# 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: