Question

The base of a triangular pyramid is a triangle with corners at `(6 ,7 )`, `(2 ,5 )`, and `(3 ,1 )`. If the pyramid has a height of `4`, what is the pyramid's volume?

Answer 1

Volume of a pyramid is `12` cubic.unit.

Explanation:

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

`(x_1,y_1)=(6,7) ,(x_2,y_2)=(2,5),(x_3,y_3)=(3,1) , h=4`

Area of Triangle is

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

`A_b = |1/2(6(5−1)+2(1−7)+3(7−5))|` or

`A_b = |1/2(24-12+6)| = | 9| =9` sq.unit.

Volume of a pyramid is `1/3*A_b*h = 1/3 *9*4 = 12` cubic.unit [Ans]

Answer 2

`"volume "=12`

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 (A) of the triangle can be found 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)=(6,7),(x_2,y_2)=(2,5),(x_3,y_3)=(3,1)`

`A=1/2|6(5-1)+2(1-7)+3(7-5)|`

`color(white)(A)=1/2|24-12+6|=1/2|18|=9`

`rArrV=1/3xx9xx4=12`