The base of a triangular pyramid is a triangle with corners at (2 ,6 )(2,6), (5 ,2 )(5,2), and (8 ,7 )(8,7). If the pyramid has a height of 18 18, what is the pyramid's volume?

1 Answer
sankarankalyanam
Dec 6, 2017

Volume of pyramid = color (blue)(81.03)=81.03 cm^3cm3

Explanation:

Volume of a triangular pyramid v = (1/3) * base area * pyramid height.
Pyramid height = 18 cm
Coordinates of the triangular base (2,6), (5,2), (8,7)(2,6),(5,2),(8,7)

Area of triangular base = sqrt(s (s-a) (s-b) (s-c))√s(s−a)(s−b)(s−c)
where a, b, c are the three sides of the triangular base and s is the semi perimeter of the base
s = (a+b+c)/2s=a+b+c2

To find triangle sides :

a = sqrt((5-2)^2 + (2-6)^2) = sqrt(9 +16) = 5a=√(5−2)2+(2−6)2=√9+16=5

b =sqrt ((8-5)^2 + (7-2)^2) = sqrt 34 = 5.831b=√(8−5)2+(7−2)2=√34=5.831

c = sqrt((7-6)^2 + (8-2)^2) = sqrt37 = 6.0828c=√(7−6)2+(8−2)2=√37=6.0828

s = (5 + 5.831 + 6.0828) / 2 = 8.4569s=5+5.831+6.08282=8.4569

s-a = 8.4579 - 5 = 3.4579s−a=8.4579−5=3.4579
s-b = 8.4579 - 5.831 = 2.6259s−b=8.4579−5.831=2.6259
s-c = 8.4579 - 6.0828 = 2.3751s−c=8.4579−6.0828=2.3751

Area of base = sqrt (8.4569*3.4579*2.6259*2.3751)=√8.4569⋅3.4579⋅2.6259⋅2.3751
Area of triangular base = 13.505 cm^2=13.505cm2

Volume of pyramid = (1/3)*13.505*18 = 81.03 cm^3=(13)⋅13.505⋅18=81.03cm3

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