We often come across with prisms in mathematics and optics. A prism is a three-dimensional transparent object which has flat surfaces. It is made up of at least three flat and polished surfaces*.* In prism, there are at least two surfaces that are inclined at an angle from each other. In geometry, there are different types of prisms - **Triangular prisms, rectangular prisms, polygonal prisms **etc. Even, **cubes **and **cuboids **are examples of prisms. Here, we are going to study about triangular prisms.

A traditional triangular prism is a solid figure which is made up of two triangular bases and three rectangular faces. The two triangular bases face each other and are joined together with the help of three rectangles. Triangular prism is the most useful prism in geometry as well as in optical physics. Prisms are usually made from any transparent material such as - plastic, fluorite and glass.

A prism does have a property of reflecting and refracting. It decomposes sunlight into 7 spectral colors. This property is known as **dispersion **of light. Due of these properties, triangular prisms do have several applications in different fields. A triangular prism may look similar to the image below -

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A triangular prism is a prism designed of two triangular bases and three rectangular sides. Triangular prism is a polyhedron with two congruent and parallel bases. All cross-sections parallel to the base faces are the same as triangle.

There are a number of properties of a triangular prism. Some of them are as follows,

In triangular prism,

### Volume of a Triangular Prism Formula:

### Triangular Prism Surface Area Formula:

Area of base = $\frac{1}{2}$ $\times$ Base $\times$ Height

= $\frac{1}{2}$ bh

Total surface area of prism shown in figure = ( (a + b + c)H + 2 . $\frac{1}{2}$ bh ) square units

= ( (a + b + c)H + bh ) square units

### Picture of a Triangular Prism:

In this figure: The parallel faces PQR and P'Q'R' are bases of the prism. The rectangular faces PP'Q'Q, QQ'R'R and RR'P'P are the lateral faces of the prism. A common line segment between two lateral faces is called a lateral edge. Here PP', QQ' and RR' are the lateral edge of the prism.

In short, we can say that:

Triangular prism faces = 5

Triangular prism edges = 9

Triangular prism vertices = 6

Base of a triangular prism = 2

### Surface Area of Right Triangular Prism:

Let a, b, c be the sides of the triangular base and H is the height of the prism.

Lateral surface area of the prism = Sum of the areas of 3 triangles

= (a + b + c)H

= Perimeter of base $\times$ height of the prism

and, area of one base = $\frac{1}{2}$ bh

Here, b - Base side of triangle and h - height of triangle

Now,

Total surface area of the triangular prism = Lateral surface area of the prism + Area of two bases

= ( (a + b + c)H + bh ) square units

### Volume of a Right Triangular Prism:

Volume of a Triangular Prism = Area of base $\times$ Height of prism

If b is the side of the base, h is the height of the triangle and H is the height of the prism, then

Volume formula for triangular prism = $\frac{1}{2}$ bhH cubic units Prisms are solids whose two ends are exactly the same shape. A prism is named after the shape of the faces at each end.

To draw a triangular prism, follow the following steps:

**Step 1:** First draw a triangle (Triangle may be right, isosceles or equilateral). Lets choose equilateral triangle.

**Step 2:** Move across and up a little and draw the same triangle.

**Step 3:** Join the corners of the two triangles.

**Step 4:** Darken the lines that you can see and draw the hidden parts as faint lines.

Given below are some examples for finding the volume of the triangular prism.### Solved Examples

**Question 1: **Find the volume of the triangular prism with base 8.2 cm, height 7.4 cm, and length is 10.5 cm.

** Solution: **

**Question 2: **The base of a right prism is a triangle whose sides measure 10, 11, 15. If the height of the prism be 8. Find its approximate total surface and volume.

** Solution: **
**Step 1: **Let the sides of the triangle base be : a = 10, b = 11 and c = 15

Then, S = $\frac{a + b + c}{2}$ = $\frac{10 + 11 + 15}{2}$ = $\frac{36}{2}$ = 18

Area of base = $\sqrt{S(S - a)(S - b)(S - c)}$

= $\sqrt{18(18 - 10)(18 - 11)(18 - 15)}$

= $\sqrt{18 \times 8 \times 7 \times 3}$

= $\sqrt{3024}$

= 55 (approx)

Also, height of prism = 8

**Step 2: **Lateral surface area = Perimeter of base $\times$ height of prism

= (10 + 11 + 15) $\times$ 8 = 288

Total surface area = Lateral surface area + 2(Area of base)

= 288 + 2 $\times$ 55

= 398

Total surface area of the prism is 398 square units

**Step 3: **Volume = Area of base $\times$ Height of prism

= 55 $\times$ 8

= 440

Therefore, the volume of the prism is 440 cubic units

There are a number of properties of a triangular prism. Some of them are as follows,

In triangular prism,

- A regular triangular prism has 9 edges.
- It has
**5 faces.** - It consists of
**6 vertices**. **Triangular prism**- If the base face of a
**triangular prism**is equilateral triangle and the other faces are squares, then the triangular prism is said to be semiregular.

