The rectangle with length 6 and width 2 has the same perimeter and area as the triangle whose sides have lengths 5, 5, and 6.

Is there a triangle with the same perimeter and area as the rectangle that has length 3 and width 2 ? If so, find the lengths of its three sides. If not, explain why not.

The perimeter would be 10 and the area would have to be 6.

The triangle with a given perimeter would have the maximum area if the sides were all the same length--an equilateral triangle.  In this case each side would be 10/3. Its height would be 10sqrt(3)/6, and its area would be 100sqrt(3)/36 ~= 4.811252243246881. No way of rearranging the sides would make it any larger in area, so it could not get up to 6.

Posted by Charlie on 2007-01-02 13:50:19