Question

The base of a triangular pyramid is a triangle with corners at `(6 ,7 )`, `(4 ,5 )`, and `(8 ,7 )`. If the pyramid has a height of `6`, what is the pyramid's volume?

Answer 1

`4`

Explanation:

The area of a triangle with vertices `(x_1, y_1)`, `(x_2, y_2)`, `(x_3, y_3)` is given by the formula:

`A = 1/2 abs(x_1y_2+x_2y_3+x_3y_1-x_1y_3-x_2y_1-x_3y_2)`

Letting `(x_1, y_1) = (6, 7)`, `(x_2, y_2) = (4, 5)` and `(x_3, y_3) = (8, 7)` we find that the area of the base of the given pyramid is:

`1/2abs(color(blue)(6) * color(blue)(5)+color(blue)(4) * color(blue)(7) + color(blue)(8) * color(blue)(7) - color(blue)(6) * color(blue)(7) - color(blue)(4) * color(blue)(7) - color(blue)(8) * color(blue)(5))`

`=1/2abs(30+28+56-42-28-40) = 2`

Another way of seeing this is by considering the points `(6, 7)` and `(8, 7)` as the base of the triangle, which is of length `2`. Then the apex of the triangle at `(4, 5)` is at height `2` above the base. Then the area of the triangle is:

`1/2 * "base" * "height" = 1/2 * color(blue)(2) * color(blue)(2) = 2`

Then the volume of a pyramid is:

`1/3 * "base" * "height" = 1/3 * color(blue)(2) * color(blue)(6) = 4`