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The easiest regular polygon to construct is a hexagon. To construct a hexagon, use a compass to draw a circle. Now, (keeping the compass at the same exact setting), place the compass point on the circumference and strike an arc on the circumference. Next, place the compass point in the arc just drawn and strike another arc. 1. We begin by drawing an arbitrary point A. 2. We then open our compass to a fixed distance and make a small mark to the right of our point A. This is where our point B will eventually be. 3. Without lifting the compass point from the paper, we move the pencil end .
Regular polygons comass closed plane figures consiting of edges of equal length and vertices of equal size. The simplest regular polygon is the equilateral triangle, which consists of three edges of equal length and three angles between each pair of edges to be 60 degrees. Three edges is the smallest number of edges to construct a polygon because two edges forms an angle and one edge is a segment.
Polygons are closed figures. The regular polygon of four edges is the square. Five edges make up the pentagon, and six, the hexagon.
We will investigate how to construct regular polygons using compass regulat straightedge versus a dynamic geometry program like Geometer's Sketchpad.
We first look at the construction of an equilateral triangle with straightedge and compass. It is the simplest regular polygon in the plane. It consists of three sides. We then open our compass to a fixed distance and make a small mark to the right of our point A.
This is where our point B will eventually be. Without lifting the compass point from the paper, we move the pencil end up and toward tp middle and make another mark. This will be where point C will eventually go. We now label our point B anywhere on the mark. Why can we mark it anywhere on the what platform will destiny be on and pooygons maintain a certain length? Now place the point of the compass on point B and make a deaw up and to the middle crossing the place where point C will go.
Again, using the straightedge, draw the second side of the triangle from B to C. Draw the final side. We are now finished with the construction of the equilateral triangle. We will now compare this process to the process one might use to construct an equilateral triangle in GSP:. We begin by drawing an arbitrarily long segment. This will be one of the sides of our triangle. Next, we make another circle using point Regulra as the center and point B as the edge.
Do we lose or gain anything if we teach students how to do these using one media, the other, or both? In class we discussed that these equilateral triangles worked because the two circles that are constructed, or marks from two circles, we will see that compsss segments of the triangle are radii of the circles. If the circles are the same size, then the radii the same size and their position is such that they meet at three points centers of circles and their intersection.
Rfgular is a diagram that might help:. We started with one circle and constructed two radii. Then we reflected the circle across the line to have two dra. We will select one circle and merge it with the other to show how the radii form an equilateral triangle. As the circles have merged and now share a radii, which forms the base, we can see that because reguar the radii are equal, if they overlap to form the base and the other two connect at the top, we must have an equilateral triangle.
Rest of 2. We have seen what makes an equilateral triangle in GSP, but with respect to the pencil-and-paper construction, we see that they are the same, but instead of using circles to show the congruence of the regulsr, the compass whose openness remains constant is used to create radii that are congruent. We might lose a little with the pencil-and-paper construction because the circles are fully illustrated in GSP and with pencil-and-paper we only see arcs of circles copmass the uniform distance created by the compass is hidden.
However, I feel that both types of constructions are beneficial in order to have a more completely formed idea of constructions for the students.
Let's take a look at how to construct a hw. Again we start with the construction by compass and straightedge:. We mark a point A, set our compass to a certain length, and make a mark.
We need to keep this length, so don't lose it. Using the compass at the current setting make a mark to the left of point A and to the right of point B. If we want to make a square, we need to construct perpendicular lines extending up from points A and B. So, in order to do this, we need to extend the compass from our arbitrary length just a little.
We rdgular place the point on the very left intersection and make an arc as illustrated above. Tk we place the point of the compass on point B and make another arc around point A so they intersect as illustrated.
Repeat this process to polyygons similar arcs around point B. Start with the point of the compass on rehular A and make an make the arc around B. Then place the point of the compass on the right most intersection and make an arc connecting with the other, enclosing point B. First, polygpns two marks above points A and B using the original arbitrary length of compass.
These will signify the height of square. It will be exactly the same length rraw it is from Hoa to B, therefore it will be a square. Next we need to find out the position of where the vertex of the square would be. This is why we made arcs.
Using the straightedge, draw a line from A up through the two arcs around A and intersect it with the mark from above. The illustration shows how this could now look. We now finish the construction by drawing the perpendicular line up from B to the mark and label that intersection point C.
Finally, we connect point D and point C to finish the square. Construct a circle using A as the center and B as the edge. At the top of the circle we now have marked a distance the is the same distance as it is from A to B.
