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About the polygons
- A polygon is convex if all its interior angles are convex.
- A polygon is concave if it has a concave interior angle. (Figure 49)
We talk about convexity and concavity not only in the case of planar polygons, but also generally in the case of planar figures and spatial solids.
Figure 49
DEFINITION: A (planar or spatial) figure is convex if the points of the line segment connecting any two points of the figure, these two points inclusive, belong to the figure too. (Figure 50)
DEFINITION: A (planar or spatial) figure is concave if it is not convex, i.e. there is a line segment – connecting two points of the figure – which does not completely belong to the figure. (Figure 51)
Figure 50
Note: The latter general definitions of convexity and concavity have the same meaning for the polygons as the determinations given with the help of interior angles at the beginning of the chapter. So in the case of polygons the corresponding definitions are equivalent.
Figure 51
Example 1
Let us give the number of diagonals of a convex polygon with \latex{n} sides.
Figure 52
Solution
No diagonals can be drawn from a vertex of a convex polygon with n sides to itself and to the two adjacent vertices; therefore \latex{n-3} diagonals can be drawn from any of its vertices. (Figure 52) Considering all of its vertices it would mean \latex{n\times(n-3)} diagonals, but since every diagonal was counted from both of its end-points, the actual number of diagonals is half of it. With this we have proven that the number of diagonals of a convex polygon with n sides is: \latex{\frac{n\times(n-3)}{2}}.
The NUMBER OF DIAGONALS
of a convex polygon
with \latex{n} sides is:
of a convex polygon
with \latex{n} sides is:
\latex{\frac{n\times(n-3)}{2}}
Example 2
Let us find the sum of the interior angles of a convex polygon with n sides.
Solution
We have just seen that \latex{n-3} diagonals can be drawn from one vertex of the polygon. These \latex{n-3} diagonals divide the polygon into \latex{n-2} pieces of triangles. When adding the interior angles of these triangles we get the sum of the interior angles of the polygon (Figure 53). Based on these the sum of the interior angles of a convex polygon with n sides is: \latex{(n-2) \times180º}.
Figure 53
Note: If a concave polygon has only one concave angle, then starting from the concave vertex it can be proven in the same way that the sum of the interior angles can be given with the above expression. In the case of several concave angles the proof is more complicated, but it can be shown that the above formula is also true for any arbitrary polygon with n sides, i.e. the sum of the interior angles is \latex{(n-2) × 180º}.
The SUM OF THE INTERIOR
ANGLES of a convex polygon
with n sides is:
ANGLES of a convex polygon
with n sides is:
\latex{(n-2)\times180^{\circ}}
DEFINITION: A polygon is regular if all its sides are of equal length and all its angles are of equal size. (Figure 54)
Example 3
What is the size of one interior angle of a regular polygon with n sides?
regular pentagon
regular octagon
Figure 54
Solution
The sum of the interior angles of a polygon with n sides is \latex{(n-2) \times180º}. Since all the interior angles are equal, the size of one interior angle is:
\latex{\frac{(n-2) \times180º}{n}}.
ONE INTERIOR ANGLE
of a regular polygon
with n sides:
of a regular polygon
with n sides:
\latex{\frac{(n-2)\times 180^{\circ}}{n}}
Exercises
{{exercise_number}}. How many diagonals does a convex polygon have if the number of sides is
- \latex{5;}
- \latex{7;}
- \latex{8;}
- \latex{12;}
- \latex{29?}
{{exercise_number}}. Calculate the sum of the interior angles of a polygon if the number of sides is
- \latex{5;}
- \latex{7;}
- \latex{8;}
- \latex{12;}
- \latex{29}.
{{exercise_number}}. Calculate one interior angle of a regular polygon if the number of sides is
- \latex{5};
- \latex{7};
- \latex{8};
- \latex{12};
- \latex{29}.
{{exercise_number}}. How many sides does a convex polygon have if the number of diagonals that can be drawn from one of its vertices is
- \latex{5;}
- \latex{7;}
- \latex{13;}
- \latex{18;}
- \latex{31?}
{{exercise_number}}. How many sides does the convex polygon have in which the sum of the interior angles is
- \latex{540^{\circ};}
- \latex{900^{\circ};}
- \latex{1,620^{\circ};}
- \latex{3,060^{\circ}?}
{{exercise_number}}. How many diagonals does a convex polygon have if the number of its sides is
- \latex{5;}
- \latex{8;}
- \latex{12;}
- \latex{23?}
{{exercise_number}}. Prove that the sum of the exterior angles of any convex polygon is \latex{360^{\circ}}.
{{exercise_number}}. How many sides does the convex polygon have in which the sum of the interior angles is
- twice;
- four times;
- \latex{7.5} times;
- \latex{17} times the sum of the exterior angles?
{{exercise_number}}. Calculate one interior angle and one exterior angle of a regular polygon if the number of its sides is
- \latex{6;}
- \latex{10;}
- \latex{16;}
- \latex{21}.
{{exercise_number}}. How many sides does the regular polygon have in which the size of one interior angle is
- \latex{60^{\circ};}
- \latex{180^{\circ};}
- \latex{140^{\circ};}
- \latex{157.5^{\circ}?}
{{exercise_number}}. How many sides does the convex polygon have in which the number of diagonals is
- \latex{2;}
- \latex{5;}
- \latex{14;}
- \latex{27?}
{{exercise_number}}. Calculate the size of the interior angles of the pentagram or “star pentagon” (it is actually an equilateral concave decagon the convex vertices of which are the vertices of a regular pentagon) shown in the figure.
