Prisms are three-dimensional shapes which are made up of polygonal surfaces. They have two polygonal bases that are connected with each other by rectangular (right prisms) or parallelogram surfaces (oblique prisms). A prism is named after the shape of its base. Have a look at following image which demonstrates different kinds of prisms, such as a triangular prism, square prism, pentagonal prism, hexagonal prism, heptagonal prism etc.

In this article, we are going to focus on triangular prisms only. Basically, we shall discuss the surface area of a triangular prism. We will understand the method of calculating lateral surface area and total surface area of triangular prisms with the help of solved examples.

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A triangular prism is said to be a prism having triangular bases. It has two triangular-shaped surfaces facing each other. It has three lateral faces that join these two bases.

A right triangular prism in which lateral surfaces are rectangles and bases are exactly one upon another, is shown in the diagram below.

An oblique triangular prism is that in which lateral surfaces are parallelograms. Its shape is tilted towards one side as demonstrated in the image below.

From the surface area of a triangular prism (or any other 3D shape), we mean to refer to the area covered by its outer surface. Recall that there are two types of surface areas - lateral surface area and total surface area. The lateral surface area of a triangular prism includes the area of its three lateral faces or side faces. While the total surface area is said to be the area of side faces and the area of base triangles as well.**Lateral Surface Area**

**LSA = P h**

**LSA = (a + b + c) h**

**LSA = 3a h**

**Total Surface Area**

**TSA = Ph + 2A**

**TSA = 3 a h + $\frac{\sqrt{3} a^{2}}{2}$**

**Below are the steps to find the lateral surface area of a triangular prism:**

**Step 1 :** Calculate the perimeter of the base by using appropriate formula or simply by adding all three sides of the base.

**Step 2 :** Determine the height of the triangular prism.

**Step 3 :** Check for the units of height and perimeter. They must be same (m, cm, mm etc). If not, convert and make them same.

**Step 4 :** Multiply perimeter of base and height of the prism. This followed by the appropriate unit (unit$^{2}$) will be your required lateral surface area.
**Below are the steps for calculating the total surface area of a triangular prism:**

**Step 1 :** Determine the height and perimeter of the triangular base. Look for the units, they should be same.

**Step 2 :** Calculate lateral surface area by finding the product of perimeter of base and height of prism as discussed in above section.

**Step 3 :** Calculate the area of the base triangle. It can be done using following formula :

Area of triangle, A = $\frac{1}{2}$ $\times$ base $\times$ height.

In case of equilateral triangle

A = $\frac{\sqrt{3}a^{2}}{4}$

**Step 4 :** Substitute these values in the formula :

**TSA = LSA + 2 A**

and find the value and write appropriate unit (unit$^{2}$). This will be the required total surface area. Have a look at the following examples.

**Example 1 :** Find the total surface area of a triangular prism shown in the adjacent image.

**Solution :**

Determine Height

Height of the prism = perpendicular distance between two bases = 7 inches = h

**Find Perimeter of Base**

Perimeter of base = sum of three sides = 3 + 4 + 5 = 12 cm = p

**Find Area of Base**

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

A = $\frac{1}{2}$ $\times$ 3 $\times$ 4

A = 3 $\times$ 2 = 6 square inches

**Calculate TSA**

TSA = p h + 2 A

= 12 $\times$ 7 + 2 $\times$ 6

= 84 + 12 = 96 square inches**Example 2 :** Calculate the total surface area of a triangular prism whose height is 28 cm and base is an equilateral triangle of side 16 cm.

Solution :

Determine Height

Height of the prism, h = 28 cm

**Find Perimeter of Base**

Side of base, a = 16 cm

Perimeter of base = 3a = 3 $\times$ 16 = 48 cm

**Find Area of Base**

A = $\frac{\sqrt{3}a^{2}}{4}$

A = $\frac{\sqrt{3} \times16^{2}}{4}$

= $\frac{\sqrt{3} \times 256}{4}$

= $\sqrt{3} \times 64$

= $1.732 \times 64$

= 110.85 cm$^{2}$

**Calculate TSA**

TSA = p h + 2 A

= 48 x 28 + 2 x 110.85

= 1344 + 221.7 = 1565.7 cm$^{2}$

A right triangular prism in which lateral surfaces are rectangles and bases are exactly one upon another, is shown in the diagram below.

An oblique triangular prism is that in which lateral surfaces are parallelograms. Its shape is tilted towards one side as demonstrated in the image below.

From the surface area of a triangular prism (or any other 3D shape), we mean to refer to the area covered by its outer surface. Recall that there are two types of surface areas - lateral surface area and total surface area. The lateral surface area of a triangular prism includes the area of its three lateral faces or side faces. While the total surface area is said to be the area of side faces and the area of base triangles as well.

The formula for calculating lateral surface area of a right triangular prism is given by :

Where LSA denotes lateral surface area, P stands for the perimeter of base and h means the height of the prism.

For a prism with scalene triangle (all sides of different measures) as its base, the formula will take the following form :

Where, a, b and c be the sides of the triangular base.

For a prism with equilateral (all sides of same measure) triangular base, the formula will be as under :

Where a denotes the side of the base.

The formula for the total surface area (includes lateral surface area and area of bases) of a right triangular prism is given by :

where TSA eventually represents the total surface area, P is the perimeter of the base, h is the height of prism and A is the area of the base.

For an equilateral triangular base, the total surface area of a prism is given by :

Area of triangle, A = $\frac{1}{2}$ $\times$ base $\times$ height.

In case of equilateral triangle

A = $\frac{\sqrt{3}a^{2}}{4}$

and find the value and write appropriate unit (unit$^{2}$). This will be the required total surface area. Have a look at the following examples.

Determine Height

Height of the prism = perpendicular distance between two bases = 7 inches = h

Perimeter of base = sum of three sides = 3 + 4 + 5 = 12 cm = p

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

A = $\frac{1}{2}$ $\times$ 3 $\times$ 4

A = 3 $\times$ 2 = 6 square inches

TSA = p h + 2 A

= 12 $\times$ 7 + 2 $\times$ 6

= 84 + 12 = 96 square inches

Solution :

Determine Height

Height of the prism, h = 28 cm

Side of base, a = 16 cm

Perimeter of base = 3a = 3 $\times$ 16 = 48 cm

A = $\frac{\sqrt{3}a^{2}}{4}$

A = $\frac{\sqrt{3} \times16^{2}}{4}$

= $\frac{\sqrt{3} \times 256}{4}$

= $\sqrt{3} \times 64$

= $1.732 \times 64$

= 110.85 cm$^{2}$

TSA = p h + 2 A

= 48 x 28 + 2 x 110.85

= 1344 + 221.7 = 1565.7 cm$^{2}$

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