How to construct a Star - Home.ne.jp

Q6: What is the measure of an interior angle of a polygon?The formula for the sum of interior angles of polygon whose number of sides is n is180°(n-2). When n=5, 180°(5-2)=180°(3)=540°. 540°/5=108°.Q7: What is the measure of an interior angle of a star?108°÷3 = 36°Since an interior angle of a star is 36°, it’s very difficult to construct a star without the useof a protractor.But we are not allowed to use a protractor in constructing. We are allowed to use only acompass and a straightedge.Do you think that a star is a beautiful figure?The hint to construct a star exists in the beautyof it.Namely, it is “The Golden Ratio”.Let us introduce “The Golden Rectangle (Ratio, Section)” to you!This can be the hint in order to construct a star.If you have a cell card, please look at the shape ofit carefully. As you see, the shape of a cell card is arectangle. The measure of sizes are as shown in thepicture on the right.Likewise, the ratio of width to height of cash cardand calling card is also close to 1: 1.6. We can getthis ratio from the proportion as shown below.5.35cm1 : x = x : ( 1 + x ) (x>1)2x − x −1= 01+ 5x =28 .5 ÷ 5.35 = 1.598.5cmThe ratio of a shorter length toa longer length is equal to alonger length to a whole length.This ratio “1:1.618” is called “the golden ratio”. It has enchanted peoples’aesthetic sense by the beauty of its balance beyond the centuries. The golden ratio isalso seen to the pyramid in Egypt and pictures drawn by Leonardo Da Vinci.In this page, we introduced only the summary of “The golden ratio”. We suggestyou to research it more. It is worth studying.