ICSE Class 9 Maths Chapter 14 Polygons

14 Polygons

Points To Remember

Polygon: A closed plane figure bounded by three or more line segments is called a polygon. These line segments are called its sides and by joining two non-consecutive vertices of a polygon is called its diagonal and the point of intersection of two consecutive sides of a polygon is called a vertex.

Convex Polygon: If each angle of a polygons is less than 180°, then it is called a convex polygon.

Concave Polygon: If at least one angle of a polygon is a reflex angle i.e. more than 180°, then it is called a concave polygon.

Regular Polygon: If all sides of a polygon are equal and all angles are equal, then it is called a regular polygon.

Theorems

The sum of all the interior angles of a convex polygon of n sides is (2n - 4) right angles.

The sum of all the exterior angles of a convex polygon is 4 right angles.

Some More Results

For All Convex Polygons

Sum of all interior angles of a polygon of n sides = (2n - 4) right angles.

Sum of all exterior angles of a polygon of n sides = 4 right angles.

At each vertex of a polygon, we have: Interior Angle + Exterior Angle = 180°.

For Regular Polygons

Each interior angle of a regular polygon of n sides = \[\frac{2n-4}{n} \text{ rt. } \angle s = \left[\frac{(2n-4) \times 90°}{n}\right]\]

Each exterior angle of a regular polygon of n sides = \[\left(\frac{360}{n}\right)°\]

If each exterior angle of a regular polygon is x°, then number of its sides = \[\left(\frac{360}{x}\right)\]

Greater is the number of sides in a regular polygon, greater is the value of its interior angle and smaller is the value of its each exterior angle.

Number of diagonals in a polygon of n sides = \[\left[\frac{n(n-1)}{2} - n\right]\]

Exercise 14 (A)

Question 1

Write in degrees the sum of all interior angles of a:

(i) Hexagon (ii) Septagon

(iii) Nonagon (iv) 15-gon

Solution

(i) Sum of interior angles of a hexagon = (2n - 4) right angles = (2 × 6 - 4) × 90° = (12 - 4) × 90° = 8 × 90° = 720° Ans.

(ii) Sum of interior angles of a septagon = (2n - 4) right angles = (2 × 7 - 4) × 90° = (14 - 4) × 90° = 10 × 90° = 900° Ans.

(iii) Sum of interior angles of nonagon = (2n - 4) right angles = (2 × 9 - 4) × 90° = (18 - 4) × 90° = 14 × 90° = 1260°

(iv) Sum of interior angles of a 15-gon = (2n - 4) right angles = (2 × 15 - 4) × 90° = (30 - 4) × 90° = 26 × 90° = 2340° Ans.

Question 2

Find the measure, in degrees, of each interior angle of a regular:

(i) Pentagon (ii) Octagon

(iii) Decagon (iv) 16-gon

Solution

We know that each interior angle of a regular polygon of n sides = \[\frac{(2n-4)}{n}\] right angles.

(i) Each interior angle of pentagon = \[\frac{(2n-4)}{n}\] right angles = \[\frac{2 \times 5 - 4}{5} \times 90°\] = \[\frac{10-4}{5} \times 90° = \frac{6}{5} \times 90° = 108°\] Ans.

(ii) Each interior angle of octagon = \[\frac{2n - 4}{n}\] rt. angles. = \[\frac{2 \times 8 - 4}{8} \times 90°\] = \[\frac{16-4}{8} \times 90° = \frac{12}{8} \times 90°\] = 135° Ans.

(iii) Each interior angle of decagon = \[\frac{2n - 4}{n}\] rt. angles. = \[\frac{2 \times 10 - 4}{10} \times 90°\] = \[\frac{20-4}{10} \times 90° = \frac{16}{10} \times 90°\] = 144° Ans.

(iv) Each interior angle of 16-gon = \[\frac{2n - 4}{n}\] rt. angles. = \[\frac{2 \times 16 - 4}{16} \times 98 = \frac{32-4}{16} \times 90°\] = \[\frac{28}{16} \times 90° = \frac{315}{2}\] = 157.5° Ans.

Question 3

Find the measure, in degrees, of each exterior angle of a regular polygon containing:

(i) 6 sides (ii) 8 sides

(iii) 15 sides (iv) 20 sides

Solution

We know that each exterior angle of a polygon of n sides = \[\frac{360°}{n}\]

(i) Each exterior angle of 6 sided polygon = \[\frac{360°}{6} = 60°\]

(ii) Each exterior angle of 8 sided polygon = \[\frac{360°}{8} = 45°\]

(iii) Each exterior angle of 15 sided polygon = \[\frac{360°}{15} = 24°\]

(iv) Each exterior angle of 20 sided polygon = \[\frac{360°}{20} = 18°\] Ans.

Teacher's Note

Regular polygons appear everywhere in nature and architecture - from honeycomb hexagons to the stop sign octagon you encounter daily on roads.

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