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

1 Answer
Sep 7, 2016

Volume of pyramid is 1/3xx6xx8.0004=16.0008

Explanation:

Find the distance between corners should give us three sides and using Heron's formula we can then find area of base triangle.

Delta=sqrt(s(s-a)(s-b)(s-c)), where s=1/2(a+b+c), where sides of a triangle are a, b and c.

Then area can be multiplied by height and divided by 3, which will give us volume of pyramid.

The sides of triangle formed by (6,2), (4,5) and (8,7) are

a=sqrt((4-6)^2+(5-2)^2)=sqrt(4+9)=sqrt13=3.6056

b=sqrt((8-4)^2+(7-5)^2)=sqrt(16+4)=sqrt20=4.4721 and

c=sqrt((8-6)^2+(7-2)^2)=sqrt(4+25)=sqrt29=5.3852

Hence s=1/2(3.6056+4.4721+5.3852)=1/2xx13.4629=6.7315

and Delta=sqrt(6.7315xx(6.7315-3.6056)xx(6.7315-4.4721)xx(6.7315-5.3852)

= sqrt(6.7315xx3.1259xx2.2594xx1.3463)=sqrt64.0062=8.0004

Hence volume of pyramid is 1/3xx6xx8.0004=16.0008