Area of Prism: N-sided Base, Types and Formula

Prism is a polyhedron with an n-sided base, having both bases parallel and equal in shape and size. The surface area of a prism is the sum of the area of all the sides of the prism. To find the surface of any prism, we use the general formula, which is the sum of the area of the lateral faces and the two bases of the prism.

Key Takeaways: Prism, Area of a prism, Polyhedron, Total surface area, area of the base, height of the prism

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Area of a Prism with an n-sided Base

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Area of a prism is the sum of the area of all its lateral surfaces and the two equal bases.

Area of Prism

Let's say that the base of the prism has n sides, say, S₁, S₂, S₃…, S_n respectively, and the height of the prism is h.

So, the area of the lateral surface is= S₁xh +S₂xh +S₃xh + .........+ S_nxh

= h (S₁+ S₂+ S₃…+ S_n)

= height x perimeter of the base

Area of both the bases = 2 x Area of the base

So, the total area of the prism = (height x perimeter of the base) + (2 x Area of the base)

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Different Types of Prisms and Formulae

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There are different types of prisms:

1) Triangular prism

2) Square prism

3) Rectangular prism

4) Trapezoidal prism

As we already know the general formula of a prism, we will now apply this formula to find the formulae of the area of these different types of prisms.

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Triangular Prism 

Triangular prism has a triangular base.

Let's say that the sides of the triangular base are 'a' each and the height of the prism is h.

The area of the triangular base = a²3/4

The area of the lateral surfaces = Perimeter of the base x height

= 3ah

So, the total area of the prism = (height x perimeter of the base) + (2 x Area of the base)

= 3ah + a²sq. root(3) x2/4

= 3ah + a² sq.root(3) /2

Triangular Prism

Square Prism

Square prism has a square base.

Let's say that the sides of the square base are 'a' each and the height of the prism is h.

The area of the square base = a²

The area of the lateral surfaces = Perimeter of the base x height

= 4ah

So, the total area of the prism = (height x perimeter of the base) + (2 x Area of the base)

= 4ah + a² x2

= 2a(a+h)

Square Prism

Rectangular Prism

Rectangular prism has a rectangular base.

Let's say that the sides of the rectangular base are 'a' and 'b' and the height of the prism is h.

The area of the base = ab

The area of the lateral surfaces = Perimeter of the base x height

= 2(a+b)h

So, the total area of the prism = (height x perimeter of the base) + (2 x Area of the base)

= 2(a+b)h + abx2

= 2(ah+bh+ab)

Rectangular Prism

Trapezoidal Prism

Trapezoidal prism has a square base.

Let's say that the parallel sides of the trapezoidal bases are 'a' and 'b' respectively, the non-parallel sides are 'c' and 'd', the perpendicular distance between the parallel sides is 'l' and the height of the prism is h.

The area of the base = (a+b)l/2

The area of the lateral surfaces = Perimeter of the base x height

= (a+b+c+d)h

Read More: Maxima and Minima

So, the total area of the prism = (height x perimeter of the base) + (2 x Area of the base)

= (a+b+c+d)h + (a+b)lx2/2

= (a+b+c+d)h + (a+b)l

= (a+b)(l+h)+ (c+d)

Trapezoidal Prism

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Things to Remember

• Prism is a polyhedron with an n-sided base. It has parallel bases equal in shape and size. 
• Area of a prism is the sum of the area of all its lateral surfaces and its two equal bases.
• The total area of the prism = (height x perimeter of the base) + (2 x Area of the base
• There are different types of prisms such as Triangular Prism, Square Prism, Rectangular Prism and Trapezoidal Prism.

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