area of the base is equal to x^2
area of one of the triangles is 1/3 * x^2
total surface area of the pyramid is x^2 + 4 * 1/3 * x^2
you are given that the total surface area of the pyramid is equal to 432.
consequently, you get 432 = x^2 + 4 * 1/3 * x^2
combine like terms to get 432 = 7/3 * x^2
solve for x^2 to get x^2 = 432 * 3 / 7
solve for x to get x = sqrt(432 * 3/7)
you get x = 13.60672103
area of the base = x^2 = 185.1428571
eare of each triangle face = 1/3 * x^2 = 61.71428571.
areaof each of the triangle faces = 1/2 * x * h
that becomes 61.71428571 = 1/2 * x * h
solve for h to get h = 61.71428571 * 2 / x
you get h = 9.071147352
that's the height of each of the triangle faces.
the height of the pyramid is equal to the square root of (9.071147352^2 + (13.60672103/2)^2) = 6.
the volume of the pyramidis equal to x^2 * h / 3 = 185.1428571 * 6 / 3 = 370.2857143.
i confirme with a pyramid calculator that this is correct.
i used the calculator with side length = 13.60672103 and triangle face height = 9.041147352.
i also used the dame calculator with side length = 13.60672103 and pyramid height = 6.
i got the same answer, as i should have.
your solution is that the volume of the pyramid is e3qual to 370.2857143 cubic units.
here is the results of using the calculator with the height of the triangular face.
a = the lenght of a side of the square base.
s = the height of the triangular face.
here is the results of using the calculator with the height of the pyramid.
a = the length of a side of the square base.
h = the height of the pyramid.
here's some pictures that might help you visualize the pyramid.
h1 = the height of the pyramid = 6
h2 = the height of a triangular face = 9.041147352
x = the length of a side of the base = 13.60672103
the third diagram is a picure of the triangle formed by conneting the bottom of the height of the trianglular face to the bottom of the height of the pyramid.