The Construction of a Regular Pentagon
The Golden Ratio
The golden ratio is defined as:
,
Let . Rearranging term and applying the quadratic formula we can solve for directly:
Note that because we consider to be a ratio between strictly positive quantities, we will only take the positive form of this value. Hence .
The Golden Ratio in a Pentagon
Consider a regular pentagon of side length 1, . Draw lines connecting each interior vertex to form a pentagram in the center of the shape. Observe the smaller pentagon contained within , denoted .
Let the distance be . Constructing a circle of radius centered at shows that .
Let the distance of be . Note that by constructing a circle of radius centered at , we can see that is also equal to .
Consider the two triangles and . The triangle proportionality theorem states that if a line is parallel to one side of a triangle and intersects the other two sides, then it divides those sides proportionally. Given that , we know that is proportional to . Hence,
What this informs us of is that the ratio of a diagonal of a regular triangle, , and a side length, , is the golden ratio.
Constructing the Golden Ratio
Given this representation we can think of the golden ratio as plus the hypotenuse of a right triangle with side lengths and .
Begin the construction by drawing a line segment. For the purposes of our construction, we will consider this length to be the unit length of . Take this line segment and construct a square. Find the midpoint of the base of the square, and draw a segment from the midpoint to the upper right corner of the square. This is the hypotenuse of a by triangle and is thus equal to .
Construct a circle centered at the midpoint, of radius equal to the constructed line segment. Extend the original base of the square to intersect with the circle. This distance will be of .
Constructing a Pentagon out of the Golden Ratio
Take the above construction of the golden ratio, and construct two circles of radius centered at and . Their upper intersection will form the top of the pentagon.
Draw two new circles of radius centered at and . Their intersections with the original circles represent the bottom diagonal lines for the pentagon, and form the last two vertices of the pentagon.
Thus a regular pentagon constructed with only Euclidean construction.