ICSE Class 8 Maths Chapter 26 Polygon

Chapter 26: Polygon

Introduction

A polygon is a closed plane figure, bounded by straight-line segments.

The line segments forming a polygon intersect only at end-points and each end-point is shared by only two line segments.

1. The figure (i), given above, represents a polygon.

2. In figure (ii), the line segments AC and BD intersect at point P which is not an end-point, therefore, the figure does not represent a polygon.

3. The figure (iii), given above, does not represent a polygon as points A and B are the end-points of three line segments.

4. The figure (iv), given above, does not represent a polygon as it is not a closed figure. Also, the end-points A and E are not shared by two line segments.

The segments which make up a polygon are called the sides of the polygon and the end-points of the segments are called the vertices of the polygon.

Polygons are named according to the number of sides they contain.

For example:

(i) Triangle - 3 sides

(ii) Quadrilateral - 4 sides

(iii) Pentagon - 5 sides

(iv) Hexagon - 6 sides

(v) Octagon - 8 sides

(vi) Decagon - 10 sides, etc.

Teacher's Note

Polygons are everywhere in daily life - from the hexagonal cells in a honeycomb to the octagonal stop signs on roads, understanding their properties helps us appreciate the geometry in nature and design.

Types Of Polygons

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

2. Concave polygon: If at least one angle of a polygon is more than 180°, it is called a concave or re-entrant polygon.

Unless it is stated, a polygon means a convex polygon.

Remember

A line segment joining any two non-consecutive vertices of a polygon is called its diagonal.

In the adjoining figure, AC is a diagonal of pentagon ABCDE as it joins two non-consecutive vertices A and C of the pentagon. Similarly, AD is also a diagonal. More diagonals can be drawn through the vertices B, C, D and E of the pentagon ABCDE.

Teacher's Note

Drawing diagonals in polygons helps visualize interior divisions - like how a city planner might divide regions or how architects design building layouts with multiple interior spaces.

Sum Of Angles Of A Polygon

Draw all possible diagonals through a single vertex of a polygon to form as many triangles as possible.

From the diagrams shown:

A 4-sided polygon forms 2 triangles.

A 5-sided polygon forms 3 triangles.

A 6-sided polygon forms 4 triangles.

An 8-sided polygon forms 6 triangles.

It is observed that the number of triangles formed is two less than the number of sides in the polygon.

So, if a polygon has n sides, the number of triangles formed will be n - 2.

Since, the sum of angles of a triangle = 180°

The sum of angles of (n - 2) triangles = (n - 2) × 180°

Therefore, sum of angles (interior angles) of a polygon with n sides = (n - 2) × 180°

= (2n - 4) × 90°

= (2n - 4) right angles

Test Yourself

1. If a polygon has 7 sides, it has ........... vertices.

2. From each vertex of a ten-sided polygon, ........... diagonals can be drawn.

3. If one angle of a polygon is 190°, the polygon is called a ........................ polygon.

4. A hexagon has ...... sides, and the sum of its interior angles is (2n - 4) × 90° = ......................

= ........................ = ........... .

5. By drawing maximum number of diagonals from one vertex of an n-sided polygon; ........... triangles are formed and sum of the interior angles of these triangles ..........................

Teacher's Note

Understanding angle sums in polygons is practical when designing roof trusses or decorative tiles - knowing the interior angles ensures pieces fit together perfectly.

Sum Of Exterior Angles Of A Polygon

If the sides of a polygon are produced in order, the sum of exterior angles so formed is always 4 right angles i.e. 360°.

The diagrams show a 5-sided polygon and a 7-sided polygon with exterior angles marked.

For the 5-sided polygon: z1 + z2 + z3 + z4 + z5 = 360°

For the 7-sided polygon: z1 + z2 + ... + z7 = 360°

If a man walks along the sides of a polygon and each side of the polygon is produced in the direction of motion of the man, the sides of the polygon are said to be produced in order.

For example: In each diagram shown, the direction of motion of the man is represented by arrows.

Example 1

Is it possible to have a polygon, the sum of whose interior angles is 9 right angles.

Solution: Let the number of sides be n.

The sum of its interior angles = (2n - 4) × 90°

According to the given statement: (2n - 4) × 90° = 9 × 90°

Therefore, 2n - 4 = 9

Therefore, n = 6.5; which is not possible

Note: The number of sides in a polygon is always a natural number and is never in fraction or decimals. The smallest number of sides in a polygon is 3, which is in case of a triangle.

Example 2

The sides of a pentagon are produced in order. If the measures of exterior angles so obtained are x°, (2x)°, (3x)°, (4x)° and (5x)°, find all the exterior angles.

Solution: Since, the sum of exterior angles obtained in the above case = 360°

Therefore, x° + (2x)° + (3x)° + (4x)° + (5x)° = 360°

Therefore, (15x)° = 360° i.e. x = 360/15 = 24

Exterior angles = 24°, (2 × 24)°, (3 × 24)°, (4 × 24)° and (5 × 24)°

= 24°, 48°, 72°, 96° and 120°

Example 3

One angle of a seven-sided polygon is 114° and each of the other six angles is x°. Find the magnitude of x°.

Solution: Since, each of the other six angles is x°

Therefore, sum of these six angles = 6x°

Therefore, sum of all the seven angles = 114° + 6x° ... I

According to the formula: Sum of interior angles of the seven-side polygon = (2n - 4) × 90°

= (2 × 7 - 4) × 90° = 900° ... II

From I and II: 114° + 6x° = 900°

Therefore, 6x° = 900° - 114° = 786° => x° = 131°

Test Yourself

6. The angles of a quadrilateral are in the ratio 5 : 6 : 3 : 4. The smallest angle of this quadrilateral is ......................

7. The angles of a pentagon are in the ratio 7 : 6 : 5 : 4 : 5; its largest angle is ......................

= ..........

8. Two angles of a quadrilateral are 68° and 107° and the other two angles are in the ratio 2 : 3. Since, 360° - 68° - 107° = .........., the smallest of other two angles is ......................

= ...........

9. One angle of a quadrilateral is 120° and the remaining angles are equal; each of the equal angles is .............................

Teacher's Note

The constant sum of exterior angles (360°) explains why a person walking around any polygon returns to their starting point facing the original direction - a principle used in navigation and surveying.

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