# 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 *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 *cylinder* is called an *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 *cylinder* is called a *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 *cone* is called an *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.