
Geometry
 Plane Figures Problems and Solutions 


Plane figures, solved problems, examples 





Example:
The area of a circle is 6 cm^{2} greater then the area of the square inscribed into the circle.
Then, the area of the circle
is? 
Solution: Given
A_{circle}
= A_{square }+ 6.
A_{circle} =
? 


Example:
Into a quarter of a circle with the radius
1 + Ö2
inscribed is a circle, find the radius of the
inscribed circle. 
Solution: Given
R = 1 + Ö2.
r
= ? 


Example:
Which polygon
have two times more diagonals then sides? 
Solution: The
number of diagonals in a polygon,



where
n
is the number of polygon sides. 



Example:
Given is a regular hexagon with the side
a
into which is inscribed a circle, find the area between the hexagon and
the circle. 
Solution: Given a.
A
= ? 


Example:
If the ratio of the greater angle between diagonals of a rectangle to the smaller angle is
2 :
1 and a >
b then, the
ratio a
: b of sides of the rectangle is? 
Solution: Given
j
: (180°

j)
= 2 : 1
and a >
b. a
: b
= ? 


Example:
Into a circle inscribed is deltoid (kite). If one of its angles between equal sides subtends
60°
then, the relation of the longer to the shorter diagonal
is? 
Solution: Given a =
60°. e
: f
= ? 


Example:
Determine the least
n
(the number of sides of a regular polygon), for which the interior angle of a regular polygon is greater then
163°. 
Solution: Given a
> 163°.
n
= ? 
The interior angle of a regular
polygon, 




Example:
Deltoid with sides length of 1 and 2 is inscribed into a circle. The ratio of greater to the shorter
diagonal is? 

Solution: Given
a = 2
and b
= 1.
e
: f
= ? 


Example:
One side of a parallelogram is
4 cm, other
6 cm, and the diagonal
4Ö2 cm,
length of the other diagonal is? 
Solution: Given
a = 6
cm, b
= 4 cm and d_{1}
= 4Ö2 cm
. d_{2}
= ? 


Example:
Find the area of the rhomb (diamond) with the side of length 20
cm and the diagonal of length 32 cm. 
Solution: Given a =
20 cm and
e = 32 cm.
A _{rhombus}
= ? 


Example:
Length of the side of the right decagon is 12. Find the radius of its
circumcircle. 
Solution: Given
n = 10 and
a = 12.
r
= ? 


Example:
In a circle with the radius 2
cm, on its chord
2Ö2 cm
long taken as a diameter, constructed is another circle. The
area of the part of that circle, which is outside of the greater circle, is? 
Solution: Given
r = 2 cm and
t = 2Ö2 cm.
A
= ? 









Geometry
and use of trigonometry contents  A 




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