
Geometry  solid figures
solved problems 



Solid figures, solved problems examples 





Example:
In a cube inscribed is a regular tetrahedron whose edges are face’s diagonals of the cube, the
ratio of volumes of
the cube and the tetrahedron is? 
Solution: Given
a cube.
V_{cube}
: V_{tetrahedron}
= ?


Example:
At what distance from the center of a nottransparent sphere of radius
4
cm^{}
should be placed
the light to brighten
up one third of the surface of the sphere. 
Solution: Given
r =
4
cm. SO
= ? 

Example:
In an oblique circular cone the longest slant height
s_{1}=
6Ö3
and the shortest s_{2}=
6
subtend, at the vertex of the
cone, the angle of 30°, find the volume of the cone. 
Solution: Given
s_{1}=
6Ö3
and s_{2}=
6. V_{cone} = ? 

Example:
A plane passes through three vertices of a cube and cuts of it a tetrahedron.
The ratio of the volumes of obtained
solids is? 
Solution: Given a cube.
V_{tetrahedron}
:
(V_{cube}
 V_{tetrahedron}
) = ?


Example:
Around a cone of slant height that equal the diameter of the base circumscribed is a sphere.
The
ratio of surfaces of the cone
and the sphere is? 
Solution: Given
s = 2r.
S_{cone}
: S
_{sphere
}= ? 

Example:
A point source of light is 4
m distant from the center of a sphere of radius
2
m^{}. What surface area of the sphere
is brightened? 
Solution: Given
R
= 2 m and
SA
= 4 m. S
_{cap}
= ? 

Example:
Edges of a cuboid (rectangular prism) are in ratio 1
: 2 : 5 and length of its diagonal
5Ö6. Surface of the cuboid is? 
Solution: Given
a
: b
: c
= 1
: 2
: 5 and D_{
}= 5Ö6
. S
_{rectangular
prism} = ? 

Example:
The base of a pyramid is regular pentagon and the lateral faces are equilateral triangles. The basetoface dihedral
angle is? 
Solution: Given a
regular
pentagonal pyramid.
a
= ? 

Example:
On a horizontal surface placed are four spheres of the same radius
R
such that they touch each other and their
centers form square. Onto these four spheres placed is the fifth sphere of the same radius
which touches all the
four. The distance of the center of the fifth sphere from the horizontal surface is? 
Solution: Given
R.
x
= ? 









Geometry
and use of trigonometry contents  B 



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