
Geometry  solid figures
solved problems 



Solid figures, solved problems examples 





Example:
A right triangle with legs a
and b
and the hypotenuse c
first rotates around the leg a
and then around the leg b.
Find the ratio of volumes of generated solids of revolution. 
Solution: Given a
and b.
V_{s.r.}_{
 }_{a}
: V_{s.r.
 }_{b}
= ? 

Example:
Sides of a triangle are 34,
20
and 18
cm^{} long. Find the volume of a solid generated by rotation
of
the triangle
around a line passing through the vertex opposite of triangles' longest side and which is parallel to
it. 
Solution: Given
a
= 34 cm, b
= 20 cm
and c
= 18 cm. V_{s.r.}
= ? 


Example:
A triangle with sides 5,
4
and 3
cm, rotates around the longest side, find the volume of the solid
of revolution. 
Solution: Given
a
= 5, b
= 4 and c
= 3. V_{s.r.}
= ? 


Example:
An isosceles trapezium with longer base 9
cm, shorter base
5 cm
and with leg 3
cm rotates around shorter base.
Find the volume of the solid of revolution. 
Solution: Given
a
= 9
cm, c
= 5 cm and b
= 3 cm. V_{s.r.}
= ? 

Example:
An isosceles trapezium with longer base 9 cm, shorter base
1 cm
and with leg 5 cm
rotates around longer base.
Find the surface of the solid of revolution. 
Solution: Given
a
= 9
cm, c
= 1 cm and b
= 5 cm. S_{s.r.}
= ? 

Example:
A rhombus with longer diagonal f
and shorter diagonal e
rotates around the shorter diagonal. If (
f
/
6 )^{2}
= 1/e, find the volume of the generated solid of revolution.

Solution: Given
(
f
/
6 )^{2}
= 1/e.
V_{s.r.}
= ? 

Example:
An isosceles triangle of the base a
and the leg b, rotates around the base
a. If the altitude to the
base a
is twice the length of the altitude to the leg b, find the surface of the solid of revolution. 
Solution: Given
a and
h_{a}
= 2h_{b}.
S_{s.r.}
= ? 

Example:
Find the surface of a regular hexagonal pyramid whose height is four times longer than the altitude of any of six
equilateral triangles that form hexagonal base. 
Solution: Given
h_{1}
and h
= 4h_{1}.
S
= ? 









Geometry
and use of trigonometry contents  B 



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