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altitude axis base bisects Book bounded chord circle circumference circumscribed coincide common cone of revolution construct contains corresponding cube cylinder cylinder of revolution denoted determine diagonal diameter dihedral angle dimensions distance divided draw drawn element equal equidistant equilateral equivalent EXERCISES faces figure Find find the ratio Find the volume formed formula four frustum geometry given line given point greater GROUP Hence included inscribed intersection lateral area lateral edges limit measure meet method opposite parallel parallel planes parallelogram pass pass a plane perpendicular plane MN polygon polyhedron prism Proof properties PROPOSITION prove prove Given pyramid radii radius regular respectively segment Show sides similar solid space sphere spherical triangle square straight line surface symmetrical tangent tetrahedron THEOREM third triangular trihedral vertex vertices volume zone
Page 492 - Euclid's, and show by construction that its truth was known to us ; to demonstrate, for example, that the angles at the base of an isosceles triangle are equal...
Page 416 - Every section of a circular cone made by a plane parallel to the base is a circle. Let the section abcd of the circular cone S-ABCD be parallel to the base. To prove that abcd is a circle.
Page 353 - The sum of the face angles of any convex polyhedral angle is less than four right angles.
Page 367 - An oblique prism is equivalent to a right prism whose base is a right section of the oblique prism, and whose altitude is equal to a lateral edge of the oblique prism. Hyp. OM is a right section of oblique prism AD', and OM ' a right prism whose altitude is equal to a lateral edge of AD'. To prove AD' =0= GM' . Proof. The lateral edges of GM
Page 421 - The volume of a frustum of a circular cone is equivalent to the sum of the volumes of three cones whose common altitude is the altitude of the frustum and whose bases are the lower base, the upper base, and the mean proportional between the bases of the frustum. Let V denote the volume, B the lewer base, b the upper base, H the altitude of a frustum of a circular cone.
Page 311 - The line joining the midpoints of two sides of a triangle is parallel to the third side and equal to one-half of it.
Page 343 - COR. 1. If two planes are perpendicular to each other, a perpendicular to one of them at any point of their intersection will lie in the other plane.
Page 373 - COR. 2. The volume of a rectangular parallelopiped is equal to the product of its base by its altitude.