The base of a triangular pyramid is a triangle with corners at (5 ,8 ), (6 ,7 ), and (2 ,3 ). If the pyramid has a height of 2 , what is the pyramid's volume?

1 Answer
May 21, 2018

color(maroon)(V = (1/3) * A_t * h = 2.65 cubic units

Explanation:

Given : A (5,8), B (6,7), C (2,3), h = 2#

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((6-2)^2+(7-3)^2) = sqrt32 = 5.66

b = sqrt((5-2)^2+(8-3)^2) = sqrt34 = 5.83

c = sqrt((5-6)^2+(8-7)^2) = sqrt2 = 1.41

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

s = (5.66 + 5.83 + 1.41)/2 = 12.9/2 = 6.45

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

A_t = sqrt(6.45 * (6.45-5.66)*(6.45-5.83)*(6.45-1.41))

A_t = 3.97

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

color(maroon)(V = (1/3) * 3.97 * 2 = 2.65 cubic units