Unizor - Geometry3D - Cylinders - Easy Problems
Channel: Zor Shekhtman on UNIZOR Education
Date: 2015-10-09

Unizor - Creative Minds through Art of Mathematics - Math4Teens

Notes to a video lecture on http://www.unizor.com

Cylinders - Easy Problems

Problem 1

Prove that an intersection of a cylinder with any plane, that is parallel to its bases and is located in between them, is a circle.

Problem 2

Prove that an intersection of a cylinder with any plane that contains a generatrix at its two positions around the base is a rectangle.

Problem 3

A cube with an edge length of d is inscribed into a cylinder such that one face of a cube (a square) is inscribed into one base of a cylinder (a circle) and an opposite face of a cube is inscribed into an opposite base of a cylinder.
What is the total area and a volume of a cylinder?

Answer:
Area = 2πd²(1+√2)/2
Volume = πd³/2

Problem 4

Prove that the line connecting two centers of bases of a cylinder (center line) is its axis of symmetry.

Problem 5
Given a cylinder of a radius R and height H. A plane is drawn parallel to its center line on a distance d from it (less than a radius).
What is the area of a section of a cylinder cut by this plane?

Answer:
2H√R²−d²

Problem 6

A regular tetrahedron with all edges equal to d is inscribed into a cylinder such that one of its faces (an equilateral triangle) is inscribed into one base of a cylinder (a circle) and an opposite vertex of a tetrahedron coincides with a center of an opposite base of a cylinder.
What is the total area and a volume of a cylinder?

Answer:
Area = 2πd²(1+√2)/3
Volume = πd³√6/9