triangular Prism – Its 2 bases are not equilateral triangles.

## Right and Oblique Triangular Prisms

1. Right Triangular Prism â€“ It has all the lateral faces perpendicular to the bases. Thus every lateral face is rectangular.
2. Oblique Triangular Prism â€“ Its lateral faces are not perpendicular to its bases. So, every lateral face is parallelogram-shaped.

Like all other polyhedrons, we can calculate the surface area and volume of a triangular prism.

## Formulas

### Surface Area

The surface area of a triangular prism is the entire space occupied by its outermost layer (or faces). It is expressed in square units such as m2, cm2, mm2, and in2. It has 2 types of surface areas.

#### Lateral Surface Area

The lateral surface area (LSA) of a triangular prism is the sum of the surface area of all its faces except the bases. The formula to calculate the total and lateral surface area of a triangular prism is given below:

Lateral Surface Area (LSA) = (s1 + s2 + s3) Ã— lhere, s1, s2, and s3 are the base edges, l = length

#### Total Surface Area

The total surface area (TSA) of a triangular prism is the sum of the lateral surface area and twice the base area. The formula to calculate the TSA of a triangular prism is given below:

Total Surface Area (TSA) = (b Ã— h) + (s1 + s2 + s3) Ã— l, here, s1, s2, and s3 are the base edges, h = height, l = length

Let us solve some examples to understand the concept better.

Find the surface area of a triangular prism with a triangular base of 7 cm, 6 cm, and 4 cm, and height 6 cm, and length 9 cm.

Solution:

As we know,
Total Surface Area (TSA) = b Ã— h + (s1 + s2 + s3) Ã— l, here s1 = 7 cm, s2 = 6 cm, s3 = 4 cm, h = 6 cm, l = 9 cm
âˆ´ TSA = 6 Ã— 6 + (7 + 6 + 4) Ã— 9
= 189 cm2

Find the lateral and total surface area of a triangular prism with an equilateral triangular base of 6.5 cm, and length is 10.5 cm.

Solution:

As we know,
The prism is an equilateral triangular prism, so height is not required,
Total Surface Area (TSA) = 2B + (s1 + s2 + s3) Ã— l, here B = base area, s1 = s2 = s3 = 6.5 cm, l = 10.5 cm
Applying the formula to find the area of an equilateral triangle,
âˆ´ Â B = ${\dfrac{\sqrt{3}}{4}a^{2}}$
â‰ˆ 18.3 cm2
Now, TSA = 2 Ã— 18.3 + (6.5 + 6.5 + 6.5) Ã— 10.5
= 241.35 cm2
Lateral Surface Area (LSA) = (s1 + s2 + s3) Ã— l
âˆ´ Â LSA = (6.5 + 6.5 + 6.5) Ã— 10.5
= 204.75 cm2

### Volume

The volume of a triangular prism is its space in the three-dimensional plane. The formula to calculate the volume of a triangular prism is given below:

Volume (V) = 1/2 Ã— b Ã— h Ã— l, here b = base edge, h = height, l = length

Let us solve an example to understand the concept better.

Find the volume of a triangular prism whose base is 16 cm, height is 9 cm, and length is 21 cm.

Solution:

As we know,
Volume (V) = 1/2 Ã— b Ã— h Ã— l, here b = 12 cm, h = height, l = 21 cm
âˆ´ Â V = 1/2 Ã— 16 Ã— 9 Ã— 21
= 1512 cm3

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