A solid oblique pyramid has an equilateral triangle as a base with an edge length of 4 cm and an area of 12 cm2. What is the volume of the pyramid?

Solution:Let me define pyramid first.A pyramid is a Polyhedron , whose base  can be any polygon and it's faces are triangular which meet at a point called vertex.Volume of Pyramid =  [tex]\frac{1}{2}\times{\text{Area of Base}\times{\text{Height}[/tex]Side of base which is an equilateral triangle= 4 cmArea of whole pyramid = 12 square cmArea of Equilateral triangle having side 4 cm=[tex]\frac{\sqrt3}{4}\times(Side)^2=\frac{\sqrt3}{4}\times(4)^2=4 \sqrt3[/tex] cm²So, Area of three triangle which are faces of pyramid = (12 - 4 √3)cm²Area of triangle = [tex]\frac{1}{2} \times (Base)\times(Height)[/tex]12 - 4 √3= [tex]\frac{1}{2} \times (4)\times(Height)[/tex]Height =[tex]\frac{12-4\sqrt3}{2}[/tex] Height=(6-2√3) cmVolume of pyramid = Area of Equilateral triangle which is it's Base × Height of Pyramid           =4√3 × (6-2√3) cm³            =(24 √3 -24)            = 24 (1.732-1)           =24 × 0.732           =17.568 cm³