Question

The base of a triangular pyramid is a triangle with corners at `(6 ,2 )`, `(3 ,7 )`, and `(4 ,8 )`. If the pyramid has a height of `6`, what is the pyramid's volume?

Answer 1

Volume of the pyramid `color(blue)(= 8 cm^3`

Explanation:

Volume of a triangular pyramid v = (1/3) * base area * pyramid height.
Pyramid height = 6 cm
Coordinates of the triangular base `(6,2), (3,7), (4,8)`

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((3-6)^2 + (7-2)^2) = sqrt(9 +25) = 5.831`

`b =sqrt ((4-3)^2 + (8-7)^2) = sqrt 2 = 1.4142`

`c = sqrt((4-6)^2 + (8-2)^2) = sqrt40 = 6.3246`

`s = (5.831 + 1.4142 + 6.3246) / 2 = 6.7849`

`s-a = 6.7849 - 5.831 = 0.9539`
`s-b = 6.7849 - 1.4142 = 5.3707`
`s-c = 6.7849 - 6.3246 = 0.4603`

Area of base `= sqrt (6.7849*0.9539*5.3707*0.4603)`
Area of triangular base `= 4 cm^2`

Volume of pyramid `= (1/3)*4*6 = 8 cm^3`