A GEOMETRIC SHAPE Is a Shape That Follows a Mathematical Rule
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UNIT 2: GEOMETRIC SHAPES
A SHAPE is a 2-dimensional area that is defined in some way. It could be by a line, space, color, texture or a variety of other ways. There are two types of shapes: geometric and free-form (organic)
A GEOMETRIC SHAPE is a shape that follows a mathematical rule.
These are the most important ones: POLYGONS
DEFINITION: A polygon is a shape closed by straight lines
REGULAR polygons are those which have EQUAL SIDES AND ANGLES TRIANGLES
QUADRILATERALS REGULAR POLYGONS INSCRIBED IN A CIRCUMFERENCE
Definition: They are the regular polygons that have all their vertices (the plural of vertex is vertices)
ON the circumference (not inside the circle) In this unit we will learn how to make:
1. An EQUILATERAL TRIANGLE INSCRIBED IN A CIRCUMFERENCE 2. AN ISOSCELES TRIANGLE INSCRIBED IN A CIRCUMFERENCE (THIS IS NOT REGULAR) 3. A SQUARE INSCRIBED IN A CIRCUMFERENCE 4. A PENTAGON INSCRIBED IN A CIRCUMFERENCE 5. A HEXAGON INSCRIBED IN A CIRCUMFERENCE 6. A HEPTAGON INSCRIBED IN A CIRCUMFERENCE 7. AN OCTAGON INSCRIBED IN A CIRCUMFERENCE 1.- EQUILATERAL TRIANGLE. INSCRIBED IN A CIRCUMFERENCE
st 1 Draw a circumference centered in o
nd 2 Draw any diameter AP .
rd 3 Draw an arch centered in O and with OP radius until it cuts the circumference in B and C.
4st Link the vertices A, B and C.
2.- AN ISOSCELES TRIANGLE INSCRIBED IN A CIRCUMFERENCE (THIS IS NOT REGULAR) If when drawing the arch you do not pass through the center you will obtain an isosceles triangle 3.- A SQUARE INSCRIBED IN A CIRCUMFERENCE st 1 Draw a circumference centered in O
nd 2 Draw any diameter AP .
rd 3 Draw its perpendicular bisector
A
C O D
th 4 Link the vertices A, C, B and D. B
4.- A PENTAGON INSCRIBED IN A CIRCUMFERENCE
ST 1 Follow the steps 1 2 and 3 as to make a square
A
C O D
nd 2 Draw the bisector of OD (you will obtain M)
A B
B
C O M D nd 3 Draw an arch centered in M with MA radius, from A till you reach CO (you will obtain L)
A
A
C O M D th L 4 Transferring the measure AL through the circumference we will obtain the vertices of the pentagon.
C O M D L
B
B 5.- A HEXAGON INSCRIBED IN A CIRCUMFERENCE
st 1 Draw two arches with the same radius than the circumference (OB) centered in A and in B until they cut the circumference.
You will obtain D, C, E and F
2º Link A, E, D, B, C and F.
6.- A HEPTAGON INSCRIBED IN A CIRCUMFERENCE Following the GENERAL METHOD
Following the GENERAL METHOD METHOD for how TO DRAW ANY REGULAR POLYGON (INSCRIBED IN A
CIRCUMFERENCE) (WITH ANY NUMBER OF SIDES)
This is a general method and has a small error, so if you know the specific method for the polygon, this will be more accurate.
1º Draw any diameter (AB) A
B 2º Divide this diameter into the number of equal parts you want to obtain in the polygon; I want to get the heptagon, (7 sided polygon) so I divide the diameter into 7 equal parts. We will do this by applying the Thales theorem.
A
B
rd 3 Trace two arcs centred in each side of the diameter, and with the diameter as the radius (of the arcs). Where these two arcs intersect we will obtain the point
A
P
4th Trace a straight line starting at P till the SECOND MARK OF THE DIAMETER (NOT THE AUXILIARY LINE) and
B continue it until it reaches the circumference. We will obtain point L.
A L
P
5th The segment AL, will be the measure we have to mark consecutivelyA through the circumference to obtain the vertexes of the L polygon. B P
B 6º By joining these points with straight lines you will get the polygon (the heptagon in this particular case)
A L
P
B
7. - AN OCTAGON INSCRIBED IN A CIRCUMFERENCE ST 1 Follow the steps 1st and 2nd two draw a square
A
C O D nd 2 Then draw the bisectors of the 4 right angles. That way we obtain E, F, G y H.
B
th 3 Join the points A, E,C,G,B,F,D and H.