File:FS CF dia.png
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Captions
Summary[edit]
DescriptionFS CF dia.png |
English: Largest fan (circular sector of 120degrees) within a circle
Deutsch: Größter Fächer (Drittelkreis) in einem Kreis |
Date | |
Source | Own work |
Author | Hans G. Oberlack |
Elements[edit]
Base is the circle of given radius around point
Inscribed is the largest possible fan (circular sector with
.
General case[edit]
Segments in the general case[edit]
0) The radius of the base circle:
1) The radius of the inscribed fan: , see Calculation 1
Perimeters in the general case[edit]
0) Perimeter of base circle
1) Perimeter of the inscribed fan: see calculation 2
Areas in the general case[edit]
0) Area of the base circle
1) Area of the inscribed fan: , see Calculation 3
Centroids in the general case[edit]
The centroid positions of the following shapes will be expressed orientated so that the first shape n with will be of type with . The graphical representation does correspond to the mathematical expression.
0) Centroid position of the base circle:
1) Centroid position of the inscribed fan: , see calculation 4
Normalised case[edit]
In the normalised case the area of the base is set to 1.
Segments in the normalised case[edit]
0) Radius of the base circle:
1) Radius of the inscribed fan:
Perimeters in the normalised case[edit]
0) Perimeter of base square:
1) Perimeter of the fan:
Areas in the normalised case[edit]
0) Area of the base circle:
1) Area of the fan:
Covered surface of the base shape[edit]
Centroids in the normalised case[edit]
Centroid positions are measured from the centroid point of the base shape.
0) Centroid of the base circle:
1) Centroid of the inscribed fan:
Calculations[edit]
Equations of given elements and relations[edit]
(1)
(2)
(3) , since the angle in
(4) , since the apothem is perpendicular to the chord .
Calculation 1[edit]
The radius is calculated:
, applying the law of sines on the triangle
, applying equation (2)
, applying equation (3)
, eliminating 1 off the denominator
, applying equation (1)
, applying the sin formula
, rearranging
Calculation 2[edit]
Perimeter of the inscribed fan:
, applying calculation 1
, rearranging
, extending the fraction
, multiplying
Calculation 3[edit]
Area of the inscribed fan:
, applying calculation 1
, rearranging
, squaring
, rearranging
Calculation 4[edit]
Centroid of the fan relative to the centroid of the circle.The centroid positions of the following shapes will be expressed orientated so that the first shape n with will be of type with . The graphical representation does correspond to the mathematical expression.
, rearranging
, rearranging
, since
, applying the sine formula
, applying the centroid formula
, extracting
, applying the sine formula
, shortening the fraction
, applying equation 3
, shortening the fraction
, rearranging
, applying calculation 1
, 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.
File history
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Date/Time | Thumbnail | Dimensions | User | Comment | |
---|---|---|---|---|---|
current | 12:02, 4 February 2024 | 1,383 × 1,277 (88 KB) | Hans G. Oberlack (talk | contribs) | Uploaded own work with UploadWizard |
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Vertical resolution | 129.92 dpc |