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

1 Answer
Mar 2, 2016

V ~~3.35

Explanation:

The volume of a pyramid is given by V = 1/3*Base*Height
so the first step is to find the area of the base.

The simplest way to do that, given the three vertices, is to find the lengths of the three sides and use the area formula A = sqrt(s(s-a)(s-b)(s-c)) where s is the semi-perimeter and a,b and c are the three sides of the base.
Pyramid basePyramid base

a^2 = (4-2)^2 + (7-4)^2 4+9 = 13
a =sqrt(13) ~~3.6

b^2 = (6-4)^2 + (7-5)^2 =4 4 = 8
b = 2sqrt(2) ~~2.83

c = (6-2)^2 + (5-4)^2 = 16+1 = 17
c = sqrt(17) ~~4.12

s = (sqrt(13) + 2sqrt(2) + sqrt(17))/2 ~~5.28

A = sqrt(5.28(5.28 - 3.6)(5.28-2.83)(5.28-4.12))~~5.02

Then the volume is V = (5.02*2)/3 ~~3.35