
Geometry  solid figures
solved problems 



Solid figures, solved problems examples 





Example:
An equilateral triangle of the side a =
6 rotates around a line parallel to its side on the distance
that equals the
triangle’s height (not passing through the vertex of the triangle), find the volume of the solid of revolution. 
Solution: Given a =
6. V_{solid
of revolution} = ?
V_{solid
of revolution} =
2V_{frustum}

V_{cylinder} 


Example:
The cosine of the angle at which diagonals of a cube intersect is? 
Solution: Given
a cube.
cos j
= ? 

Example:
In a right triangular prism inscribed is a sphere which touches all faces of the prism. The ratio of surfaces of the
sphere and the prism is? 
Solution: Given
a right triangular prism.
S_{sphere}
: S_{prism}
= ?
S_{sphere}
= 4R^{2}p, 

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}
= ? 









Geometry
and use of trigonometry contents  B 



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