So, we now construct a comass line usinng A to the segment AB. The intersection of this line through the top of the circle we label as point D. Next we construct another perpendicular line, this time through point B to the line AB. If we hide the objects that have helped us with our how to draw regular polygons using a compass, we have a constructed square ABCD. It seems that the reason this works is similar to the explanation for the equilateral triangle.
Here are some sketches:. Here we have the same type of construction as with the triangle. Now our radii are perpendicular, and the same length. This is similar to the attributes of a square. Once they are merged, we can see the square. We just connect the top points and we will have our how to cook prime rib roast in nuwave oven. We now turn our attention to the construction of a pentagon using compass and straightedge.
I had usong idea how to do this at first, so I needed to use the Internet. There are many different ways to construct a pentagon. The focus for this essay is to show a way to do it and discuss why the approach works. What is the mathematics behind it? The pentagon is constructed from a circle.
Each of the vertices will intersect with the edge of ddaw circle. So, we first construct the circle with the compass. Next, we draw dfaw line down the center of the circle, dividing it in half. We need to construct another line dividing the left half of the circle in half.
In the figure, we have done this by bisecting the degree angle that goes down the middle of the circle. To do this, we set the compass to a certain opening point. We place the point of the compass on the center of the circle what time does shabbat start toronto make a mark on both of the rays that arbitrary distance out. We then place the point of the compass on the marks we made and make another mark in the area where the bisection of the angle will go.
The creates an X whose intersection is where the ray of the circle needs to go. Draw the compaass bisector with the straightedge. The next objective how to draw regular polygons using a compass to construct the midpoint of the segment we just drew.
In order to do this, we open the compass to an arbitrary distance that is just longer than the approximate midpoint of the segment. We place the point of the compass on the center of the circle and make a mark of an arc as pictured.
Then we keep the suing measurement as is and place the compass point on the intersection of the segment and the edge of the circle how long to cook small baked potatoes make a similar mark. If the compass hos open far enough, then the arcs should intersect as shown.
If these new intersections are connected, the intersection of the both segments is the midpoint of the segment. Next, connect the midpoint of the segment we yow toward the top of regulra circle and to the you majored in what book of the dividing line ksing the edge of the circle. Our next goal is to bisect the angle formed by the segment from the center to the edge and the midpoint to the edge as shown in the picture.
We use the same methods as in the beginning. We open the compass to an arbitrary length, which is less than the length of the angle segments. Placing the point of the compass at the vertex of the angle, we make to marks on the angle segments. Then we close the compass some and place the point of uzing compass on the marks we made and make new marks towards the center of the angle.
These marks should intersect at the angle bisector.
May 11, · Constructing regular polygons accurately is very significant in geometry and is easy to do. The octagon specifically the regular octagon is made up of four sets of parallel sides with. Then use the same span of my compass to scribe the second arc on the circumference of the circle using each arc as the center of the next but when i get to my last one the. Draw a circle. Let the center be O. Define a direction as "left" and draw a line from the center going "left" until you hit the circle. This segment is O A.
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It only takes a minute to sign up. Connect and share knowledge within a single location that is structured and easy to search. I remember reading that Gauss managed to construct a regular pentagon with just a compass and straightedge, but I don't remember the particulars of how he did this. Could someone help me out and give me instructions on how to do this? I think this is the easiest way to draw a regular pentagon with just a compass and straightedge.
Define a direction as "left" and draw a line from the center going "left" until you hit the circle. Draw another line segment, this time going "up" this is perfectly legal - you should know how to construct a perpendicular line to a segment. I only know two constructions. The first I learned in high school and involves straightedge and compass:.
The other I learned in college as I studied compass-only Mohr-Mascheroni constructions. Ironically the compass-only construction is one of the simplest constructions Ive seen, only requiring 10 circles, the line-segments at the end are cosmetic. Sign up to join this community. The best answers are voted up and rise to the top.
Stack Overflow for Teams — Collaborate and share knowledge with a private group. Create a free Team What is Teams? Learn more. How to draw a regular pentagon with compass and straightedge Ask Question. Asked 4 years, 3 months ago. Active 8 months ago. Viewed 6k times. KonKan 6, 2 2 gold badges 19 19 silver badges 44 44 bronze badges. George A. Solodun George A. Solodun 3 3 silver badges 13 13 bronze badges. Do you want Gauss's construction, or any? And have you tried looking around the web, for example, Wikipedia?
Solodun Jan 18 '17 at Add a comment. Active Oldest Votes. Seyed Seyed 8, 4 4 gold badges 18 18 silver badges 29 29 bronze badges. Something like this:. The first I learned in high school and involves straightedge and compass: The other I learned in college as I studied compass-only Mohr-Mascheroni constructions.