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Volume of a Pyramid

Volume of a pyramid_1The volume of a pyramid is one third the volume of a prism with the same base and perpendicular height.

V  =  1/3x Area of base x Height

V  =  1/3Ah

 

For example, a pyramid has a rectangular base 30 cm x 10 cm and perpendicular height 25 cm.Volume of a pyramid_2

Area of base  =  30 x 10  =  300 cm2

Volume  =  1/3x Ah

=  1/3x 300 x 25

=  3750 cm3

 

Example 1 : Calculate the volume of a rectangular pyramid with base 30 cm x 15 cm and a perpendicular height of 16 cm.

Volume  =  1/3x Ah

=  1/3x 30 x 15 x 16

=  2400 cm3

 

Example 2 : Calculate the volume of a square pyramid (i.e. pyramid with a square base) with base edge 9 cm and perpendicular height 14 cm.

Volume  =  1/3x Ah

=  1/3x 9 x 9 x 14

=  378 cm3

 

Volume of a pyramid_3Sometimes instead of the height, you are given the slant height, i.e. the length along the side face of the pyramid.

In that situation, you will need to use Pythagoras Theorem to first find the height of the pyramid, and then apply it to the volume formula to obtain the volume of the pyramid.

 

 

Example 3 : Calculate the volume of a square pyramid with base edge 24 cm and slant height 20 cm.

Here we need to first find the height of the pyramid.

Height  =  sqrt{slant height^2~-~base^2}

=  sqrt{20^2~-~12^2}   (the base of this right-angled triangle is half the length of the base of the pyramid)

=  sqrt{400~-~144}

=  16 cm

Volume of a square pyramid  =  1/3x Ah

=  1/3x 24 x 24 x 16

=  3072 cm3

 

Example 4 : Calculate the volume of a rectangular pyramid with the dimensions in this diagram (correct to 1 decimal place).Volume of a pyramid_4

Given slant height 14 cm, we have to first work out the height of the pyramid.

 

Note the base of the triangle is half the length of the side (here 30 cm).

Height of pyramid  =  sqrt{26^2~-~15^2}Volume of a pyramid_5

=  sqrt{451}

=  21.2 cm

Volume of pyramid  =  1/3x Ah

=  1/3x 14 x 30 x 21.2

=  2968 cm3