Geometry: What is the formula to calculate the area of a flat surface of a sphere?

There is no formula, because there is no flat surface.  A sphere is curved everywhere. Edit: It seems the asker now wants to know about an imperfect sphere (for example Earth, or a steel ball that would fit on a desk).  To estimate the flat area of these objects -- and even defining "flat" is nontrivial for real physical things -- would require a lot of data and detailed, computer-assisted calculation.  Again there is no general formula. One more addendum to clear up a tangential (pun not intended) misconception: If somehow you set a perfect sphere (made of hypothetical infinitely small atoms) on a perfectly flat desk, it would not roll even though no part of the sphere is flat, assuming the only force on the system is Earth's gravity acting perpendicular to the table.

I think the question is related to the idea of regular polyhedron.  If a regular polyhedron has enough tiny flat surface, maybe hundreds of thousand, it can be regarded as a sphere.  A sphere basically does not have any flat area on it unless there is a huge pressure forcing it to become a combination of flat and curve. It is hard to give a exact formula to calculate the flat since it is hard to describe the shape of this flat area -- It might be a small circular or just irregular.  I wonder if the differential method can be applied.

There will be an empirical formula. The area would depend on the type of sphere material, surface material and possibly gravity. If the material is made up of rubber, then surely the area in contact will be more. But in case the material is made up of graphite, then surely the area is going to be less. The above paragraph is applicable for real life spheres. However, for a theoretical/ mathematical sphere, theoretically, it touches the surface exactly at a point. The first sphere is a real life sphere with a real surface area in contact. Second one is mathematical/ideal/theoretical sphere with just a point in contact. Thanks for the correction

"Contact" is electrostatic repulsion between the outer electrons of the atoms at the surface. In the inside, it's backed up by internal electrostatic repulsion. Electrons stay in place because of the energy of their orbitals and the Pauli exclusion principle. You don't get beyond that unless you're in a neutron star or something. So even if you say the ball is perfectly spherical, there's going to be some bit of "skooshiness" at its surface. The contact area isn't well defined. A reasonable heuristic, probably good enough for most purposes, would be to take the average size of the atoms that make up the ball and describe a circle such that at the circumference the distance between the ideal sphere and ideal surface is about the size of one of these atoms and zero at the surface. That would, of course, require knowing also the radius of the ball.

Your question refers to a "flat area" which confuses me. If we really were considering a perfect sphere, there would be no flat area; it would be equally curved throughout. If we brought this perfect sphere in contact with a perfectly flat surface, they would touch at precisely one point. (Or, if you like, we might say they would touch throughout an infinitesimal neighborhood of one point, the infinitesimal deviations from that point having zero squared magnitude.).

"Infinitesimal Calculus" deals with problems like this http://www.google.com/url?sa=t&rct=j&q=&esrc=s&frm=1&source=web&cd=6&ved=0CFkQFjAF&url=http%3A%2F%2Fhomepage.math.uiowa.edu%2F%7Estroyan%2FInfsmlCalculus%2FFoundInfsmlCalc.pdf&ei=lceIU-vTNuPfsATyloHIAg&usg=AFQjCNHc47ihYoKxNQ7MReD5LkRAPm5QtA&sig2=mgAeLfXgl0UXYuRDS_ZAag

The flat area of sphere you are talking about is observed in spheres made of materials that yield under pressure.  That is under pressure near the point of contact, the materials deform and become "flatter" and the flatness stops when an equilibrium is reached. A giant rigid smooth sphere as large as earth "Will not" have any flat surface - because its not being subjected to contact pressure. On earth - every object is subjected to strong gravity and deformation occurs. Interestingly it's not the sphere alone  that would deform, its the contact surface too. Now, the physics of yielding and shearing are not so simple. There are various formula for simple shaped objects (like linear rod , cylinders) but it gets really complicated when you apply it to real life objects : sphere, material properties. Spheres of Water, wet clay would yield more and be more flatter (they would hardly be sphere anymore) Spheres of dough, wet earth, rubber would yield a little and be somewhat flatter Spheres of materials like iron, metal etc would be very rigid and practically have point contact for calculations. However, a rubber sphere on a surface made of dough will not yield much (instead the dough would deform and) and sphere as such wouldn't get flatter  - the surface gets curved (and still increases contact area) A rubber sphere  placed on a hot metal surface would yield much more (as the contact melts a little) and have larger contact area A rubber sphere placed on another rigid surface for long long time would slightly yield more have more contact area. To summarise: flatness (or contact area of sphere on a surface) depends on "material of sphere, weight of sphere (which in turn depends on gravity), material of contact, temperature and duration of contact" Unfortunately , all of these complex interactions make it tough to derive a simple formula for yield and area of contact. May be you can work through basic principles and make a simple approximation model In practice you need complex computational model [basic principles applied to tiny pieces and then integrated over complex shape] to determine the area of contact (flatness) accurately (almost). I'll try to find a simple approximate equation and reply back if I can formulate such an equation. ----------------------- Some common everyday examples of yielding behavior of spheres on contact:  Rain drops are spherical while falling down down but loose their shape and ultimately become flat on earth / hard surface - they yield way too easily. Tyres - the safety of tyre is based on the principle that tyres would yield under pressure, get a larger contact area and when brakes are applied would stop due to higher friction (because of larger contact area).   You might have noticed this - (much easily if it's a bicycle - because you need to pump your muscles) - that driving/riding becomes so easy with tyres with higher pressure - you feel it so smooth and fast and easy (because of less contact area with roads, lesser friction). However, deflated tyres easily yield under pressure (weight of you + vehicle), have more friction and harder to move - but easier to stop. The same principle works in having higher safety speed with increased load on a car (say on a highway, you can safely drive upto 120-140 km when it's full of 5 people). But it becomes very unsafe when you drive at same speed alone - lesser weight => lesser yield => lesser contact area => lesser friction => difficult to stop. Frictional force f=μ∗N f = \mu*N which you read in books are simple approximations to real life situation: it depends on the van der Waals force between the two objects in contact and varies with applied pressure, duration of contact, area of contact , temperature and molecular interaction between the materials in contact -- all those factor is conveniently characterized as μ\mu for simple approximately correct newtonian model of a system of objects.

In maths? A point. In the real world, they never actually touch.

Sorry but this begs the question "Depends on what you mean by a flat surface."  If you mean one which could be extended to infinity in any of its 4 directions, then we have an ideal which could never exist in reality: However if the surface is e.g. the top of the typical dining room tablem then the table is a tangent plane to the sphere, cutting it in the points of a circle radius 0.  There is fact no such thing as a flat surface of a sphere, since any three points on the surface ( the minimum to define a surface, or plane m) must all be joined by curved lines: Again as the point to point distances tend to zero, this will become a point, not a pllane, which by euclid's basic axiom has no size but marks position