# Areas of plane figures

*Areas of plane figures: square, rectangle, rhombus, parallelogram,*

trapezoid, quadrangle, right-angled triangle, isosceles triangle,

equilateral triangle, arbitrary triangle, polygon, regular hexagon,

circle, sector, segment of a circle. Heron’s formula.

trapezoid, quadrangle, right-angled triangle, isosceles triangle,

equilateral triangle, arbitrary triangle, polygon, regular hexagon,

circle, sector, segment of a circle. Heron’s formula.

*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 A*m*BO 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 A*m*B = 60 deg – about 1.5% ; at A*m*B = 30 deg ~0.3%.

E x a m p l e . Calculate areas of the sector A*m*BO ( Fig.68 ) and the segment

A*m*B

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: