# The base of a triangular pyramid is a triangle with corners at (7 ,5 ), (6 ,9 ), and (3 ,4 ). If the pyramid has a height of 4 , what is the pyramid's volume?

Jun 9, 2018

color(crimson)("Volume of Pyramid " V_p = (1/3) * A_b * h = 11.36 " cubic units"

#### Explanation:

color(violet)("Volume of Pyramid " V_p = (1/3) * A_b * h

$A r e a o f b a s e \triangle \text{ A_b = sqrt(s (s-a) (s-b) (s-c)), } u \sin g H e r o n ' s f \mathmr{and} \mu l a$

$A \left(7 , 5\right) , B \left(6 , 9\right) , C \left(3 , 4\right) , h = 4$

$a = \sqrt{{\left(6 - 3\right)}^{2} + {\left(9 - 4\right)}^{2}} = 5.83$

$b = \sqrt{{\left(7 - 3\right)}^{2} + {\left(5 - 4\right)}^{2}} = 4.12$

$c = \sqrt{{\left(6 - 7\right)}^{2} + {\left(9 - 5\right)}^{2}} = 4.12$

It's an isosceles triangle with sides b & c equal.

$\text{Semi-perimeter } s = \frac{a + b + c}{2}$

$s = \frac{5.83 + 4.12 + 4.12}{2} \approx 7.04$

${A}_{b} = \sqrt{7.04 \cdot \left(7.04 - 5.83\right) \cdot \left(7.04 - 4.12\right) \cdot \left(7.04 - 4.12\right)} = 8.52$

color(crimson)("Volume of Pyramid " V_p = (1/3) * A_b * h = (1/3) * 8.52 * 4 = 11.36 " cubic units"