File:FS EV dia.png
![File:FS EV dia.png](https://upload.wikimedia.org/wikipedia/commons/thumb/4/46/FS_EV_dia.png/678px-FS_EV_dia.png?20230416161412)
Original file (1,889 × 1,670 pixels, file size: 86 KB, MIME type: image/png)
Captions
Captions
Summary
editDescriptionFS EV dia.png |
English: Largest quarter circle inscribed into an equilateral triangle
Deutsch: Größter Viertelkreis, der in ein gleichseitiges Dreieck eingeschrieben ist |
Date | |
Source | Own work |
Author | Hans G. Oberlack |
![](https://upload.wikimedia.org/wikipedia/commons/thumb/4/46/FS_EV_dia.png/220px-FS_EV_dia.png)
The equilateral triangle as base element. Inscribed is the largest quarter circle.
General case
editSegments in the general case
edit0) The side length of the equilateral base triangle is:
1) The radius of the inscribed quarter circle is: , see calculation 4
Perimeters in the general case
edit0) Perimeter of equilateral base triangle:
1) Perimeter of inscribed quarter circle:
Areas in the general case
edit0) Area of the equilateral base triangle:
1) Area of the inscribed quarter circle:
Centroids in the general case
edit0) By definition the centroid point of a base shape is
1) The centroid point of the inscribed quarter circle relative to the centroid of the base shape is: , see calculation (5)
![](https://upload.wikimedia.org/wikipedia/commons/thumb/f/fc/FS_EV.png/220px-FS_EV.png)
Normalised case
editIn the normalised case the area of the base shape is set to 1.
So
Segments in the normalised case
edit0) Side length of the triangle
1) The radius of the inscribed quarter circle is: ,
Perimeters in the normalised case
edit0) Perimeter of base triangle:
1) Perimeter of inscribed quarter circle:
S) Sum of perimeters:
Areas in the normalised case
edit0) Area of the base triangle is by definition
1) Area of the inscribed quarter circle:
Centroids in the normalised case
edit0) By definition the centroid point of a base shape is
1) The centroid point of the inscribed quarter circle relative to the centroid of the base shape is:
Distances of centroids
editThe distance between the centroid of the base triangle and the centroid of the semicircle is:
Sum of distances:
Identifying number
editApart 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
editKnown elements
edit(0) Given is the side length of the equilateral triangle:
(1)
(2)
(3)
(4)
(5), because
is equilateral
(6), because the quarter circle is tangent to
so that
is perpendicular.
Calculation 1
editThe angle is calculated:
,applying the sum of angles in the triangle
, rearranging
, applying equation (6)
, applying equation (5)
Calculation 2
editThe length of in relation to
is calculated:
, since
is a right triangle
, applying equation (5)
, applying the tan-function
, applying the tan-function
, rearranging
, applying equation(3)
Calculation 3
editThe length of in relation to
is calculated:
, since
is a right triangle
, applying calculation 1
, applying the cos-function
, applying equation (3)
, rearranging
Calculation 4
editThe length of in relation to
is calculated:
, since
, applying equation (4)
, applying calculation 3
, applying calculation 2
, rearranging
, rearranging
, rearranging
Calculation 5
editThe position of the centroid of the inscribed quarter circle in relation to the centroid
of the equilateral triangle is calculated:
, applying the centroid formula for quarter circles with
, simplifying the terms
, applying formula for centroids of triangles
, applying formula for height of triangles
, simplifying the term
, reducing to known elements
, applying equation (2)
, simplifying term
, applying calculation (2)
, applying calculation (4)
, simplifying term
, applying calculation (4)
, rearranging
, rearranging
, rearranging
, rearranging
, rearranging
, rearranging
, rearranging
, rearranging
, rearranging
, rearranging
, rearranging
, rearranging
Licensing
edit![w:en:Creative Commons](https://upload.wikimedia.org/wikipedia/commons/thumb/7/79/CC_some_rights_reserved.svg/90px-CC_some_rights_reserved.svg.png)
![attribution](https://upload.wikimedia.org/wikipedia/commons/thumb/1/11/Cc-by_new_white.svg/24px-Cc-by_new_white.svg.png)
![share alike](https://upload.wikimedia.org/wikipedia/commons/thumb/d/df/Cc-sa_white.svg/24px-Cc-sa_white.svg.png)
- 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
Click on a date/time to view the file as it appeared at that time.
Date/Time | Thumbnail | Dimensions | User | Comment | |
---|---|---|---|---|---|
current | 16:14, 16 April 2023 | ![]() | 1,889 × 1,670 (86 KB) | Hans G. Oberlack (talk | contribs) | updated |
15:47, 16 April 2023 | ![]() | 1,889 × 1,670 (84 KB) | Hans G. Oberlack (talk | contribs) | Uploaded own work with UploadWizard |
You cannot overwrite this file.
File usage on Commons
The following 2 pages use this file:
Metadata
This file contains additional information such as Exif metadata which may have been added by the digital camera, scanner, or software program used to create or digitize it. If the file has been modified from its original state, some details such as the timestamp may not fully reflect those of the original file. The timestamp is only as accurate as the clock in the camera, and it may be completely wrong.
Horizontal resolution | 129.92 dpc |
---|---|
Vertical resolution | 129.92 dpc |