File:FS HR dia.png

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Summary

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Description
English: Largest right triangle in a semi-circle
Deutsch: Größtes rechtwinkliges Dreieck in einem Halbkreis
Date
Source Own work
Author Hans G. Oberlack

The semicircle as base element with given radius
. Inscribed is the largest right triangle.

General case

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Segments in the general case

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0) Radius of the semicircle:
1) Side length of the right triangle: , see Calculation 1

Perimeters in the general case

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0) Perimeter of base semicircle:
1) Perimeter of inscribed right triangle:
S) Sum of perimeters:

Areas in the general case

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0) Area of the base semicircle
1) Area of the inscribed right triangle

Centroids in the general case

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0) By definition the centroid point of a base shape is
1) The centroid of the inscribed right triangle relative to the base centroid is: , see Calculation 2


Normalised case

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In the normalised case the area of the base semicircle is set to 1.
So

Segments in the normalised case

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0) Radius of the base semicircle
1) Side length of inscribed right triangle

Perimeter in the normalised case

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0) Perimeter of base semicircle:
1) Perimeter of inscribed right triangle:
S) Sum of perimeters:

Areas in the normalised case

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0) Area of the base semicircle is by definition
1) Area of the inscribed right triangle

Centroids in the normalised case

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0)
1)

Distances of centroids

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The distance between the centroid of the base semicircle and the centroid of the circle is:

Sum of distances:

Identifying number

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Apart of the base element there is only one shape allocated. Therefore the integer part of the identifying number is 1.
The decimal part of the identifying number is the decimal part of the sum of the perimeters and the distances of the centroids in the normalised case.



So the identifying number is:


Calculations

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Known elements

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(0) Given is the radius of the base semicircle
(1)
(2)
(3)

Calculation 1

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, applying the pythagorean theoreme on the rectangular triangle
, applying equation (1)
, applying equation (1)
, multiplying
, applying equation (2)

Calculation 2

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, applying the formulas for centroids of semicircles and triangles
, applying equation (1)
, summing the real and the imaginary terms





Licensing

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I, the copyright holder of this work, hereby publish it under the following license:
w:en:Creative Commons
attribution share alike
This file is licensed under the Creative Commons Attribution-Share Alike 4.0 International license.
You are free:
  • to share – to copy, distribute and transmit the work
  • to remix – to adapt the work
Under the following conditions:
  • attribution – You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.
  • share alike – If you remix, transform, or build upon the material, you must distribute your contributions under the same or compatible license as the original.

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Date/TimeThumbnailDimensionsUserComment
current20:18, 14 April 2023Thumbnail for version as of 20:18, 14 April 20231,534 × 928 (40 KB)Hans G. Oberlack (talk | contribs)description enhanced
19:58, 14 April 2023Thumbnail for version as of 19:58, 14 April 20231,534 × 928 (37 KB)Hans G. Oberlack (talk | contribs)Uploaded own work with UploadWizard

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