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Everything about the circle Definition, tests to calculate the area, approximations to the number pi, formulas [Home] [Search] [Forum] [Links] [Contact] [Imprint] [Sitemap]
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One of the biggest Greek problems occupied itself with trying to figure out the area of a circle. The are of a circle is the first and easiest area, that does not have steight sides/borders. All straight sided figures let themselves be segmented into triangles which can then be calculated!
Definition of circles A circle is the amount of all points that have the same distance from the center point. This distance is defined as the radius (r).
Test, calculate the area of a circle with the radius = 1 cm. First test
The circle is divided into eight equally large “pie
pieces” and lie these together such a way, that they would vaguely become
the form of a parallelogram (the curves are ignored).
The area can only be calculated with the formula of a parallelogram, A = g ∙ h (A = the half circle [p] · 1). To become a more exact result, you can divide the circle into infinity of sections and then make them a parallelogram. The bigger the number of sections of “pie pieces” becomes, the more exact the result becomes but will never be truly exact. There are also other predictions of how to calculate the area of a circle.
2. Second test A square is put around the square:
The side length of the inside square is
Useing figures with many sides (hexagon, octagon, decagon,...), the area can be calculated more exactly. There is one more posibility to figure out the area of a circle that is much better than the two that have already been reviewed. With this methode, it can be calculated much more elaboratly. 3. Third test
A circle is cut into strips, that all have the same
width. Only the strips of one fourth of the circle will be investigated. In
the example there are four. Consequently, every rectangle is
The theorem of Pythagoras can be used for every rectangle in the fourth of the circle:
A(R1) =
A(R2) =
A(R3) =
According to this, the area of a circle with the radius r = 1 is:
A circle =
A circle =
A circle =
A circle =
A circle =
A circle = 2,495709068
Formula for p (p is actually the half circle, but in the unit circle it = the area) Now, the decomposition of these four strips is not a very exact matter. The circle can also be cut into several strips, 100 for example. Of 100 strips, 99 can be calculated, and from 1000, 999 can be calculated, so on and so forth. But a certain number can not be named, but any number n. A formula can now be made from it (the radius of a circle is 1, but it will firstly be enlarged to the number of strips):
A circle =
Formulas that can be used for the circle Perimeter (U): U = p · d = 2 p r Area (A): A = p · r˛ The area of a section of a circle AS with the center point angle a:
AS =
For the associated length of an arc of circle (b) gilts:
b = |
