Question

How can I find the missing length of this triangular prism that's inside a sphere?

The triangular prism is inside a sphere, the sphere's surface area is 385.3475938, the volume is 711.2999702. I calculated each side of the triangle already and the opposite is 8.499996983, the adjacent is 7.100000001, and the hypotenuse is 11.0751952. The two angles of the triangle are 39.87181897 and 50.12818103. When I visualize the triangle inside the sphere, I think about each side of the triangle but I am not sure how to calculate length though I know the other 3 sides of the triangular prism already.

Answer 1

The triangular prism basis has a circumcircle which is the sphere's meridian. So the prism height is null.

Explanation:

The triangular prism basis has a circumcircle which is the sphere's meridian. So the prism height is null.

Calling `a,b,c` the base sides,

`a = 8.499996983;`
`b = 7.100000001;`
`c = 11.0751952;`

they verify

`a^2+b^2=c^2`

The basis circumclicle is the circle

`(x-x_0)^2+(y-y_0)^2=r^2`

passing on the points

`p_1={0,0}`
`p_2={a,0}`
`p_3={0,b}`

This condition gives the equations

#{ (x_0^2 + y_0^2 = r^2), (a^2 - 2 a x_0 + x_0^2 + y_0^2 = r^2), (b^2 + x_0^2 - 2 b y_0 + y_0^2 = r^2) :}#

Solving for `x_0,y_0,r` gives

`x_0 = 4.25, y_0 = 3.55, r = 5.5376`

but the sphere volume data says that

`V = 4/3 pi R^3 = 711.2999702`

so

`R = 5.5376 = r`