Geometry - solid figures solved problems
      Solid figures, solved problems examples
Example:   In a cube inscribed is a regular tetrahedron whose edges are faces diagonals of the cube, the ratio of volumes of the cube and the tetrahedron is?
Solution:  Given a cube.    Vcube : Vtetrahedron = ?
 
Example:   At what distance from the center of a not-transparent 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 s1= 63 and the shortest s2= 6 subtend, at the vertex of the cone, the angle of 30, find the volume of the cone.
Solution:  Given  s1= 63 and s2= 6.    Vcone = ?
 
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.    Vtetrahedron : (Vcube - Vtetrahedron ) = ?
 
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.    Scone : 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 56. Surface of the cuboid is?
Solution:  Given  a : b : c = 1 : 2 : 5 and D = 56 .    S rectangular prism = ?
 
Example:   The base of a pyramid is regular pentagon and the lateral faces are equilateral triangles. The base-to-face 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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