File:FS QC.1E dia.png
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Contents
Summary
[edit]DescriptionFS QC.1E dia.png |
English: Largest equilateral triangle inscribed in the largest circle within a square
Deutsch: Größtes gleichseitiges Dreieck innerhalb des größten Kreises in einem Quadrat |
Date | |
Source | Own work |
Author | Hans G. Oberlack |
![](https://upload.wikimedia.org/wikipedia/commons/thumb/9/98/FS_QC.1E_dia.png/220px-FS_QC.1E_dia.png)
Task
[edit]The largest equilateral triangle inscribed in the largest circle that is inscribed in a square of side length
General case
[edit]Segments in the general case
[edit]0) The side length of the base square:
1) The radius of the inscribed circle:
2) Side of the inscribed triangle: , see Calculation 1
Perimeters in the general case
[edit]0) Perimeter of the base square:
1) Perimeter of inscribed circle
2) Perimeter of the inscribed triangle
Areas in the general case
[edit]0) Area of the base square:
1) Area of the inscribed circle
2) Area of the inscribed triangle , see Calculation 2
Centroids in the general case
[edit]Centroid positions are measured from the centroid point of the base shape
0) Centroid position of the base square:
1) Centroid position of the inscribed circle:
2) Centroid position of the inscribed triangle:
W) Weighted centroid:
Normalised case
[edit]![](https://upload.wikimedia.org/wikipedia/commons/thumb/e/e5/FS_QC.1E.png/220px-FS_QC.1E.png)
In the normalised case the area of the base is set to 1.
Segments in the normalised case
[edit]0) Side length of the base square:
1) Radius of the inscribed circle
2) Side length of the inscribed triangle
Perimeters in the normalised case
[edit]0) Perimeter of the base square:
1) Perimeter of inscribed square
2) Perimeter of the inscribed triangle
S) Sum of perimeters
Areas in the normalised case
[edit]0) Area of the base square
1) Area of the inscribed circle
2) Area of the inscribed triangle
Centroids in the normalised case
[edit]Centroid positions are measured from the centroid point of the base shape.
0) Centroid of the base square:
1) Centroid of the inscribed circle:
2) Centroid of the inscribed triangle:
W) Weighted centroid:
Distances of centroids
[edit]The distance between the centroid of the base element and the centroid of the triangle is:
Identifying number
[edit]Apart of the base element there are two shapes allocated. Therefore the integer part of the identifying number is 2.
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]Given elements
[edit](0) , because
is a square
(1) , since the inscribed triangle
is an equilateral triangle
(2) , since the inscribed triangle
is an equilateral triangle
(3) , since the inscribed triangle
is an equilateral triangle
(4) , since the inscribed triangle
is an equilateral triangle
(5) , since the circle around
is th largest circle inscribed the square
Calculation 1
[edit], since
is a right triangle
, applying equations (1) and (2)
, rearranging
, applying equation (3)
, applying equation (4)
, since
is a right triangle
, rearranging
, squaring both sides
, applying equation (2)
, multiplying
, applying binomial formula
, shortening
, rearranging
, rearranging
, reducing
, dividing
, rearranging
, applying equation (5)
Calculation 2
[edit]
, applying equation (2)
, applying the Pythagorean theoreme
, applying equation (1)
, applying equation (2)
, multiplying
, rooting
, multiplying
, applying calculation 1
, multiplying
, multiplying
Licensing
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Date/Time | Thumbnail | Dimensions | User | Comment | |
---|---|---|---|---|---|
current | 14:11, 4 June 2023 | ![]() | 1,293 × 1,336 (80 KB) | Hans G. Oberlack (talk | contribs) | Uploaded own work with UploadWizard |
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