2
Polygons
Textbook: pp. 7–10
GoalLearn what a polygon is, its elements and diagonals, and what a convex polygon is.
New words
polygon · ko‘pburchakdiagonal · diagonalconvex polygon · qavariq ko‘pburchakadjacent vertices · qo‘shni uchlar
Explanation
A closed broken line that does not cross itself is called a simple closed broken line. The part of the plane made up of such a line and its inside is called a polygon. A polygon has equally many sides, vertices and angles. Two vertices on the same side are adjacent vertices, and a segment joining two non-adjacent vertices is a diagonal. A polygon is convex if it lies on one side of the line through any of its sides. A convex n-gon has n(n − 3) : 2 diagonals, because n − 3 diagonals leave each vertex and every diagonal is counted twice.
Worked examples
In a convex heptagon, 7 − 3 = 4 diagonals leave each vertex, so there are 7 · 4 : 2 = 14 diagonals in all.
A convex polygon has 20 diagonals. n(n − 3) : 2 = 20, so n(n − 3) = 40. For n = 8, 8 · 5 = 40, so it is an octagon.
Class activity
“Count the diagonals”: students draw convex polygons with 5, 6 and 7 sides, draw and count the diagonals and compare with the formula.
Practice
1
How many diagonals leave one vertex of a convex 12-gon?
9
2
How many diagonals does a convex 10-gon have in all?
35
3
A convex polygon has 54 diagonals. How many sides does it have?
12
4
Why does a quadrilateral have only 2 diagonals?
Only 4 − 3 = 1 diagonal leaves each vertex, and each diagonal joins two vertices, so 4 · 1 : 2 = 2.