ICSE Class 8 Maths Geometry Chapter 03 Polygons

Polygons

A closed plane shape bounded by three or more line segments is called a polygon. Some polygons have names based on the number of sides, as shown in the table. Otherwise, a polygon with n sides is called n-gon. For example a polygon that has 15 sides is called 15-gon.

Names Of Polygons

When all the sides of a polygon are equal in length and all its angles are equal in magnitude, it is called a regular polygon.

An equilateral triangle and a square are regular polygons.

Teacher's Note

Regular polygons appear in nature and design - from honeycomb hexagons to the octagonal stop signs on roads. Understanding their properties helps us recognize patterns in everyday structures.

Angles Of A Polygon

A polygon has as many angles as it has sides. For example, in the adjoining figure, the hexagon ABCDEF has six (interior) angles \(\angle A, \angle B, \angle C, \angle D, \angle E\) and \(\angle F\).

If we extend the side ED to X, we get the exterior angle CDX at the vertex D. And if we extend the side CD to Y, we get the exterior angle EDY at D, and \(\angle CDX =\) vertically opposite \(\angle EDY\).

Also, \(\angle CDX + \angle CDE = \angle EDY + \angle CDE = 180°\) (straight angle).

We can generalise this for any polygon as follows.

At each vertex of a polygon, there are two equal exterior angles.

The sum of an exterior angle and adjacent interior angle at a vertex = 180°.

Teacher's Note

When you turn at a corner while walking, the exterior angle represents how much you change direction. The full 360-degree turn brings you back to your original heading.

Convex Polygon

If all the interior angles of a polygon are less than 180°, the polygon is said to be convex. All regular polygons are convex.

Concave (Or Re-entrant) Polygon

If one or more of the interior angles of a polygon is greater than 180° (that is, a reflex angle), the polygon is said to be concave.

Teacher's Note

A convex lens bulges outward like a convex polygon, while a concave polygon has an inward-pointing angle similar to a concave mirror's shape.

Angle Properties Of A Polygon

Property 1 The sum of the interior angles of a polygon with n sides = (2n - 4) right angles.

(i) The sum of the (interior) angles of a triangle = (2 \(\times\) 3 - 4) right angles = 2 right angles = 180°, which we know is true.

(ii) The sum of the (interior) angles of a quadrilateral = (2 \(\times\) 4 - 4) right angles = 4 right angles.

In the adjoining figure, the sum of the angles of the quadrilateral ABCD = the sum of the angles of \(\triangle ABD +\) the sum of the angles of \(\triangle BCD\) = 2 right angles + 2 right angles = 4 right angles.

So, the property is true for a quadrilateral.

(iii) The sum of the (interior) angles of a hexagon = (2 \(\times\) 6 - 4) right angles = 8 right angles.

In the adjoining figure, the sum of the angles of the hexagon ABCDEF = the sum of the angles of the triangles ABC, ACD, ADE and AEF = 4 \(\times\) 2 right angles = 8 right angles.

So, the property holds for a hexagon.

It can be verified easily that it holds for other polygons as well.

Example Find the magnitude of each interior angle of a regular octagon.

Solution The sum of the eight angles of an octagon = (2 \(\times\) 8 - 4) right angles = 12 \(\times\) 90°.

Now, in a regular polygon, all the angles are equal.

Each angle of a regular octagon = \(\frac{12 \times 90°}{8} = 135°\). We can generalise this:

An interior angle of a regular polygon of n sides = \(\frac{1}{n}(2n - 4)\) right angles

Note When we speak of angle of a polygon we mean interior angle.

Teacher's Note

The interior angles of a regular polygon determine how tightly the shape closes - an octagon with its 135° angles is closer to a circle than a square's 90° angles.

Property 2

The sum of the exterior angles of a convex polygon = 360°.

(i) In the adjoining figure,

\(\angle a + \angle 1 = 180°, \angle b + \angle 2 = 180°,\) \(\angle c + \angle 3 = 180°.\)

\(\therefore \angle a + \angle 1 + \angle b + \angle 2 + \angle c + \angle 3\) = 180° + 180° + 180° = 540°

\(\Rightarrow \angle a + \angle b + \angle c + (\angle 1 + \angle 2 + \angle 3) = 540°.\)

But, \(\angle 1 + \angle 2 + \angle 3 = 180°\), being angles of a triangle.

\(\therefore \angle a + \angle b + \angle c = 540° - 180° = 360°.\)

So, the property is true for a triangle.

(ii) In the adjoining figure,

\(\angle a + \angle 1 = 180°, \angle b + \angle 2 = 180°,\) \(\angle c + \angle 3 = 180°, \angle d + \angle 4 = 180°.\)

\(\therefore \angle a + \angle 1 + \angle b + \angle 2 + \angle c + \angle 3 + \angle d + \angle 4\) = 180° + 180° + 180° + 180° = 720°

\(\Rightarrow \angle a + \angle b + \angle c + \angle d + (\angle 1 + \angle 2 + \angle 3 + \angle 4) = 720°.\)

But the sum of the interior angles of a quadrilateral = 360°.

\(\therefore \angle a + \angle b + \angle c + \angle d = 720° - 360° = 360°.\)

So, the property holds for a quadrilateral.

(iii) In the adjoining figure,

\(\angle a + \angle 1 = 180°, \angle b + \angle 2 = 180°,\) \(\angle c + \angle 3 = 180°, \angle d + \angle 4 = 180°,\) \(\angle e + \angle 5 = 180°.\)

\(\therefore \angle a + \angle 1 + \angle b + \angle 2 + \angle c + \angle 3 + \angle d + \angle 4\) + \(\angle e + \angle 5 = 5 \times 180° = 900°.\)

\(\Rightarrow \angle a + \angle b + \angle c + \angle d + \angle e + (\angle 1 + \angle 2 + \angle 3 + \angle 4 + \angle 5) = 900°.\)

But the sum of interior angles of a pentagon = (2 \(\times\) 5 - 4) right angles = 6 \(\times\) 90° = 540°.

\(\therefore \angle a + \angle b + \angle c + \angle d + \angle e = 900° - 540° = 360°.\)

So, the property is true for a pentagon.

You can easily verify that the property holds for other polygons as well.

Example 1 Find the measure of each exterior angle of a regular hexagon.

Solution The sum of the exterior angles of a polygon = 360°.

The interior angles of a regular polygon are equal, so its exterior angles are also equal.

The number of exterior angles in a hexagon = 6.

So, each exterior angle of a regular hexagon = \(\frac{360°}{6} = 60°\).

Teacher's Note

Exterior angles always sum to 360° because as you walk around any polygon and turn at each corner, you make one complete rotation before returning to your starting position and direction.

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