
Geometry  solid figures
solved problems 



Solid figures, solved problems examples 





Example:
The midpoint of the altitude of a tetrahedron of the edge
a
is connected with all three vertices of the base, find the
angle these connections make. 
Solution: Given
a.
a
= ? 

Example:
Given is a cube of edge
a. Midpoints of its faces are vertices of a regular octahedron. Find the
volume of the
octahedron. 
Solution: Given
a.
V_{octahedron}
= ? 

Example:
By rolling the cone about its apex obtained is lateral surface area of a cone in a form of a semicircle of radius
R = 2Ö3
m. Find the volume of the cone. 
Solution: Given
R = 2Ö3
m. V_{cone}
= ? 


Example:
Volume of a circular cone is V. Through the midpoint of the altitude of the cone passes a plane parallel to the base
of the cone, find the volume of the part of the cone between base and the plane. 
Solution: Given
V. V_{conical
frustum} = ? 

Example:
In a cone of base B
and the height h
inscribed is a cone with the vertex at the center of the
given cone and whose
height is half of the given cone. Find the volume of the inscribed cone. 
Solution: Given
B
and h.
V_{inscribed cone} = ? 

Example:
An isosceles
triangle, of leg b
= 2
that is inclined to the base at a
= 30°,
rotates around its base
bisector. Find the volume of obtained solid of revolution. 
Solution: Given
b
= 2 and a
= 30°.
V_{solid
of revolution} = ? 

Example:
Find the volume of the solid generated by rotating an equilateral triangle
ABC
of the side a
= 2 around the axis
which passes through vertex C parallel to base
AB. 
Solution: Given
a
= 2.
V_{solid
of revolution} = ? 

Example:
In a regular square pyramid of height 2
and side of the base 1
inscribed is a cube. The side of
the cube
is? 
Solution: Given
h
= 2 and
a = 1.
x
= ? 

Example:
Height of a cone is 12 and the volume
324p. Find the central angle of the sector to which is unrolled the side
surface of the cone. 
Solution: Given
h
= 12 and
V = 324p.
a
= ? 










Geometry
and use of trigonometry contents  B 



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