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Surface Area of a Trapezoidal Prism

There are a number of solid (three-dimensional) shapes studied in geometry. Prism is one of the most useful ones. A prism may be defined as a 3D shape having two bases in the shape of a polygon. Both bases are connected with each other with the help of rectangular or parallelogram-shaped lateral faces. A right prism has same cross section throughout and the line joining centers of both bases is perpendicular to them. The prisms are named after the polygonal bases they have. The following diagram demonstrates triangular and rectangular right prisms.

Surface Area of a Trapezoidal prism

In this article, we will learn about trapezoidal prisms. We shall throw light especially on the surface area of a trapezoidal prism. Let us go ahead and understand the definition, formula, and method of finding the surface area of trapezoidal prisms with the help of some reasonable examples.

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Definition

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A trapezoidal prism consists of two opposite faces or bases in the shape of trapeziums or trapezoids. Recall that a trapezoid is a plane figure having one pair of opposite sides to be parallel. The distance between parallel sides is known as the height of the trapezium. It is shown below :

Definition of Traezoid

Any trapezoidal prism is made up of 6 faces (2 bases and 4 lateral surfaces), 12 edges and 8 vertices. 
In a right trapezoidal prism, the opposite trapezoid-shaped bases are congruent and are connected together by the means of four rectangles. The diagram of a right trapezoidal prism is given as under :

Definition of Trapezoidal prism

Formula

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There are two forms of surface areas of solid shapes - lateral surface area and total surface area. Let us suppose that there be a right trapezoidal prism as shown in the figure below.

Surface Area of a Trapezoidal prism Formula

The height of the prism be "H". Let the parallel sides of the trapezoid-shaped base measure "a" and "b". The height of the trapezoid is supposed to be "h".
 
Lateral Surface Area
It includes the surface area of the lateral side of a prism. The formula for lateral surface area of the trapezoidal prism is given by :

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

or

LSA = (a + b + c + d) $\times$ H

where, c and d be the length of non-parallel sides of the base.
Total Surface Area

It includes surface area lateral sides as well as the bases. The formula for total surface area will eventually be written as :

TSA = LSA + 2 $\times$ Area of base

or

TSA = (a + b + c + d) H + 2 $\frac{a+b}{2}$ h


or

TSA = (a + b + c + d) H + (a + b) h

How To Calculate Surface Area of a Trapezoidal Prism

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For calculating the surface area, the following steps can be adopted.

Lateral Surface Area

Step 1 : Find the perimeter of the trapezoidal base by adding measures all four sides together.

Step 2 : Determine height of the prism.

Step 3 :
 Calculate the product of both quantities. It gives required lateral surface area.
Total Surface Area

Step 1 :
 Find the lateral surface area of the prism by using method discussed above.

Step 2 :
 Calculate area of the trapezoidal base. The formula is : Area of trapezoid = $\frac{a+b}{2}$ $\times\ h$where, h is the height of trapezoid and a, b are the lengths of parallel sides.

Step 3 :
 Double the area of the trapezoid as there are two bases.

Step 4 :
 Find the sum of the lateral surface area and twice of the area of the base. This will be the required total surface area.

Examples

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The example problems based on finding the surface area of the trapezoidal prism are given below.

Example 1 : Calculate the total surface area of following trapezoidal prism.

Total Surface Area of a Trapezoidal prism Examples

Solution : Four sides of the trapezoidal base are
a = 4 cm, b = 10 cm, c = 3 cm = d

Height of the trapezoidal base, h = 2 cm

Perimeter of base, P = 4 + 10 + 3 + 3 = 20 cm

Height of the prism, H = 7 cm

Area of base = $\frac{a + b}{2}$ $\times\ h$ = $\frac{14}{2}$ $\times\  2$ = 14 sq cm

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

Total surface area = (P $\times$ H)  + 2 Area of base

= 20 $\times$ 7 + 2 $\times$ 14

= 140 + 28 = 168 sq cm.
Example 2 : Take a look at the image below :

Trapezoidal prism Example

Calculate its surface area.

Solution :
 Sides of the trapezoidal base are

a = 5 cm, b = 8 cm, c = 6 cm , d = ?

For finding the fourth side, we use Pythagoras theorem in the right triangle

$d^{2} = (8 - 5)^{2} + 6^{2}$

$d^{2} = 3^{2} + 6^{2}$ = 9 + 36 = 45

d = $\sqrt{45}$ $\approx$ 6.7 cm

Perimeter of base, P = 5 + 8 + 6 + 6.7 = 25.7 cm

Height of the trapezoidal base, h = 6 cm

Height of the prism, H = 12 cm

Area of base = $\frac{a + b}{2}$ $\times\ h$ = $\frac{13}{2}$ $\times\  6$

= 13 $\times$ 3 = 39 sq cm

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

Total surface area = (P $\times$ H)  + 2 Area of base

= 25.7 $\times$ 12 + 2 $\times$ 39

= 308.4 + 78 = 386.4 sq cm.
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