**The volume of a triangular prism** can be defined as the product of base area and the height of the prism.

The volume of a triangular prism formula is as follows,

Volume of a triangular prism = Area of base $\times$ Height of prism

Volume of triangular prism formula * = *$\frac{1}{2}$

**Where ,** b – triangle base length,

h - Triangle height,

l - Length between triangles.

Let us find the volume of the triangular prism with base is 9 feet, height is 15 feet, and length is 17 feet.

**Volume of the triangular prism**** = **$\frac{1}{2}$** bhl**

Here, l = 16 feet, h = 10 feet, b = 11 feet

**Volume** =** **$\frac{1}{2}$** (**11 $\times$ 10 $\times$ 16)

= $\frac{1}{2}$** (**1760)

**= $\frac{1760}{2}$**

=** **880

Therefore, the **volume of the triangular prism is 880 feet**^{3}

**Surface area of triangular prism ****= 2A + PH**

where, A - Base area, P - Perimeter of a base of the prism and H - Height of prism.

The formula for surface area of a
prism whose sides of the bases are a, b, c units and the height of the
prism is H units is given by the following formula,

** **

**Lateral area of a triangular prism = P H sq. unit, **

Where, P = a + b + c

P is the perimeter of the prism and H is height of the prismArea of base = $\frac{1}{2}$ $\times$ Base $\times$ Height

= $\frac{1}{2}$ bh

Total surface area of prism shown in figure = ( (a + b + c)H + 2 . $\frac{1}{2}$ bh ) square units

= ( (a + b + c)H + bh ) square units

The surface area of a triangular prism formula is as follows,

** Area of a Triangular Prism = ( (a + b + c)H + bh ) square units**

where, P = a + b + c = Perimeter of the base

h - Height of the triangular base

and H - Height of the prism

A right triangular prism has a triangular base in which the joining edges and faces are perpendicular to the base edges. In a right triangular prism, all the lateral faces are rectangles and are perpendicular to the bases.In this figure: The parallel faces PQR and P'Q'R' are bases of the prism. The rectangular faces PP'Q'Q, QQ'R'R and RR'P'P are the lateral faces of the prism. A common line segment between two lateral faces is called a lateral edge. Here PP', QQ' and RR' are the lateral edge of the prism.

In short, we can say that:

Triangular prism faces = 5

Triangular prism edges = 9

Triangular prism vertices = 6

Base of a triangular prism = 2

Let a, b, c be the sides of the triangular base and H is the height of the prism.

Lateral surface area of the prism = Sum of the areas of 3 triangles

= (a + b + c)H

= Perimeter of base $\times$ height of the prism

and, area of one base = $\frac{1}{2}$ bh

Here, b - Base side of triangle and h - height of triangle

Now,

Total surface area of the triangular prism = Lateral surface area of the prism + Area of two bases

= ( (a + b + c)H + bh ) square units

Volume of a Triangular Prism = Area of base $\times$ Height of prism

If b is the side of the base, h is the height of the triangle and H is the height of the prism, then

Volume formula for triangular prism = $\frac{1}{2}$ bhH cubic units Prisms are solids whose two ends are exactly the same shape. A prism is named after the shape of the faces at each end.

To draw a triangular prism, follow the following steps:

Given below are some examples for finding the volume of the triangular prism.

Volume of the triangular prism = $\frac{1}{2}$ bhl

Here, l = 10.5 cm, h = 7.4 cm, b = 8.2 cm

Volume = $\frac{1}{2}$ (8.2 $\times$ 7.4 $\times$ 10.5)

= $\frac{1}{2}$ (637.14)

= $\frac{637.14}{2}$

= 318.57

Therefore, the **volume of the triangular prism is 318.57 cm**^{3}

Then, S = $\frac{a + b + c}{2}$ = $\frac{10 + 11 + 15}{2}$ = $\frac{36}{2}$ = 18

Area of base = $\sqrt{S(S - a)(S - b)(S - c)}$

= $\sqrt{18(18 - 10)(18 - 11)(18 - 15)}$

= $\sqrt{18 \times 8 \times 7 \times 3}$

= $\sqrt{3024}$

= 55 (approx)

Also, height of prism = 8

= (10 + 11 + 15) $\times$ 8 = 288

Total surface area = Lateral surface area + 2(Area of base)

= 288 + 2 $\times$ 55

= 398

Total surface area of the prism is 398 square units

= 55 $\times$ 8

= 440

Therefore, the volume of the prism is 440 cubic units

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