How to draw a pentagon using a compass
Video: Draw pentagon with compass, inscribe a given circle.(Step-by-Step) 2024, July
The task of constructing a regular pentagon is reduced to the problem of dividing a circle into five equal parts. Since the regular pentagon is one of the figures containing the proportions of the golden ratio, painters and mathematicians have long been interested in its construction. Now several methods have been found for constructing a regular polygon inscribed in a given circle.
You will need
- - ruler
- - compass
Instruction manual
one
Obviously, if we construct a regular decagon, and then connect its vertices through one, we get a pentagon. To construct a decagon draw a circle of a given radius. Designate its center with the letter O. Draw two radii perpendicular to each other, in the figure they are indicated as OA1 and OB. Divide the OB radius in half using a ruler or by dividing the segment in half using a compass. Draw a small circle with center C in the middle of the segment OB with a radius equal to half OB.
Connect point C to point A1 on the original circle in a ruler. Segment CA1 intersects the auxiliary circle at point D. Segment DA1 is equal to the side of the regular decagon inscribed in this circle. With a compass, mark this segment on a circle, then connect the intersection points through one and you will get a regular pentagon.
2
Another way was found by the German artist Albrecht Durer. To build a pentagon by its method, start again by building a circle. Mark its center O again and draw two perpendicular radii OA and OB. Divide the OA radius in half and mark the middle with the letter C. Set the compass needle in point C and open it to point B. Draw a circle of radius BC until it intersects with the diameter of the original circle on which the radius OA lies. Designate the intersection point D. Line BD is the side of the regular pentagon. Postpone this segment five times on the original circle and connect the intersection points.
3
If you want to build a pentagon on its given side, then you need a third way. Draw the side of the pentagon along the ruler, mark this line with the letters A and B. Divide it into 6 equal parts. From the middle of segment AB, draw a beam perpendicular to the segment. Draw two circles of radius AB and centers in A and B, as if you were going to divide the line in half. These circles intersect at point C. Point C lies on the ray emanating perpendicular upward from the middle AB. Put the distance equal to 4/6 of the length AB from C upwards along this ray, mark this point D. Draw a circle of radius AB centered at point D. The intersection of this circle with two auxiliary ones constructed earlier will give the last two vertices of the pentagon.
note
The ratio of the diagonal of a regular pentagon to its side is the golden ratio (irrational number (1 + √5) / 2).
Each of the five inner corners of the pentagon is 108 °.
Useful advice
If you connect the vertices of a regular pentagon with diagonals, you get a pentagram.