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

Question

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

Ella Armstrong · Accepted Answer

Volume of pyramid #color(red)v= color(purple)(27)#

Three vertices' coordinates are A (6,3), B (4,7), and C (8,8). The pyramid's height is equal to 6.

#AB = c = sqrt((4-6)^2 + (7-3)^2) = 4.4721#

#BC = a = sqrt((8-4)^2 + (8-7)^2) = 4.1231#

#CA = b = sqrt ((6-8)^2 + (3-8)^2) = 5.3852#

Semi perimeter #s = (a+b+c)/2 = (4.1231 + 4.4721 + 5.3852) / 2 = 6.9902#

Area of base triangle #= Delta = sqrt (s (s- a) (s - b) (s - c))#

#Delta = #sqrt (6.9902 * (6.9902-4.1231) * (6.9902-4.4721) * (6.990-5.3852)) = #9

Volume of pyramid #v = (1/3) * Delta * h = (1/3) * 9 * 6 = #18

Caroline Aucoin · Answer

undefined To find the volume of the triangular pyramid, we first need to calculate the area of the base triangle and then use the formula for the volume of a pyramid, which is one-third of the product of the base area and the height.

Given the coordinates of the three vertices of the base triangle: (6, 3), (4, 7), and (8, 8), we can use the formula for the area of a triangle formed by three points in the coordinate plane.

The formula for the area of a triangle formed by three points (x1, y1), (x2, y2), and (x3, y3) is:

$$ Area = \frac{1}{2} |x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2)| $$

After calculating the area of the base triangle, we can use the formula for the volume of the pyramid:

$$ Volume = \frac{1}{3} \times \text{Base Area} \times \text{Height} $$

Substitute the calculated area of the base triangle and the given height (6) into this formula to find the volume of the pyramid.