Analytical Geometry - Equation Of Sphere Center At M And Radius R In Rectangular Coordinates

 
   
     
       

Analytic Geometry – Sphere

       

Equation of a Sphere (Center M and Radius R)

       

          A sphere in 3D space is the set of all points that are at a fixed distance (radius) from a central point (center).        

     
     
       
         
Standard Equation of a Sphere:
         

            If the center of the sphere is at \( M(x_0, y_0, z_0) \) and radius \( R \), then the equation is:          

         

            \[             (x - x_0)^2 + (y - y_0)^2 + (z - z_0)^2 = R^2             \]          

         

            Equation of Sphere          

         

Key Properties of a Sphere:

         
               
  • It is a perfectly symmetrical 3D object.
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  • Every point on the surface is equidistant from the center.
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  • The cross-section through the center is always a circle.
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  • Surface Area: \( A = 4\pi R^2 \)
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  • Volume: \( V = \frac{4}{3}\pi R^3 \)
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Applications of Spheres:

         
               
  • Used in physics to model atoms, bubbles, or planets.
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  • 3D graphics and animation (collision detection, bounding spheres).
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  • Used in CAD systems and engineering designs involving balls or domes.
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  • Navigation and geodesy (Earth modeled as a sphere).
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