§ 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 Sa Rr 
_{} _{} _{} _{} _{} _{} 
_{} _{} _{} _{} _{} _{} 
_{} _{} _{} _{} 
_{} _{} _{} Ra _{} 
_{} _{} _{} _{} _{ } _{} 
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 