§ 2    Circles and Regular Polygons

 

1.      Calculation formulas of various quantities related to circles

 

where represents the angle of the central angle ∠ AOB corresponding to the AMB arc (the same below), and C is any point on the ANB arc .

 

  [ Two secant lines and their included angle ]

AE · BE=CE · DE=ET 2

 

 

 AE · BE= CE · DE=r 2 -OE 2

 where r is the radius of the circle .

   

  [ Area S of a quadrilateral inscribed in a circle ]

   

                                  in the formula

   a,b,c,d are four sides

 

2.      Calculation formulas for the area, geometric center of gravity and moment of inertia of various figures related to circles

 

graphics

Area, Geometric Center of Gravity, and Moment of Inertia

O is the center of the circle , r is the radius , and d is the diameter

 

 

O is the center of the circle , r is the radius , and d is the diameter

 

 

perimeter  

The center of gravity G coincides with the center O of the circle  

Moment of inertia

( a ) The axis of rotation passes through the center of the circle and is perpendicular to the plane of the circle ( Figure ( a ))

      

( b ) The axis of rotation coincides with the diameter of the circle ( Figure ( b ))

      

( c ) The axis of rotation is a tangent to the circle ( Figure ( c ))

      

 

 

 

 

 

 

area  

The center of gravity G coincides with the center O of the circle  

Moment of inertia

( a ) The axis of rotation passes through the center of the circle and is perpendicular to the plane of the circle ( Figure ( a ))

      

( b ) The axis of rotation coincides with the diameter of the circle ( Figure ( b ))

      

( c ) The axis of rotation is parallel to a certain diameter of the circle , and its distance is h ( Fig. ( c ))

      

        

 

 

 

 

 

 

 

 

graphics

Area, Geometric Center of Gravity, and Moment of Inertia

r is the radius , b is the chord length , is the degree of the central angle corresponding to the arc s , which is the number of radians , and O is the center of the circle

 

area  

center of gravity  

Moment of inertia

(a)    The axis of rotation coincideswith GO (Fig.( a ))

      

(b)    The axis of rotation passes throughpoint G andis parallel to the diameter AB (Fig.( b ))

      

       

 

 

 

 

arc length  

area  

center of gravity  

         

Moment of inertia

(a)    The axis of rotation passes through point G on the graphics planeandis perpendicular to GO (Fig.( a ))

 

(b)    The axis of rotation coincideswith GO (Fig.( b ))

 

 ( At that time , it was a quarter circle )

 

 

 

 

 

 

graphics

Area, Geometric Center of Gravity, and Moment of Inertia

r is the radius , b is the chord length ( b=2a ), h is the arch height , is the number of the central angle, is the radian of the central angle , s is the arc length , and O is the center of the circle

 

 

 

 

 

 

 

R is the outer radius , r is the inner radius , D is the outer diameter , d is the inner diameter , and O is the center of the circle

Chord length  

       

vault  

area  

        

  

center of gravity 

( At that time , the bow was a semicircle )

Moment of inertia

(a)  The axis of rotation coincideswith GO (Fig.( a ))

 

(b)  The axis of rotation passes through the center of gravity G andis parallel to the chord(Fig.( b ))

area 

       

     where t=Rr is the ring width , 

           is the average diameter

The center of gravity G coincides with the center O of the circle  

Moment of inertia

The axis of rotation is on the graphics plane and passes through point G

( Figure ( a ))

    

    

      

 

 

 

 

 

graphics

Area, Geometric Center of Gravity, and Moment of Inertia

 

Same as before , it is the degree of the corresponding central angle, which is the number of radians

 

 

 

 

 

 

 

 

r is the radius , d is the diameter , l is the distance from the center of the circle , , is the opening angle of the crescent , and is the number of radians

area  

        

center of gravity  

         

The moment of inertia axis coincides with GO ( Fig. ( a )) 

    

      

 

 

area  

        

        

     in the formula 

 

center of gravity  

      

 

0.1

0.2

0.3

0.4

0.399

0.795

1.182

1.556

 

 

0.5

0.6

0.7

0.8

0.9

1.913

2.247

2.551

2.815

3.024

 

 

 

 

3.        Conversion formulas and proportional coefficients of regular polygons

 

n is the number of sides R is the radius of the circumcircle                   

a is the side length r is the radius of the inscribed circle                    

is the central angle S is the area of ​​the polygon      

The center of gravity G coincides with the center O of the circumcircle

 

 

 

Regular polygon conversion formula table

each amount

equilateral triangle

square

regular pentagon

hexagon

regular n -gon

picture

 

 

shape

 

S

 

 

 

 

 

 

a

 

 

R

 

 

 

r

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

R

 

 

a

 

 

 

 

 

 

 

 

 

Regular polygon scale coefficient table

 

n

a/R

R/a

r/a

3

4

5

6

7

8

9

10

 

12

15

16

20

 

0.4330

1.0000

1.7205

2.5981

3.6339

4.8284

6.1818

7.6942

 

11.196

17.642

20.109

31.569

 

1.2990

2.0000

2.3776

2.5981

2.7364

2.8284

2.8925

2.9389

 

3.0000

3.0505

3.0615

3.0902

 

5.1962

4.0000

3.6327

3.4641

3.3710

3.3137

3.2757

3.2492

 

3.2154

3.1883

3.1826

3.1677

 

1.7321

1.4142

1.1756

1.0000

0.8678

0.7654

0.6840

0.6180

 

0.5176

0.4158

0.3902

0.3129

 

0.5774

0.7071

0.8507

1.0000

1.1524

1.3066

1.4619

1.6180

 

1.9319

2.4049

2.5629

3.1962

 

0.2887

0.5000

0.6882

0.8660

1.0383

1.2071

1.3737

1.5388

 

1.8660

2.5323

2.5137

3.1569

 

n

a/R

R/a

r/a

twenty four

32

48

64

45.575

81.225

183.08

325.69

3.1058

3.1214

3.1326

3.1366

3.1597

3.1517

3.1461

3.1441

0.2611

0.1960

0.1308

0.0981

3.8306

5.1012

7.6449

10.190

3.7979

5.0766

7.6285

10.178

 

 

Original text

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