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Circumference

Lecture






With an unlimited increase in the sides of a regular polygon, its perimeter approaches the perimeter of the circle.

  Circumference

Theorem

The ratio of the circumference of a circle to its radius does not depend on the circle.

Evidence.

Take two arbitrary circles with radii R1 and R2 and lengths l1 and l2. Let's pretend that

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We write in the circle the correct n-gons. N is so large that the perimeters p1 and p2 of regular polygons approach the lengths of circles l1 and l2. Therefore, we replace the circumference lengths by the perimeters, then

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But the perimeters of regular convex n-gons are referred to as the radii of the circumscribed circles:

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That contradicts the assumption. The theorem is proved.

The ratio of the circumference to diameter is denoted by the Greek letter π.

  Circumference

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Planometry

Terms: Planometry