File:FS EQ dia.png
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Summary[edit]
DescriptionFS EQ dia.png |
English: Largest square in an equilateral triangle
Deutsch: Größtes Quadrat in einem gleichseitigen Dreieck |
Date | |
Source | Own work |
Author | Hans G. Oberlack |
The equilateral triangle as base element. Inscribed is the largest square.
General case[edit]
Segments in the general case[edit]
0) The side length of the equilateral base triangle is:
1) The side length of the inscribed square is: , see calculation 3
Perimeters in the general case[edit]
0) Perimeter of equilateral base triangle:
1) Perimeter of inscribed square:
Areas in the general case[edit]
0) Area of the equilateral base triangle: , see calculation (2)
1) Area of the inscribed square:
Centroids in the general case[edit]
0) By definition the centroid point of a base shape is
1) The centroid point of the inscribed square relative to the centroid of the base shape is: , see calculation (4)
Normalised case[edit]
In the normalised case the area of the base shape is set to 1.
So
Segments in the normalised case[edit]
0) Side length of the triangle
1) The side length of the square is: ,
Perimeters in the normalised case[edit]
0) Perimeter of base triangle:
1) Perimeter of inscribed square:
S) Sum of perimeters:
Areas in the normalised case[edit]
0) Area of the base triangle is by definition
1) Area of the inscribed square:
Centroids in the normalised case[edit]
0) By definition the centroid point of a base shape is
1) The centroid point of the inscribed square relative to the centroid of the base shape is:
Distances of centroids[edit]
The distance between the centroid of the base triangle and the centroid of the semicircle is:
Sum of distances:
Identifying number[edit]
Apart of the base element there is one other 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 side length of the equilateral triangle:
(1)
(2)
(3)
(4)
(5), because is equilateral
Calculation 1[edit]
The height is calculated:
,applying the Pythagorean theorem on the rectangular triangle
, applying equation (2)
, applying equation (1)
, rearranging
, rearranging
, rearranging
Calculation 2[edit]
, applying equation (2)
, applying result of calculation (2)
Calculation 3[edit]
, applying equation (5)
, applying equation (3)
, rearranging
, applying the tan-formula
, applying the tan-formula
, applying equation (2)
, applying equation (4)
, rearranging
, rearranging
, rearranging
, rearranging
, rearranging
, rearranging
, rearranging
Calculation 4[edit]
, adding the real and the complex terms
, rearranging
, rearranging
, centroid of square
, centroid of triangle
, applying result of calculation (1)
, applying result of calculation (3)
, rearranging
, rearranging
, rearranging
, rearranging
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 | |
---|---|---|---|---|---|
current | 21:53, 13 June 2022 | 722 × 649 (25 KB) | Hans G. Oberlack (talk | contribs) | Uploaded own work with UploadWizard |
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