The base of a triangular pyramid is a triangle with corners at (8 ,5 ), (7 ,2 ), and (4 ,6 ). If the pyramid has a height of 7 , what is the pyramid's volume?

1 Answer
Jul 30, 2018

color(green)(V = 1/3 A_b h = (1/3) * 7.1171 * 7 = 16.6065, " cubic units"

Explanation:

Given : A (8,5), B (7,2), C (4,6), h = 7#

Using distance formula we can calculate the lengths of sides a, b, c.

d = sqrt((x_2-x_1)^2 + (y_2-y_1)^2)

a = sqrt((7-4)^2+(2-6)^2) = 5

b = sqrt((4-8)^2+(6-5)^2) = sqrt 17

c = sqrt((8-7)^2+(5-2)^2) = sqrt 10

Semi perimeter of base triangle s = (a+b+c)/2

s = (5 + sqrt 17 + sqrt 10)/2 = 12.2854/2 ~~ 6.2427

Area of triangle A_t = sqrt(s (s-a) (s-b) (s-c))

A_t = sqrt(6.2427 * (6.2427-5)*(6.2427- sqrt 17)*(6.2427-sqrt 10))

#A_t ~~ 7.1171

Volume of pyramid V = (1/3) * A_t * h

color(green)(V = (1/3) * 7.1171 * 7 = 16.6065, " cubic units"