File:FS HCCC dia.png
FS_HCCC_dia.png (680 × 425 pixels, file size: 35 KB, MIME type: image/png)
Captions
Captions
Summary
[edit]DescriptionFS HCCC dia.png |
English: Semicircle with largest inscribed circle and the two next largest circles
Deutsch: Halbkreis mit größtem eingeschriebenen Kreis und den zwei nächstgrößeren Kreisen. |
Date | |
Source | Own work |
Author | Hans G. Oberlack |
![](https://upload.wikimedia.org/wikipedia/commons/thumb/5/54/FS_HCCC_dia.png/220px-FS_HCCC_dia.png)
The semicircle as base element. And the largest inscribed circle. And the next largest inscribed circle. And the symmetrically placed next largest circle.
General case
[edit]Segments in the general case
[edit]0) The radius of the semicircle around
1) The radius of the inscribed circle around
2) The radius of the next inscribed circle around , see calculation 1
3) The radius of the next inscribed circle around , for symmetrical reasons
Perimeters in the general case
[edit]0) Perimeter of base semicircle:
1) Perimeter of inscribed circle:
2) Perimeter of next inscribed circle:
3) Perimeter of next inscribed circle:
Areas in the general case
[edit]0) Area of the base semicircle
1) Area of the inscribed circle
2) Area of the next inscribed circle
3) Area of the next inscribed circle
Centroids in the general case
[edit]0) By definition the centroid point of a base shape is
1) The centroid of the inscribed circle relative to the base centroid is: , see Calculation 2
2) The centroid of the next inscribed circle relative to the base centroid is: , see Calculation 3
3) The centroid of the next inscribed circle relative to the base centroid is: , for symmetrical reasons
Normalised case
[edit]![](https://upload.wikimedia.org/wikipedia/commons/thumb/e/e8/FS_HCCC.png/220px-FS_HCCC.png)
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) Radius of the inscribed circle
2) Radius of the next inscribed circle
3) Radius of the next inscribed circle
Perimeter in the normalised case
[edit]0) Perimeter of base semicircle:
1) Perimeter of inscribed circle:
2) Perimeter of next inscribed circle:
3) Perimeter of next inscribed circle:
S) Sum of perimeters:
Area in the normalised case
[edit]0) Area of the base semicircle is by definition
1) Area of the inscribed circle
2) Area of the next inscribed circle
3) Area of the next inscribed circle
Centroids in the normalised case
[edit]0)
1)
2)
3)
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 are three shapes allocated. Therefore the integer part of the identifying number is 3.
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](0) Segments of given length:
Calculation 1
[edit](1) is perpendicular on
since the circle around
is tangent to
and is perpendicular on
since the circle around
is tangent to
and
(2) is perpendicular in
since the circle around
is tangent to the semicircle around
with
and
(3) is perpendicular in
since the circle around
is tangent to the circle around
with
and
(4)
, applying eqaution (1)
(5) , applying the Pythagorean theorem
, applying equation (1)
, applying equation (2)
, applying binomial theorem
, eliminating
on both sides
(6) , applying the Pythagorean theorem
, applying equation (1)
, applying equation (5)
, applying equation (4)
, applying equation (3)
, applying binomial theorem
, applying binomial theorem
, eliminating
on both sides
, adding
on both sides
, adding
on both sides
, since
Calculation 2
[edit]
Calculation 3
[edit]
, applying equation (1)
,applying equation (5)
, applying equation (6)
Licensing
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File history
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Date/Time | Thumbnail | Dimensions | User | Comment | |
---|---|---|---|---|---|
current | 22:19, 24 May 2022 | ![]() | 680 × 425 (35 KB) | Hans G. Oberlack (talk | contribs) | Uploaded own work with UploadWizard |
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