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Surface Area of a Pyramid

A pyramid has a plane (or flat) base and four congruent triangular faces.

Let us look at a square pyramid with the dimensions shown here – surface area of pyramid_1

 

 

 

 

‘Opening up’ this square pyramid will give us the following shape.surface area of pyramid_2

Surface area of a pyramid  =  area of base  +  area of 4 triangles.

=  (6 x 6)  +  (4 x 1/2x 6 x 10)

=  36 x 120

=  156 cm2

 

When you are given the height of the pyramid (instead of the slant height), use Pythagoras Theorem to first calculate the slant height, and then calculate surface area of the pyramid.surface area of pyramid_3

Here (slant height)2  =  (base)2  +  (height)2

=  32  +  42

=  9 + 16

=  25

So slant height of pyramid  =  5

Now surface area of square pyramid = area of base + (4 x area of triangular faces)

=  (6 x 6)  +  (4 x 1/2x 6 x 5)

=  36 + 60

=  96 cm2

 

Example 1 : Calculate the surface area of a square pyramid with base length 10 cm and sloping edges 18 cm long.  surface area of pyramid_example1

Surface area of square pyramid  =  area of base + area of 4 triangular faces

=  (10 x 10)  +  (4 x 1/2x 10 x 18)

=  100 + 360

=  460 cm2

 

Example 2 : What is the surface area of a square pyramid with base length of 6 cm and pyramid height 4 cm?

We first need to find the slant height of the pyramid.

Using Pythagoras theorem, slant height = sqrt{base^2~+~height^2}   (base is half the dimension of the side).

=  sqrt{3^2~+~4^2}

=  sqrt{25}

=  5 cm

So surface area  =  area of base + area of 4 triangular faces

=  (6 x 6)  +  (4 x 1/2x 6 x 5)

=  36 + 60

=  96 cm2