# 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?

Feb 18, 2018

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

$A = \sqrt{\left(s\right) \left(s - a\right) \left(s - b\right) \left(s - c\right)}$

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

Feb 18, 2018

$5$

#### Explanation:

$\text{the volume (V) of a pyramid is calculated using the formula}$

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

$\text{the area of the base (A) can be found using}$

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

$\text{let } \left({x}_{1} , {y}_{1}\right) = \left(4 , 2\right) , \left({x}_{2} , {y}_{2}\right) = \left(3 , 7\right) , \left({x}_{3} , {y}_{3}\right) = \left(5 , 3\right)$

$A = \frac{1}{2} | 4 \left(7 - 3\right) + 3 \left(3 - 2\right) + 5 \left(2 - 7\right) |$

$\textcolor{w h i t e}{A} = \frac{1}{2} | 16 + 3 - 25 | = 3$

$\Rightarrow V = \frac{1}{3} \times 3 \times 5 = 5$