Lesson Finding the distanse from a point in 3D to a plane

Finding the distanse from a point in 3D to a plane

Problem 1

Consider the plane of the triangle and the sphere of the radius R = 10 cm with the center at the point P.
This plane cuts the sphere and the triangle lies in/at the cutting section.

Now what you need is to find the radius "r" of the circle circumscribed about the triangle.  

Then the distance from the point P to the plane is 

d = .     (1)


For the radius of the circle circumscribed about the triangle, there is the formula

r = ,         (2)

where a, b, and c are the triangle's side dimensions and A is its area.

See the lesson Proof of the formula for the radius of the circumscribed circle in this site.


To calculate the area, use the Heron's formula

A =   (where s is semi-perimeter) =  =  = 9.922 cm^2 (approximately). 


Then according (2)  r =  = 3.024 cm (approximately).


Then the distance from P to the plane (which is the value under the question) is 

d =  = 9.532 cm.