
Quadrilaterals,
Polygons  Regular Polygons, Circle 


Circle

Central angle, inscribed angle

Construction of a tangent
from a point

Circumference, length of an arc

Circle and circular sector

Circular segment

Annulus and annulus segment 





Circle

A circle is a set of points that are at the fixed distance (called the
radius r) from a fixed point called the center
O. 

Central angle, inscribed angle

A
central angle is
double the inscribed angle (formed when two secant lines intersect
on the circle) subtended by the same arc. 
Proof:
Angles
b_{1}
and b_{2 }are
external angles of the isosceles triangles AOC and
BOC, hence 
b_{1}_{
}= 2a_{1},
b_{2}_{
}= 2a_{2},
a_{
}= a_{1}+
a_{2}
=>
b_{ }= b_{1}+
b_{2}_{
}= 2(a_{1}+
a_{2})_{
}= 2a 

Inscribed angles
subtended by the same arc are equal. An angle inscribed in a semicircle is a right angle. 

Construction of a tangent
from a point P
to a circle c. 

The
midpoint of the line segment OP
is the circumcenter of the quadrilateral PD_{1}OD_{2}.
The lines PD_{1}
and PD_{2}
are tangents from P
to the given circle c. 


Circumference, the length of a circle  the perimeter:





Length of an arc:



Circle and circular sector

By substituting
P = 2pr
and R
= r
in the formula for the area of a regular polygon, obtained is the formula
for the area of a circle, that is : 


Circular segment

The portion of a circle bounded by an arc and a chord is called a segment. 
Symbols
used in the formulas: c
chord, r
radius, h
height of a segment, A
area of a segment. 


Annulus and annulus segment 
Annulus
or ring is the region enclosed between two concentric circles. 
Annulus 
Annulus
segment 










Geometry
and use of trigonometry contents  A 



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