# Tangent plane of a ball, a cylinder and a cone

*Tangent planes of curved surfaces: spherical surface,*

surface of a round cylinder, surface of a round cone.

Cylinder inscribed into a prism. Cylinder circumscribed

around a prism. Cone inscribed into a pyramid. Cone

circumscribed around a pyramid.

surface of a round cylinder, surface of a round cone.

Cylinder inscribed into a prism. Cylinder circumscribed

around a prism. Cone inscribed into a pyramid. Cone

circumscribed around a pyramid.

Consider
*
three points
*
A, B, C on a some curved surface ( Fig. 94 ) and draw through them the crossing plane
*
P
*
. Two points B and C
we’ll move to point A along the two
*
different
*
directions. Then, the plane
*
P
*
will approach the some position
*
Q
*
independently
on a place, where points B and C have been taken, and a path of their moving to point A. The plane Q is called a
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tangent
*
plane in point
A. It is possible, that there is not a tangent plane in some point of a surface. For example, a conic surface has no a tangent plane in a vertex of a
cone.

The plane P, which is a
*
tangent plane of a spherical surface
*
( Fig.95 ), is perpendicular to radius OA, drawn to the tangency point A; a tangent
plane of a spherical surface has only one common point with the surface – a tangency point.

The plane P, which is a
*
tangent plane to a surface of a round cylinder
*
in the point A ( Fig.96 ), goes through the generatrix MN,
containing the point A, and a tangent line BC of a base circle, containing the point N. A plane, tangent to a surface of a round cylinder is removed from all
points of its axis by a distance, equal to radius of a cylinder base. The plane P, which is a
*
tangent plane to a surface of a round cone
*
in the point A,
which doesn’t coincide with the vertex S ( Fig.97 ), goes through the generatrix SB, containing the point A, and a tangent line MN of a base circle,
containing the point B. A
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cylinder
*
is called an
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inscribed into a prism
*
, if lateral faces of the prism are planes, tangent to the cylinder, and planes
of their bases are the same. A
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cylinder
*
is called a
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circumscribed around a prism
*
, if lateral edges of the prism are generatrices of a lateral
surface of a cylinder, and planes of their bases are the same.

A
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cone
*
is called an
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inscribed into a pyramid
*
, if lateral faces of the pyramid are planes, tangent to the cone, and planes of their bases are
the same. A
*
cone
*
is called a
*
circumscribed about a pyramid
*
, if lateral edges of the pyramid are generatrices of a lateral surface of a cone,
and planes of their bases are the same.