# Find the base, lateral area, surface area and volume of the tetrahedron below: http://www.flickr.com/photos/93084714@N07/8674224220/

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Area of an equilateral triangle = `(sqrt(3)/4)a^2`

`=sqrt(3)/4xx324`

`=81sqrt(3)`

Area of the base=`81sqrt(3)` sq. unit

Volume of a pyramid: (1/3) × Area of base × height

Height = ?

Note that the vertex at the top, one corner at the triangle at the base and the circumcentre form a right angled triangle.

The hypotenuse is 'a'

The base is a/√3

The height will be √(2/3)a

The volume will be

V = (1/3) × Area of base × height

= (1/3) × √3a²/4 × a√(2/3)

= a³/(6√2)

The height of tetrahedron= `(sqrt(2)/3)xx18`

`=6sqrt(2)`

Area of the base=`` 81 sqrt(3) sq. unit

**Surface area of tetrehedron= 3x 81 sqrt(3)**

**=243 sqrt(3) sq.unit.**

**Total Surface area of tetrehedron= 4x 81 sqrt(3)**

**=324 sqrt(3) sq.unit.**

**The volume of tetrahedron**= (1/3) x 81 sqrt(3)x 6 x sqrt(2)

`V=162sqrt(6)`