Question

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

Answer 1

Volume of pyramid `V = color(red)(5` cubic units

Explanation:

First to find the area of the triangular base.

If three sides are known, area of the triangle is given by the formula

`A = sqrt((s )(s-a) (s-b) (s-c))`

where s is the semi perimeter of the triangular base, a,b and c the sides of the base.

Using distance formula we can find the sides.

`c = sqrt((4-3)^2 + (2-7)^2) = color(brown)(5.099`

`a = sqrt(5-3)^2 + (3-7)^2) = color (brown)(4.4721`

`b = sqrt((5-4)^2+(3-2)^2) = color(brown)(1.4142`

Semi perimeter `p = (a + b + c)/2 = (5.099 + 4.4721 + 1.4142)/2 = color(purple)(5.4919`

Area of triangular base `A = sqrt(5.4919 * (5.4919-4.4721) * (5.4919-5.099) * (5.4919-1.4142)) ~~ color(green)(3`

Volume of pyramid `V = (1/3) * A * h = (1/cancel3) * cancel3 * 5= color(red)(5` cu. units

Answer 2

`5`

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) 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)=(4,2),(x_2,y_2)=(3,7),(x_3,y_3)=(5,3)`

`A=1/2|4(7-3)+3(3-2)+5(2-7)|`

`color(white)(A)=1/2|16+3-25|=3`

`rArrV=1/3xx3xx5=5`