The base of a triangular pyramid is a triangle with corners at (8 ,5 )(8,5), (6 ,2 )(6,2), and (5 ,9 )(5,9). If the pyramid has a height of 8 8, what is the pyramid's volume?

1 Answer

The volume is 22 2/32223

Explanation:

The volume is the area of the base multiplied by the height:

V = 1/3AhV=13Ah

Because we are given 3 points, the area is best computed using a determinant.

Here is a reference for the determinant that will help to compute the area given 3 points:

A = +-1/2|(x_1,y_1,1),(x_2,y_2,1),(x_3,y_3,1)|

Substituting in the 3 points:

A = +-1/2 |(8,5,1), (6,2,1), (5,9,1)|

If find it easier to demonstrate the evaluation of a 3xx3 determinant if the first two rows are repeated:

A = +-1/2 | (8,5,1,8,5), (6,2,1,6,2), (5,9,1,5,9) |

Multiply each of the major diagonals and add them:

A = +-1/2 | (color(red)(8),color(green)(5),color(blue)(1),8,5), (6,color(red)(2),color(green)(1),color(blue)(6),2), (5,9,color(red)(1),color(green)(5),color(blue)(9)) | =

color(red)((8)(2)(1)) + color(green)((5)(1)(5))+ color(blue)((1)(6)(9)) = 95

Multiply the minor diagonals and subtract them from the sum of the major diagonals:

A = +-1/2 | (8,5,color(red)(1),color(green)(8),color(blue)(5)), (6,color(red)(2),color(green)(1),color(blue)(6),2), (color(red)(5),color(green)(9),color(blue)(1),5,9) | =

95 - color(red)((1)(2)(5)) - color(green)((8)(1)(9)) - color(blue)((5)(6)(1)) = -17

Because the area cannot be negative, we choose the negative value for the +-1/2

A = -1/2(-17)

A = 17/2

Using the h = 8 we can compute the volume:

V = 1/3xx17/2xx8

V = 68/3=22 2/3