The base of a triangular pyramid is a triangle with corners at (6 ,2 ), (3 ,7 ), and (4 ,8 ). If the pyramid has a height of 6 , what is the pyramid's volume?

1 Answer
sankarankalyanam
Dec 6, 2017

Volume of the pyramid color(blue)(= 8 cm^3

Explanation:

Volume of a triangular pyramid v = (1/3) * base area * pyramid height.
Pyramid height = 6 cm
Coordinates of the triangular base (6,2), (3,7), (4,8)

Area of triangular base = sqrt(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)/2

To find triangle sides :

a = sqrt((3-6)^2 + (7-2)^2) = sqrt(9 +25) = 5.831

b =sqrt ((4-3)^2 + (8-7)^2) = sqrt 2 = 1.4142

c = sqrt((4-6)^2 + (8-2)^2) = sqrt40 = 6.3246

s = (5.831 + 1.4142 + 6.3246) / 2 = 6.7849

s-a = 6.7849 - 5.831 = 0.9539
s-b = 6.7849 - 1.4142 = 5.3707
s-c = 6.7849 - 6.3246 = 0.4603

Area of base = sqrt (6.7849*0.9539*5.3707*0.4603)
Area of triangular base = 4 cm^2

Volume of pyramid = (1/3)*4*6 = 8 cm^3

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