How to Construct a Pentagon on Sketchpad Using the Golden Ratio

Use a ruler and compass to construct the golden ratio.

The golden ratio is a measurement describing two quantities (such as lengths or shapes) in which the ratio of the larger part to the smaller part is equal to the ratio of the whole to the larger part. The golden ratio or approximations of the ratio are often present in art and architecture due to its reputation as representing perfection or balance in nature. Construct a pentagon with a ruler and compass by creating a line segment with two sub-segments in the golden ratio.

Things You'll Need

• Ruler
• Compass

Instructions

1. Construct a line segment AB with your ruler and mark a point C along the line according to the golden ration. The ratio of the longer length to the smaller length on the segment should be the same as the ratio of the entire segment to the longer length. In other words, if AC is longer than CB, then AC : CB = AB : AC. If the line segment is 1-inch long, then AB = 0.618 inches and CB = 0.382 inches.

2. Place the point of your compass at point A and extend the pencil to point B. Draw a circle around point A with a radius equal to the length AB.

3. Place the point of your compass at point C, and extend the pencil to point A. Draw an arc that intersects the circle you drew in Step 2. Label the intersection of the arc and the circle point D.

4. Draw a ray extending from point D through point C, intersecting the circle you drew in Step 2. Label the intersection of the ray and the circle point E.

5. Label the point F that forms the golden ratio on line AD, similar to the point C you drew on line AB. Because AB and AD are the same length, point F will be the same distance from A as point C (0.618 inches if AB and AD are both 1 inch).

6. Draw a ray extending from point B through point F, intersecting the circle you drew in Step 2. Label the intersection of the ray and the circle point G.

7. Draw lines connecting points A, E, B, D and G on the circle to form a regular pentagon inscribed in the circle.

References

• Mac Tutor History of Mathematics: Golden Ratio