Geometry - solid figures solved problems
      Solid figures, solved problems examples
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 =  ?
Example:   An isosceles trapezoid with bases lengths, 3n and n, whose base angle is 45°, rotates around greater base. Find the volume of the solid of revolution.
Solution:  Given bases lengths, 3n and n,  and base angle 45°.    Vsolid of revolution = ?
Example:   By rotating the rhomb around its longer diagonal generated is solid whose volume is two times smaller than the volume of the solid generated by rotating the rhomb around its shorter diagonal. The ratio of diagonals d1 : d2, (d1 > d2 ), is? 
Solution:  Given  V1 : V2 = 1 : 2.    d1 : d2 = ?
Example:   In a sphere of radius 2 cm inscribed is a cone of the height 3 cm. The ratio of volumes of the cone and the sphere is?  
Solution:  Given  R = 2 cm and h = 3 cm.    Vconic : Vsphere = ?
Example:   The slant height of a cone is 3 cm and is inclined to the base of the cone at the angle 60°. The surface of the sphere inscribed in the cone is?
Solution:  Given s = 3 cm and a = 60°.    S sphere = ?   The cross section through the apex and center of the base of the cone is the equilateral triangle.
Example:   A channel’s cross section has the form of the isosceles trapezoid of bottom side 2 m and upper parallel side 4 m and altitude 1 m. How much water is in the channel if it is fulfilled to the half of its height?
Solution:  Given,  c = 4 m, a = 2 m, h = 1 m,  and  l = 100 m.    V = ?
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.    Vsolid of revolution = ?    Vsolid of revolution = 2Vfrustum - Vcylinder
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.    Ssphere : Sprism = ?    Ssphere = 4R2p,
Geometry and use of trigonometry contents - B
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