Areas of plane figures

Any  triangle.  a, b, c – sides;  a – a base;  h – a height;  A, B, C – angles,
                        opposite to sides  a, b, c ;    p = ( a + b + c ) / 2.

The last expression is known as Heron's  formula.

A polygon, area of which we want to determine, can be divided into some triangles by its diagonals. A polygon, circumscribed around a circle  ( Fig. 67 ), can be divided by lines, going from a center of a circle to its vertices. Then we receive:

Particularly, this formula is valid for any regular polygon.

A regular hexagon.  a – a side. 

A circle. D – a diameter;  r – a radius.

A sector  ( Fig.68 ).  r – a radius; n – a degree measure of a central angle;
                                 l – a length of an arc.

A segment  ( Fig.68 ). An area of a segment is found as a difference between areas of a sector AmBO and a triangle AOB. Besides, the approximate formula for an area of a segment is:

where  a = AB ( Fig.68 ) – a base of segment;  h – its height ( h = r – OD ). A relative error of this formula is equal:  at AmB = 60 deg  – about 1.5% ;  at  AmB = 30 deg   ~0.3%.

E x a m p l e .  Calculate areas of the sector AmBO  ( Fig.68 )  and the segment
                        AmB at the following data: r = 10 cm, n = 60 deg.

S o l u t i o n .  A sector area:

                       An area of the regular triangle AOB:

                       Hence, an area of a segment:

                        Note, that in a regular triangle AOB:  AB = AO = BO = r, 
                        AD = BD = r / 2 , and  therefore a height OD according to
                        Pythagorean theorem is equal to:

                         Then, according to the approximate formula we’ll receive: