Find the base, lateral area, surface area and volume of the tetrahedron below:

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pramodpandey | College Teacher | (Level 3) Valedictorian

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