 # Triangular Prism

## Triangular Prism Definition

As the term tri refers to three, the triangular prism is defined as a three-dimensional solid with two identical ends connected by equal parallel lines. In geometry, it is also defined as a polyhedron composed of three rectangular sides and two triangular bases one on either side. The two triangular bases of this prism are congruent and parallel to each other. To understand how a triangular prism is made up of two-dimensional shapes – rectangles and triangles, there is a need to create a net for it, as shown in the image below. The top and bottom are triangles and known as bases. The three rectangles are sides, which are known as lateral faces. Two of the faces meet to form a line segment called edge and then three edges meet at a point called a vertex. And, the final shape includes 5 faces, 9 edges, and 6 vertices. The three rectangular sides play the role of the connectors by joining the vertices and edges of the bases with each other. Besides joining the edges and vertices of the bases, the rectangular sides of this prism are joint with each other also side by side. All the cross-sections parallel to the base faces appear as triangles. Another noticeable point about this prism is that the two triangular bases present in it are often equilateral triangles.

In the case of a right triangular prism, the sides are either in the rectangular shape or else can be oblique. In general, prisms are not limited to just triangular prisms. Rather, these three-dimensional shapes composed of at least three flat surfaces are available in various types. For instance, while studying maths and physics, we often come across Rectangular Prism, Polygonal Prisms, etc. Note that cubes and cuboids that we study in geometry can also be considered as prisms.

### Volume of Triangular Prism

In geometry, the volume of any triangular prism is defined as the product of its triangular base area and height. By definition, the formula to calculate the volume of a prism is given as:

Volume = Area of the base x height

Now, as we know that the base of a triangular prism is in the shape of a triangle, the area of the base is the same as that of a triangle, i.e., 1/2 x base (b) x height (h)

So, the volume of a triangular prism = ½ × b × h × l

Where,

b = the base length,

h = the height of the triangle,

l = the length between the triangular bases.

### Surface Area of a Triangular Prism

The surface area of a triangular prism is equal to the sum of its lateral surface area and twice the base area. By definition, the surface area of this figure is given as:

SA = 2b + Ph

Where,

SA = Surface area of the triangular prism

b = Base area of the one end of the triangular prism

P = Perimeter of the base

h = length of the prism, l

Now, if we let a, b, and c be the sides of the triangular bases, then the perimeter (P) of the base will be:

P = a + b+ c

Hence, by putting the value of perimeter and area in the above equation of the surface area of the triangular prism, we will get:

Surface area (SA) of the triangular prism = 2(½ × b × h) + (a + b + c)h

From this, we can also determine the formula for the area of the triangular prism, and it is:

Area of any Triangular Prism = (bh + (a + b + c)h)

### Properties of Triangular Prism

• It consists of a total of nine sides, five faces, and six vertices joined together by rectangular sides

• Its five faces include two triangular bases and three rectangular sides or we can say rectangular lateral faces

• When two faces of a triangular prism meet, they create a line segment called edge

• It is a polyhedron with congruent and parallel bases

• If the bases of any triangular prism are equilateral triangles, but the faces are in a square shape instead of rectangular, then that type of prism is known as semiregular.

### Examples of Triangular Prism

Some of the real-life examples of a triangular prism include triangular roofs, camping tents, Toblerone wrappers, and chocolate candy bars.

Solved Examples

1. Find out the volume of a triangular prism with base 4 cm, side 3 cm, and height 6 cm

Solution:

Given,

Base length, b = 4 cm

Height of the triangular base, h = 3 cm

Length between the triangular bases, l = 6 cm

According to the formula,

Volume of the triangular prism = 1/2 x b x h x l

Volume (v) = 1/2 X 4 X 3 X 6

= 36 cm3.

1. Find out the area of a triangular prism with height 4cm and base 5 cm respectively. The height of the triangular base is 10 cm. Note that this triangular prism consists of the equilateral triangular base with the length of its side measuring 6cm.

Solution:

Given,

Base, b = 5cm,

Height, h of the base = 10cm,

Length of the side of the equilateral triangular base, a = b = c = 6 cm,

Height, H of the prism = 4cm

Now, by putting these values in the formula below, we get:

Area of a triangular prism = (bh + (a + b + c)H)

= (5 x 10 + (6+6+6)4)

= (50 + (24)4)

= 50 + 96

= 146 cm2.