In geometry, a right triangular prism is a three-sided prism. A polyhedron with two congruent and parallel faces and whose lateral faces are parallelograms. The shape of the base gives the prism its name. That is, a prism with triangular bases is a triangular prism. A Triangular Prism is a polyhedron made of a triangular base and 3 faces joining corresponding sides.

## What is a Triangular Prism

A triangular prism is a prism composed of two triangular bases and three rectangular sides. A solid object that has two identical ends and all flat sides. The shape of the base give the prism a name, a prism with triangular bases is called as triangular prism. A right triangular prism is a prism whose bases are triangles. A right triangular prism, the word right describes the prism, whereas the word triangle refers to the triangular base.

## Volume of a Triangular Prism

The volume of a right triangular prism can be found by multiplying the area of base times the height.

### Formula for Volume of a Triangular Prism

We know,
The Volume of a Prism = Ah
Where, A is area of the base and h is height of the prism.

In Triangular Prism,

Area of the base (A) = $\frac{1}{2}$ * a * b

Volume of Triangular Prism = Ah = $\frac{1}{2}$ * a * b * h

where, a = altitude, b = base, h = height.

Formula:

Volume of Triangular Prism =
$\frac{1}{2}$ * a * b * h = $\frac{1}{2}$ abh.

Where, a = altitude, b = base, h = height.

Example:

Find the volume of the triangular prism whose length is 15 cm, altitude 10 cm and base is 11 cm.

Solution:
Length of the prism (h) = 15 cm
Altitude (a) = 10 cm
Base (b) = 11 cm

Step 1:

Area of the base,

A = $\frac{1}{2}$ base x height

= $\frac{1}{2}$ x 11 x 10

= 55

=> A = 55 cm2 .

Step 2:

Volume of the triangular prism (V) = Ah

=> V = 55 x 15

= 825

Hence the volume of the triangular prism is 825 cm3 .

## Surface Area of a Triangular Prism

Surface area of a right triangular prism is the sum of lateral surface area and twice the base area of the triangular prism. Square units are used to measure the surface area of the prism.

### Formula for Surface Area of a Triangular Prism

We know,
Surface Area of a Prism = L + 2B

Where, L is lateral surface area and B is base area of the prism.

In Triangular Prism,

Lateral surface area (L) = Ph = (s1 + s2 + s3)h

Area of the base (B) = $\frac{1}{2}$ * a * b

Surface Area of Prism = (s1 + s2 + s3)h + 2($\frac{1}{2}$ab) = Ph + ab
Where, a = altitude, b = base, h = height and s1, s2, s3 are the sides of the triangular prism.

Formula:

Surface Area of Triangular Prism = Ph + ab = (s1 + s2 + s3)h +
ab

Where, a = altitude, b = base, h = height and s1, s2, s3 are the sides.

Example:

Find the surface area of a triangular prism with the given altitude 3, base 4, height 5 and the sides 2, 3, 4.

Solution:
Given
Triangular prism with the altitude 3, base 4, height 5 and the sides 2, 3, 4.

Step 1:

Find the area of the base,

Area of the bases (B) = $\frac{1}{2}$ * a * b = $\frac{1}{2}$ * 3 * 4 = 6

=> Area of the base (B) = 6

Step 2:

Find the lateral surface area,

Lateral surface area (L) = Ph = (s1 + s2 + s3)h

= (2 + 3 + 4)5

= 9 * 5

= 45

=> Lateral surface area (L) = 45

Step 3:

Find the surface area of prism,

Surface Area of Prism = 2A + L = (2 * 6) + 45 = 12 + 45 = 57

Hence the area of the prism is 57 square units.

## Lateral Area of a Triangular Prism

A solid whose sides are parallelograms and bases are one pair of parallel polygon that have same size and shape. The two parallel polygons are the bases of the prism and the remaining polygons will be parallelograms are called lateral faces. A right prism has lateral faces that are rectangles and perpendicular to the bases. The lateral surface of prism is the sum of the polygon faces.

=> L = ah + bh + ch

= h(a + b + c)

= Ph
Here, a, b, c are the length of the sides of base.These dimensions are used along with height of the prism.

Formula:

The lateral area, L, of the triangular prism:

L = Ph = (a + b + c)h

Where, P is the perimeter of base of the prism and h is the height of the prism.