File:FS HE dia.png
FS_HE_dia.png (681 × 472 pixels, file size: 21 KB, MIME type: image/png)
Captions
Summary
[edit]DescriptionFS HE dia.png |
English: Largest inscribed equilateral triangle in an semicircle
Deutsch: Größtes gleichseitiges Dreieck in einem Halbkreis |
Date | |
Source | Own work |
Author | Hans G. Oberlack |
The semicircle as base element. And the largest inscribed equilateral triangle.
General case
[edit]Segments in the general case
[edit]0) The radius of the semicircle:
1) The side length of the inscribed equilateral triangle: , see Calculation 1
Perimeters in the general case
[edit]0) Perimeter of base semicircle:
1) Perimeter of inscribed equilateral triangle:
S) Sum of perimeters:
Areas in the general case
[edit]0) Area of the base semicircle
1) Area of the inscribed equilateral triangle , see Calculation 2
Centroids in the general case
[edit]0) By definition the centroid point of a base shape is
1) The centroid of the inscribed equilateral triangle relative to the base centroid is: , see Calculation 3
Normalised case
[edit]In the normalised case the area of the base semicircle is set to 1.
So
Segments in the normalised case
[edit]0) Radius of the base semicircle
1) Side length of inscribed triangle
Perimeter in the normalised case
[edit]0) Perimeter of base semicircle:
1) Perimeter of inscribed triangle:
S) Sum of perimeters:
Areas in the normalised case
[edit]0) Area of the base semicircle is by definition
1) Area of the inscribed equilateral triangle
Centroids in the normalised case
[edit]0)
1)
Distances of centroids
[edit]The distance between the centroid of the base semicircle and the centroid of the circle is:
Sum of distances:
Identifying number
[edit]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
[edit]Known elements
[edit](0) Given is the radius of the base semicircle
(1)
(2)
(3)
Calculation 1
[edit], applying the pythagorean theoreme on the rectangular triangle
, applying equation (3)
, applying equation (1)
, applying equation (2)
Calculation 2
[edit]
, applying equation (3)
, applying equation (1)
, applying Calculation 1
Calculation 3
[edit]
, applying the formulas for centroids of triangles and semicircles
, applying equation (1)
, summing the real and the complex terms
Licensing
[edit]- 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.
File history
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Date/Time | Thumbnail | Dimensions | User | Comment | |
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current | 22:41, 28 May 2022 | 681 × 472 (21 KB) | Hans G. Oberlack (talk | contribs) | upload corrected | |
22:06, 28 May 2022 | 681 × 472 (21 KB) | Hans G. Oberlack (talk | contribs) | Uploaded own work with UploadWizard |
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