Question

The base of a triangular pyramid is a triangle with corners at `(2 ,6 )`, `(5 ,2 )`, and `(8 ,7 )`. If the pyramid has a height of `18`, what is the pyramid's volume?

Answer 1

Volume of pyramid `= color (blue)(81.03)` `cm^3`

Explanation:

Volume of a triangular pyramid v = (1/3) * base area * pyramid height.
Pyramid height = 18 cm
Coordinates of the triangular base `(2,6), (5,2), (8,7)`

Area of triangular base = `sqrt(s (s-a) (s-b) (s-c))`
where a, b, c are the three sides of the triangular base and s is the semi perimeter of the base
`s = (a+b+c)/2`

To find triangle sides :

`a = sqrt((5-2)^2 + (2-6)^2) = sqrt(9 +16) = 5`

`b =sqrt ((8-5)^2 + (7-2)^2) = sqrt 34 = 5.831`

`c = sqrt((7-6)^2 + (8-2)^2) = sqrt37 = 6.0828`

`s = (5 + 5.831 + 6.0828) / 2 = 8.4569`

`s-a = 8.4579 - 5 = 3.4579`
`s-b = 8.4579 - 5.831 = 2.6259`
`s-c = 8.4579 - 6.0828 = 2.3751`

Area of base `= sqrt (8.4569*3.4579*2.6259*2.3751)`
Area of triangular base `= 13.505 cm^2`

Volume of pyramid `= (1/3)*13.505*18 = 81.03 cm^3`