Question

The base of a triangular pyramid is a triangle with corners at `(3 ,1 )`, `(2 ,7 )`, and `(3 ,6 )`. If the pyramid has a height of `4`, what is the pyramid's volume?

Answer 1

Volume of Pyramid `~~ 3.3333` cubic units.

Explanation:

Using distance formula and Heron’s formula we can calculate the base Area of the pyramid. Multiplying it by one third the height will give the volume.

Distance formula `d = sqrt((x_2-x_1)^2 + (y_2 - y_1)^2)`

Heron’s formula `A_t = sqrt(s * (s-a) * (s-b) * (s-c))`

Where s is the semiperemter of the triangle and a, b, c are the three sides of the base triangle.

Let `A (3,1), B (2,7) C (3,6)`

`a = sqrt((2-3)^2 + (7-6)^2) = 1.4142`

`b = sqrt((3-3)^2 + (6-1)^2) = 5`

`c = sqrt((3-2)^2 + (1-7)^2) = 6.0828`

Semiperemeter `s = (a+b+c)/2 = 12.497/2 = 6.2485`

`A_t = sqrt(6.2485 * (6.2485-1.4142) * (6.2485-5) * (6.2485-6.0828))`

`A_t ~~ 2.5` sq. units

Volume of pyramid `V_p =(1/3) * 2.5 * 4 ~~ 3.3333` cubic units.