SURFACE AREA OF PRISM AND PYRAMID

Both prism and pyramid are basically 3D shapes. Even though we have different formulas to find surface area of prism and pyramid, the basic idea of finding surface area is to add the areas of all the faces.

First, let us look at, how to find surface area of a prism.

Surface Area of Prism

Let us consider the rectangle prism given below.

Here is the basic idea to find surface area of the above rectangular prism.

Surface Area = Sum of areas of all six faces

Let us find the area of each face separately.

Area of the front face (red colored) = l x h

Area of the back face (blue colored) = l x h

Area of the left side face (green colored) = w x h

Area of the right side face (green colored) = w x h

Area of the top portion (purple colored) = l x w

Area of the base (purple colored) = l x w

Now,

Surface area = lh + lh + wh + wh + lw + lw

Surface area = 2lh + 2wh + 2lw

Surface area = 2(lh + wh + lw)

This is the formula to find surface area of a rectangular prism.

Note :

Rectangular prism is also known as cuboid.

We can apply the above explained basic idea to find surface area of any prism without remembering the formulas.

Let us find surface area of the cube given below.

We know that the shape of each face of a cube is a square.

In the above cube, the side length of each face is 'a'.

So, area of each face (square) = a x a = a².

Therefore,

Surface area of cube = 6 x area of each face

Surface area of cube = 6a²

Now, let us find surface area of the triangular prism given below.

In the above triangular prism, there are five faces. The shape of the base and the two slanting faces is rectangle. The shape of two faces on the left side and right side is triangle.

For the given triangular prism,

Area of the base = Lb

Area of the first slanting face = Ls

Area of the other slanting face = Ls

Area of the front face = (1/2)bh

Area of the back face = (1/2)bh

So,

surface area = sum of the area of 5 faces

surface area = Lb + 2Ls + 2 x (1/2)bh

Surface area of triangular prism = Lb+2Ls+bh

Now let us look at, how to find surface area of a prism.

Surface Area of Pyramid

Let us consider the pyramid with square base given below.

Here is the basic idea to find surface area of the above pyramid.

Surface Area = Sum of areas of all five faces (Including the base)

For any pyramid, if the shape of the base is square, then we will have four side walls. The shape of each side wall will be a triangle with equal area.

In the above pyramid, the base is a square with side length 'a' and each wall is a triangle with base 'a' and height 'h'.

Let us find the area of each face separately.

Area of the base = a x a = a²

Area of each side wall = (1/2)ah

Area of all four side walls = 4 x (1/2)ah = 2ah

Surface area of the above pyramid is

= a² + 2ah

This is the formula to find surface area of a pyramid with square base.

We can apply the above explained basic idea to find surface area of a pyramid with triangular base.

Let us find surface area of a pyramid with triangular base.

For any pyramid, if the shape of the base is equilateral triangle, then we will have three side walls. The shape of each side wall will be a triangle with equal area.

In the above pyramid, the base is an equilateral triangle with side length 'a'.

And each wall is a triangle with base 'a' and height 'h'.

Let us find the area of each face separately.

Area of the base = (√3/4)a²

Area of each side wall = (1/2)ah

Area of all 3 side walls = 3 x (1/2)ah = (3/2)ah

Surface area of the above pyramid is

= (√3/4)a² + (3/2)ah

This is the formula to find surface area of a pyramid with equilateral triangle base.

Note :

If the base is not equilateral triangle and it is either scalene triangle or isosceles triangle, then the area of side walls will not be equal. We have to find area of each side wall separately.

Solved Problems

Problem 1 :

Find the surface area of the cuboid shown below.

Solution :

Surface area of cuboid is

= Sum of areas of all six faces

In cuboid, each face is a rectangle. So we can use area of rectangle formula to get area of each face.

Area of the front face = 8 x 12 = 96 cm²

Area of the back face = 8 x 12 = 96 cm²

Area of the left side face = 4 x 8 = 32 cm²

Area of the right side face = 4 x 8 = 32 cm²

Area of the top portion = 4 x 12 = 48 cm²

Area of the base = 4 x 12 = 48 cm²

Surface area of the above cuboid is

= Sum of areas of all six faces

= 96 + 96 + 32 + 32 + 48 + 48

= 352 cm²

Alternative Method :

We can use the formula given below to find surface area of cuboid.

Formula for surface area of cuboid is

= 2(lh + wh + lw)

Substitute l = 12, w = 4 and h = 8.

= 2(12x8 + 4x8 + 12x4)

= 2(96 + 32 + 48)

= 2(176)

= 352 cm²

Problem 2 :

Find the surface area of the cube shown below.

Solution :

We know that the shape of each face of a cube is a square.

In the above cube, the side length of each face is "8".

So, area of each face (square) is

= 8 x 8

= 64 cm²

Therefore, surface area of the cube is

= 6 x area of each face

= 6 x 64

= 384 sq.cm

Problem 3 :

Find the surface area of the triangular prism shown below.

Solution :

In the above triangular prism, there are five faces. The shape of the base, vertical face and slanting face is rectangle. The shape of two faces on the left side and right side is triangle.

For the given triangular prism,

Area of the base = 7 x 4 = 28 cm²

Area of the vertical face = 3 x 7 = 21 cm²

Area of the slanting face = 5 x 7 = 35 cm²

Area of the front face = (1/2) x 4 x 3 = 6 cm²

Area of the back face = (1/2) x 4 x 3 = 6 cm²

So, surface area of the above triangular prism is

= sum of the area of 5 faces

= 28 + 21 + 35 + 6 + 6

= 96 cm²

Problem 4 :

Find the surface area of the triangular prism shown below.

Solution :

In the above triangular prism, there are five faces. The shape of the base and the two slanting faces is rectangle. The shape of two faces on the left side and right side is triangle.

For the given triangular prism,

Area of the base = 8 x 12 = 96 cm²

Area of the first slanting face = 12 x 5 = 60 cm²

Area of the other slanting face = 12 x 5 = 60 cm²

Area of the front face = (1/2) x 8 x 3 = 12 cm²

Area of the back face = (1/2) x 8 x 3 = 12 cm²

So, surface area of the above the triangular prism is

= sum of the area of 5 faces

= 96 + 60 + 60 + 12 + 12

= 240 cm²

Problem 5 :

Find the surface area of the pyramid shown below.

Solution :

Surface area of the pyramid is

= Sum of areas of all 5 faces

In the above pyramid, the base is a square with side length 5 cm and each wall is a triangle with base 5 cm and height 8 cm.

Let us find the area of each face separately.

Area of the base = 5 x 5 = 25 sq.cm

Area of each side wall = (1/2) x 5 x 8 = 20 sq.cm

Area of all 4 side walls = 4 x 20 = 80 sq.cm

Surface area of the above pyramid is

= 25 + 80

= 105 sq.cm

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SURFACE AREA OF PRISM AND PYRAMID