Question

The base of a triangular pyramid is a triangle with corners at `(1 ,2 )`, `(9 ,5 )`, and `(4 ,7 )`. If the pyramid has a height of `9`, what is the pyramid's volume?

Answer 1

Volume of a pyramid is `46.5` cubic.unit.

Explanation:

Volume of a pyramid is `1/3*`base area `*`hight.

`(x_1,y_1)=(1 ,2) ,(x_2,y_2)=(9,5),(x_3,y_3)=(4,7) , h=9`

Area of Triangle is `A_b = |1/2(x1(y2−y3)+x2(y3−y1)+x3(y1−y2))|`

`A_b = |1/2(1(5−7)+9(7−2)+4(2−5))|` or

`A_b = |1/2(-2+45-12)| = | 31/2| =31/2`sq.unit

Volume of a pyramid is `1/3*A_b*h = 1/3 *31/2*9 = 46.5`

cubic.unit [Ans]

Answer 2

`"volume "=93/2`

Explanation:

`"the volume (V) of a pyramid is calculated using"`
`"the formula"`

`•color(white)(x)V=1/3xx"area of base "xx"height"`

`"the area of the base (A) is calculated using"`

`•color(white)(x)A=1/2|x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)|`

`"let "(x_1,y_1)=(1,2),(x_2,y_2)=(9,5),(x_3,y_3)=(4,7)`

`A=1/2|1(5-7)+9(7-2)+4(2-5)|`

`color(white)(A)=1/2|-2+45-12|=31/2`

`rArrV=1/3xx31/2xx9=93/2`