What geometric figure does a regular polygon begin to resemble as the number of sides increases? How come the answer is a circle?
http://www.mathsisfun.com/geometry/polygons.html
Also --
http://www.mathopenref.com/polygonincircle.html
A regular polygon is a polygon that has equal length sides and equal interior angles. As the number of sides in a regular polygon increases, it begins to resemble a circle.
To understand why a regular polygon resembles a circle, let's examine the process of increasing the number of sides step by step:
1. Triangle (3 sides): A triangle has straight sides and angles, and it does not resemble a circle.
2. Square (4 sides): A square is still noticeably different from a circle, as it has straight sides and right angles.
3. Pentagon (5 sides): A pentagon starts to look more circular compared to a square, but it still has distinct straight sides and angles.
4. Hexagon (6 sides): As we go further and increase the number of sides, the hexagon begins to appear more circular. The sides start to curve slightly, and the angles become less noticeable.
5. Heptagon, Octagon, Nonagon, Decagon, etc.: As the number of sides continues to increase, the shape becomes even more circular. The sides become smoother curves, and the interior angles become less prominent.
As the number of sides approaches infinity, a regular polygon becomes closer and closer to a circle. A circle is a special type of regular polygon with an infinite number of sides. It has no straight sides or angles, and all points on its circumference are equidistant from the center.
This resemblance of a regular polygon to a circle becomes more apparent when considering the inscribed and circumscribed circles. In an inscribed circle, the circle is inside the polygon and touches all the sides at their midpoints. A circumscribed circle surrounds the polygon and touches all its vertices. As the number of sides increases, the inscribed and circumscribed circles become better approximations of a true circle.
So, while a single regular polygon does not perfectly resemble a circle, as the number of sides increases, its resemblance to a circle becomes remarkably close.