File:FS FJCF dia.png
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Captions
Summary edit
DescriptionFS FJCF dia.png |
English: Largest 120° circular sector (fan) in a 120° circular sector (fan) that contains the broadest isosceles triangle and the largest circle - Details: FJCF dia.png
Deutsch: Größter Drittelkreis in einem Drittelkreis (Fächer), der bereits das breitestes gleichschenkliges Dreieck und den größten Kreis enthält - Details: FJCF dia.png |
Date | |
Source | Own work |
Author | Hans G. Oberlack |
0) The 120°-degree circular sector (fan) as base element.
1) Inscribed is the broadest isosceles triangle.
2) Inscribed is the largest circle.
2) Inscribed is the largest 120°-degree circular sector (fan).
General case edit
Segments in the general case edit
0) Radius of the base circular sector:
1) side length of the inscribed triangle: , because it is the broadest triangle
2) Radius of the inscribed circle: , see calculation (5)
3) Radius of the inscribed circular sector: , applying calculation (1)
Perimeters in the general case edit
0) Perimeter of base circular sector:
1) Perimeter of inscribed triangle:
2) Perimeter of inscribed circle around :
3) Perimeter of inscribed circular sector:
Areas in the general case edit
0) Area of the base circular sector
1) Area of the inscribed triangle , see calculation (3)
2) Area of the inscribed circle
3) Area of the inscribed circular sector
Centroids in the general case edit
0) By definition the centroid point of a base shape is
1) The centroid of the inscribed triangle relative to the base centroid is: , see Calculation (4)
2) The centroid of the inscribed circle relative to the base centroid is: , see Calculation (6)
3) The centroid of the inscribed circular sector relative to the base centroid is: , see Calculation (7)
Normalised case edit
In the normalised case the area of the base circular sector is set to 1.
So
Segments in the normalised case edit
0) Radius of the base circular sector:
1) Side length of the inscribed triangle:
2) Radius of the inscribed circle:
3) Radius of the inscribed circular sector:
Perimeter in the normalised case edit
0) Perimeter of base circular sector:
1) Perimeter of inscribed triangle:
2) Perimeter of inscribed circle around :
3) Perimeter of inscribed circular sector:
S) Sum of perimeters:
Area in the normalised case edit
0) Area of the base circular sector is by definition
1) Area of the inscribed triangle
2) Area of the inscribed circle
3) Area of the inscribed circular sector
Centroids in the normalised case edit
0) Centroid of the base shape:
1) Centroid of the inscribed triangle:
2) Centroid of the inscribed circle:
3) Centroid of the inscribed circular sector:
Calculations edit
Given elements edit
(1)
(2) Angle in M:
(3) Angles in A and B of triangle :, since it is a isosceles triangle
(4) , since the isosceles triangle is symmetric
(5)
Calculation 1 edit
Calculating length of MD
, applying equation 3 and the definition of the sinus
, calculating the sine
, applying equation (1)
, rearranging
Calculation 2 edit
Calculating length of AB
, applying equation 3 and the definition of the cosinus
, calculating the sine
, applying equation (1)
, rearranging
, applying equation (4)
, rearranging
Calculation 3 edit
, calculating the area of triangle
, applying calculation (1)
, applying calculation (2)
, rearranging
Calculation 4 edit
starting from S_0
, extending to M
, since (0+0i)=0
, applying the centroid formula
, shortening
, calculating the sine
, shortening
, expressing the vector as complex number
, applying calculation (1)
, expressing the vector as complex number
, applying the centroid formular for isosceles triangles
, applying calculation (1)
, shortening
, using distributive property
, adding complex numbers
, adding
, adding
, adding
Calculation 5 edit
Calculating radius
, applying equation (1)
, applying the construction of the diagram
, applying calculation (1)
, rearranging
, applying equation (5)
, applying equation (5)
Calculation 6 edit
Calculating the centroid of the circle
, since the centroid of the base shape is (0+0i)=0
, applying equation (5)
, applying equation (1)
, applying calculation (5)
, rearranging
, applying the centroid formula for circular sectors
, rearranging
, calculating the sine
, rearranging
-->
Calculation 7 edit
Calculating the centroid of the inscribe circular sector
, since the centroid of the base shape is (0+0i)=0
, applying the centroid formular for circular sectors
, rearranging
, calculating the sine
, shortening
, applying the centroid formular for circular sectors
, rearranging
, shortening
, calculating the sine
, shortening
, applying the relation
, calculating
, rearranging
Licensing edit
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- 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.
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Date/Time | Thumbnail | Dimensions | User | Comment | |
---|---|---|---|---|---|
current | 22:31, 8 November 2023 | 1,393 × 961 (64 KB) | Hans G. Oberlack (talk | contribs) | Uploaded own work with UploadWizard |
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