Surface Area Formula

(1) Proposition (Surface Area) Assume that the function has continuous partial derivatives and in a region of the -plane. Then the portion of the surface that lies over has surface area

(2) Example (Surface Area)  Find the surface area of the part of the surface that lies above the triangular region in the with vertices and

    Solution. The region is described by



We have











(3) Example (Surface Area)  Find the surface area of the part of the paraboloid that lies under the plane
    
    Solution. The plane intersects the paraboloid in the circle Therefore, the given surface lies above the disk with center the origin and radius 3. Converting to polar coordinates, we have,

    It is worth noting the following similarities.

(4) Example (Computing Surface Area) Find the surface area of the portion of the sphere that lies inside the cylinder
    
    Solution. Let Then  




and



The projected region in polar form is Since half the surface is above the and half the surface is below, we have

(5) Example (Surface Area of a Sphere) Find the surface area of a sphere of radius   

    Solution. Let  



Then





and



In polar form, we have

(6) Example (Surface Area of a Cylinder) Find the surface area of a cylinder of radius and height
    
    Solution. Let



Then





and



We have,

(7) Proposition (Surface Area Defined Parametrically) Let be a surface defined parametrically by



on the region in the -plane, and assume that is smooth in the sense that and are continuous with on Then the surface area, is given by



The quantity is called the fundamental cross product.

(8) Example (Surface Area Defined Parametrically) Find the area of the surface given parametrically by the equation



for
    
    Solution. We have,  



Therefore,


Using polar coordinates, the surface area is











(9) Example (Surface Area Defined Parametrically) Find the surface area of the helicoid which is given parametrically by  



for and

    Solution. We have,





and



Therefore,



So the surface area is

(10) Example (Surface Area of a Torus)  Find the surface area of the torus which is given parametrically by  



for and
    
    Solution. We have,





and





Therefore,



So the surface area is

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Cite this as:
Surface Area Formula
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
http://www.libraryofmath.com/surface-area-formula.html