# Inscribed and circumscribed polygons. Regular polygons

*Inscribed polygon. Circumscribed polygon. Circumcircle*

about a polygon. Incircle into a polygon. Radius of an incircle

into a triangle. Radius of a circumcircle about a triangle.

Regular polygon. Center of a regular polygon. Apothem.

Relations between sides and radii of a regular polygon.

about a polygon. Incircle into a polygon. Radius of an incircle

into a triangle. Radius of a circumcircle about a triangle.

Regular polygon. Center of a regular polygon. Apothem.

Relations between sides and radii of a regular polygon.

*
Inscribed polygon
in a circle is a polygon, vertices of which are placed on a circumference
*
( Fig.54 )

*.*( Fig.55 ) .

**Polygon circumscribed**around a circle is a polygon, sides of which are tangents to the circumference

Correspondingly,
*
a circumference, going through vertices of a polygon
*
( Fig.54 ),
*
is called a
circumcircle around a polygon
; a circumference, for
which sides of a polygon are tangents
*
( Fig.55 ),

*is called an*For an

**incircle into a polygon.***arbitrary polygon*it is impossible to inscribe a circle in it and to circumscribe a circle around it.

*For a triangle it is always possible*. A

*radius r of an incircle*is expressed by sides

*a, b, c*of a triangle as:

*
A radius R of a circumcircle
*
is expressed by the formula:

It is possible
*
to inscribe a circle in a
*
*
quadrangle
*
, if sums of its opposite sides are the same. In case of parallelograms it is valid only for a
rhombus (a square). A center of an inscribed circle is placed in a point of intersection of diagonals. It is possible
*
to circumscribe a circle around a
quadrangle
*
, if a sum of its opposite angles is equal to 180 deg. In case of parallelograms it is valid only for a rectangular (a square). A center
of a circumscribed circle is placed in a point of intersection of diagonals. It is possible
*
to circumscribe a circle around a trapezoid, only if
it is an isosceles one.
*

**
Regular polygon
**
is

*a polygon with equal sides and angles*

On Fig.56 a regular hexagon is shown, on Fig.57 – a regular octagon. A regular quadrangle is a square; a regular triangle is an equilateral triangle. Each
angle of a regular polygon is equal to 180 (
*
n
*
– 2 ) /
*
n
*
deg,
*
*
where
*
n
*
is a number of angles. There is a point O ( Fig. 56 ) inside of a regular polygon, equally removed
from all its vertices ( OA = OB = OC = … = OF ), which is called a
*
center
*
of a regular polygon. The center is also equally removed from all the sides of
a regular polygon ( OP = OQ = OR = … ). The segments OP, OQ, OR, … are called
*
apothems
*
; the segments OA, OB, OC, …
*
– radii
*
of a regular polygon. It is
possible to inscribe a circle in a regular polygon and to circumscribe a circle around it. The centers of inscribed and circumscribed circles coincide with a center of
a regular polygon. A radius of a circumscribed circle is a radius of a regular polygon, a radius of a inscribed circle is its apothem. The following formulas are
relations between sides and radii of regular polygon:

For the most of regular polygons it is impossible to express the relation between their sides and radii by an algebraic formula.

E x a m p l e . Is it possible to cut out a square with a side 30 cm from a circle with a diameter

40 cm ?

S o l u t i o n . The biggest square, included in a circle, is an inscribed square. According

to the above mentioned formula its side is equal:

Hence, it is impossible to cut out a square with a side 30 cm from a circle with

a diameter 40 cm.