How do you construct a regular pentagon and a regular decagon? Can you construct a regular n-gon?

Question

How do you construct a regular pentagon and a regular decagon? Can you construct a regular n-gon?

Answer 1

I assume that you are wanting straightedge and compass construction.



To make a decagon, let O be the center of a circle, with BOA one diameter and COD a perpendicular diameter (if you can't construct this, you shouldn't be doing this construction.) Bisect OA at the point E. Now draw a circle centered at E of radius ED. Let this circle intersect the line BO at the point F. Then OF is the length of the side of a decagon. Mark off arcs ofthis radius around the original circle.



For a pentagon, you can either just skip every other vertex from the decagon (easy!) or it turns out that the side length is the same as the distance FD above.



The problem of which n-gons can be constructed by ruler and compass was solved by Gauss when he was 17 years old. (!) For polygons with a prime number of sides, the only ones that are constuctible are those which are one more than a power of 2 (the Fermat primes): so far the only ones known are 3,5,17,257,65537.



Then, the only constructible n-gons are ones where n is a power of 2 times a (possibly empty) product of Fermat primes with no repeats. So, 15=3*5 is possible, but 9=3*3 is not. Nor are 7,11,13,14,25, etc. However, all of 4,8,16,48,51,761, 204 are.



Hope this helps!

Answer 2

How To Construct A Pentagon

Answer 3

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Answer 4

a regular n-gon can be constructed on a circle.

Answer 5

do it urself

Answer 6

To construct a regular polygon, I guess you must first know the length of each side of it & then be able to calculate the central angle (i.e. the angle formed by the two successive vertices at the centre of the polygon).



Say, you want to construct a regular pentagon of side 4 inches; first determine the central angle of the pentagon using the formula



CA=360/no. of sides



CA = 360/5 = 72 degrees



If you notice carefully, the central angle corresponding to a side of the pentagon & that particular side itself, are parts of an isosceles triangle (the unequal angle is 72 degress).



Hence, 72 + EA + EA = 180 (EA - each equal angle of the triangle) EA = 108/2 = 54 degrees.



Now that we have got our measurements, we can start construction:



1. Draw a line segment of 4 inches length.



2. Construct the triangle that we just discussed. i.e. use a protractor, at both ends of the segment draw lines at 54 degrees which intersect at a point - this is the centre of the pentagon.



3. Either you can continue the same process for the remaining sides, or you can use a shortcut method:



Draw a circle about the centre of the pentagon with radius equal to the distance between the pentagon centre & one end of the line segment we saw above. Now, for this circle, we already have two radii lines at 72 degrees to each other - using your protractor, draw more radii lines that are 72 degrees to each other in succcession - you will have an overall of 5 such lines & therefore 5 points where they intersect the circle. Now, connect every succesive point using straight lines.



Finally, erase the circle & the radii lines. Your pentagon has been constructed.



You can do the same for a decagon - only difference is that your central angle will be CA = 360/10 = 36 degrees. Your equal angles EA = (180-36)/2 = 72 degrees each. Otherwise, the same procedure. You can repat this for any polygon.

Answer 7

for pentagon ..... u need to draw exact lines at an exact angles of 360/5=72

for decagon ...... u need to draw exact lines at an exact angle of 360/10=36

for n-gons ....... same at an angle of 360/n where n is the no. of sides



u hav to take d angles on d other side of the protractor ... i.e., anticlockwise reading or convert d answers to clockwise readings jus by (180-answer) calculation ......

Answer 8

a regular pentagon has angles of 108 degrees

a regular decagon has angles of 64 degrees



these can be derived from this formula:

Sum of interior angles=180(n-2)



degree of one interior angle=(180(n-2))/n



Just keep all the angles equal, and make sure the sides are of equal length(any length, as long as it stays the same for each side), and you've got yourself a regular n-gon!!!