Area of Sphere Calc: Archimedes Method

[mprm86]

Calculate the area of the surface of the sphere in the way Archimedes did.

 

[cronxeh]

Google it.. google it hard
http://members.fortunecity.com/kokhuitan/archimedes.html

 

[M98Ranger]

Archimedes' method for calculating the area of a sphere is a geometric approach that involves inscribing a regular polyhedron inside the sphere and then circumscribing another regular polyhedron outside the sphere. By finding the surface area of these two polyhedrons, we can approximate the surface area of the sphere.

To begin, let us consider a sphere with radius r. We can inscribe a regular polyhedron inside the sphere, such as a cube, with each face tangent to the sphere. The surface area of this cube would be 6r^2, as each face has an area of r^2.

Next, we circumscribe a regular polyhedron outside the sphere, such as a dodecahedron, with each vertex touching the sphere. The surface area of this dodecahedron can be found by multiplying the number of faces (12) by the area of each face, which is given by (3√25+10)r^2. This simplifies to 15√3r^2.

Now, let us consider the ratio of the surface area of the inscribed cube to the circumscribed dodecahedron. This ratio can be expressed as 6r^2/15√3r^2, which simplifies to 2/√3.

According to Archimedes, this ratio is also equal to the ratio of the surface area of the sphere to the surface area of the circumscribed dodecahedron. Therefore, we can find the surface area of the sphere by multiplying the surface area of the circumscribed dodecahedron by 2/√3. This gives us an approximation of the surface area of the sphere as (15√3r^2)(2/√3) = 10√3r^2.

This method may seem complex, but it is a clever way to approximate the surface area of a sphere using geometric principles. While modern methods of calculus and integration can give us a more precise value, Archimedes' method is a testament to his mathematical ingenuity and continues to be a valuable tool in understanding the properties of spheres.