Sunday, June 26, 2011

• Given a unit square, what is the ratio of the areas of two circles - the circle with maximum area lying completely inside the circle, and the area of the smallest area circle that completely encloses the square. 
• How to do the same problem with a cube and inscribing and circumscribing  sphere?
• Given a unit square, you want to place a triangle whose area is maximum that lies completely inside the square. What will be its area?
• How to do the same problem with a triangle in the cube?
• How many corns on a corn? Typically a corn is cube shaped  measuring 4mm, and a corn cone is of diameter of 7cms and has a length of 27cms.
• Show that in a circle, if we take two chords of the same length, then the angle they form at the center of the circle is the same. 
• Show that in a circle, the line joining the center of the circle to the mid point of any chord, is perpendicular to the chord.
• How many circles pass through a single point? Two points? What about three points? Show that if three points do not lie on a straight line, there is a unique circle passing through them.
• Show that the angle subtended by any arc of the circle at the center is double of the angle subtended by this arc at any point on the circle.