Question

The base of a triangular pyramid is a triangle with corners at `(1 ,2 )`, `(5 ,2 )`, and `(7 ,5 )`. If the pyramid has a height of `7`, what is the pyramid's volume?

Answer 1

Volume `=14 " "`cubic units

Explanation:

We compute for the area of the base which is a triangle first
Let `P_1(x_1, y_1)=(1, 2)`
Let `P_2(x_2, y_2)=(5, 2)`
Let `P_3(x_3, y_3)=(7, 5)`

Use the formula to compute for the area of the triangular base

Area `=1/2[(x_1,x_2,x_3,x_1),(y_1,y_2,y_3,y_1)]=1/2[(1,5,7,1),(2,2,5,2)]`

Area `=1/2*(x_1*y_2+x_2*y_3+x_3*y_1-x_2*y_1-x_3*y_2-x_1*y_3)`

Area `=1/2*(2+25+14-10-14-5)`

Area `=6`

Volume `=1/3*Area*height`

Volume `=1/3*6*7`

Volume `=14 " "`cubic units

Answer 2

`14" units"^3`

Explanation:

Area of the base is `b/2xxa" "->" " 2xx3 = 6 " units"^2`

The volume of a triangular pyramid is:

`"base area" xx 1/3 xx "height"`

The height is given as 7

`color(blue)("base area" xx 1/3 xx "height " ->" " 6xx1/3xx7 =14 " units"^3`