Area of Sphere Calc: Archimedes Method

In summary, the "Area of Sphere Calc: Archimedes Method" is a mathematical formula developed by Archimedes to calculate the surface area of a sphere. This method involves inscribing the sphere within a cylinder and measuring the areas of the base and curved surface of the cylinder. The formula for this method is A = 2πrh, where A is the area of the sphere, r is the radius of the sphere, and h is the height of the cylinder. While it may not be completely accurate, the accuracy of this method increases as the number of sides of the inscribed polygon increases. It is useful in situations where an accurate measurement of the surface area of a sphere is not necessary, such as in engineering or architecture projects
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mprm86
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Calculate the area of the surface of the sphere in the way Archimedes did.
 
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Google it.. google it hard
http://members.fortunecity.com/kokhuitan/archimedes.html [Broken]
 
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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.
 

What is the "Area of Sphere Calc: Archimedes Method"?

The "Area of Sphere Calc: Archimedes Method" is a mathematical formula developed by the ancient Greek mathematician Archimedes to calculate the surface area of a sphere.

How is the area of a sphere calculated using Archimedes Method?

To calculate the area of a sphere using Archimedes Method, the sphere is inscribed within a cylinder and the areas of the base and curved surface of the cylinder are measured. The area of the sphere is then estimated by dividing the curved surface area of the cylinder by the circumference of the cylinder's base.

What is the formula for calculating the area of a sphere using Archimedes Method?

The formula for calculating the area of a sphere using Archimedes Method is A = 2πrh, where A is the area of the sphere, r is the radius of the sphere, and h is the height of the cylinder.

How accurate is the "Area of Sphere Calc: Archimedes Method"?

The "Area of Sphere Calc: Archimedes Method" is an approximation and may not be completely accurate. However, as the number of sides of the inscribed polygon increases, the accuracy of the calculation also increases.

In what situations is the "Area of Sphere Calc: Archimedes Method" useful?

The "Area of Sphere Calc: Archimedes Method" is useful in situations where an accurate measurement of the surface area of a sphere is not necessary, such as in engineering or architecture projects. It is also useful for educational purposes to demonstrate the concept of approximation in geometry.

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