**Polygons: GSP Tools**

by

**Sarah Major**

**Square**

To construct, create a circle and find its radius. Construct perpendicular lines from the endpoints of the radius. Where one of these lines intersects the sides of the circle, construct parallel lines to the other perpendicular line. Voila!

**Pentagon**

Construct a circle and its radius. Find the midpoint of the radius and construct a perpendicular segment from the end of the radius that is the same length as half the radius. Once this is done, construct one more segment to make this into a triangle. Find the midpoint of the hypotenus and then construct a circle with a center that is the same as the original circle but passing through this midpoint. Where the circle intersects the initial radius will create the endpoint of a segment that is half the length needed for the sides of the figure. Construct another circle passing through the original triangle with the segment we just found so that the diameter will be the side length needed. Continue to construct circles whose radius is the diameter of the previous circle until the whole pentagon is complete.

**Pentagon (given a radius)**

Draw a circle with the given radius. Construct a line perpendicular to the radius through the center. Where the perpendicular intersects the top of the circle will be one of the vertices of the pentagon. Find the midpoint of the radius and construct a circle with half of the original radius. Draw a line through the point where the perpendicular intersects the bottom of the circle and the small circle. With the intersection at the bottom of the circle as the center, draw a circle with a radius that is the length from the bottom of the circle to the intersection point with the small circle. The two intersection points with the original circle will be two more of the vertices of the pentagon. By drawing a segment between the top vertex and one of the other vertices, draw circles of this radius to find the other vertices (it will be where the circles intersect the original circle).

**Hexagon**

Construct a circle and its diameter. Split the diameter into its two radii. Find the midpoints of these two radii. Construct perpendicular lines through these midpoints. Where these perpendicular lines intersect the circle creates the segments needed for the sides of the hexagon. The other intersection needed is where the points from the endpoints of the diameter are.

**Hexagon (given side)**

Given the side, draw a circle with a radius that is the side. Find the midpoint of the side and construct a perpendicular. Where this intersects the circle above the radius, this will be the center of a circle with the same radius. Draw a line parallel to the radius through the center. The two intersections with the circle will be two of the vertices. Draw two circles with the same radius with centers at these vertices to find the other vertices.

**Octagon**

Construct a square (see above) and its diagonals. Construct four circles with radii that are half the length of the diagonals and pass through each corner. Where the circles intersect the square will create the segments for the sides of the figure. Connect these points to complete the octagon.

**Octagon (given side)**

Given the side, draw a circle with a radius that is the side. Draw the whole diameter. Draw a line perpendicular to the radius passing through the center. Draw a segment between the intersection of the perpendicular with the top of the circle and the right endpoint of the diameter. Find the midpoint of this segment and draw a line through it and the center of the circle. Where this line intersects the upper part of the circle creates the second side of the octagon. Continue to draw circles of this same radius to complete the octagon, draw lines parallel to the original radius to make it easier to find the intersections once one side is finished.

**Decagon**

Construct a circle, its radius, and then the midpoint of the radius. Then construct a perpendicular segment from the endpoint of the radius where it intersects the circle that is the same length as half of the radius. Connect the endpoint of this segment to the center of the circle to make a triangle. Find the midpoint of the hypotenuse of this triangle and construct a circle withe center that is the same as the first circle but passing through this midpoint. Where the circle intersects the initial radius will be the length of the segment needed for the sides of the figure. Construct the same size circles all around the figure to complete the figure.

Dodecagon

Construct a circle and its radius. Find the midpoint of this radius. Construct perpendicular lines through the center and the midpoint. Where these lines intersect the circle will create the ednpoints for the segments needed for the sides of the figure. Keep constructing small circles passing through the original circle with the same radii (the length of the side of the figure) to complete the dodecagon.