A closed bounded geometrical figure with three or more sides is called a **polygon**.
There are specific names of the polygon according to the number of sides in
the polygon. They are as follows:

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**Interior and Exterior
Angle of a Polygon**

An angle that forms inside the polygon is called an **interior angle**. When a side of a
polygon is produced, the angle that forms outside of the polygon is called an **exterior angle**.

The sum of interior angles of a polygon is given by the formula **(n – 2) × 180°** where n is the number of
sides of the polygon.

i.e. **Sum of interior
angles of a polygon = (n – 2) × 180°**

Therefore, by using this formula,

Sum of interior angles of a triangle = 180°

Sum of interior angles of a quadrilateral = 360°

Sum of interior angles of a pentagon = 540°

Sum of interior angles of a hexagon = 720°

Sum of interior angles of a heptagon = 900°

Sum of interior angles of an octagon = 1080°

Sum of interior angles of a nonagon = 1260°

Sum of interior angles of a decagon = 1440°

Sum of interior angles of a hendecagon = 1620°

Sum of interior angles of a dodecagon = 1800°

**Regular Polygon**

A polygon having all the sides equal and each interior angle
also equal is called a **regular polygon**. The Interior and exterior angles of a regular polygon is given by the formula,

**Construction of Regular
Polygon**

**Example 1:** Construct a regular pentagon of side 4 cm.

**Solution:** Here,

Length of side = 4 cm

No. of side (n) = 5

Interior angle = (n – 2)180°/n

= (5 – 2)180°/5

= 108°

**Steps of construction:**

**Step 1:** Draw a baseline XY and cut an arc of 4 cm on the line XY and
mark it AB.

**Step 2:** Draw 108° angle at points A and B using a protractor.

**Step 3:** Cut 4 cm arc AF and BG and mark as E and C.

**Step 4:** Make an angle of 108° at points E and C.

**Step 5:** Draw EH and CF and mark point D at the cross point.

Hence, ABCDE is a regular pentagon of side 4 cm.

**Example 2:** Construct a regular hexagon of side 3 cm.

**Solution:** Here,

Length of side = 3 cm

No. of side (n) = 6

Interior angle = (n – 2)180°/n

= (6 – 2)180°/6

= 120°

**Steps of construction:**

**Step 1:** Draw a baseline XY and cut an arc AB of 3 cm on XY.

**Step 2:** Make an angle 120° at each vertex and cut an arc of 3 cm in
each. Join the line segment.

Hence, ABCDEF is a regular hexagon of side 3 cm.

**Example 3:** Construct a regular octagon of side 3.5 cm.

**Solution:** Here,

Length of side = 3.5 cm

No. of side (n) = 8

Interior angle = (n – 2)180°/n

= (8 – 2)180°/8

= 135°

**Steps of construction:**

**Step 1:** Draw a baseline XY and cut an arc AB of 3.5 cm on XY.

**Step 2:** Make an angle 135° at each vertex and cut an arc of 3.5 cm in
each. Join the line segment.

Hence, ABCDEFGH is a regular octagon of side 3.5 cm.

**Worked Out Examples**

**Example 4:** Find the value of x.

**Solution:** Here,

From the figure,

No. of sides (n) = 5

Sum of interior angles = (n – 2)180°

i.e. 3x + 4x + 2x + 2x + x = (5 – 2)180°

or, 12x = 540°

or, x = 540°/12

or, x = 45° Ans.

**Example 5:** Find the value of x.

**Solution:** Here,

From the figure,

No. of sides (n) = 6

Sum of interior angles = (n – 2)180°

i.e. ∠A + ∠B + ∠C + ∠D + ∠E + ∠F = (6 – 2)180°

or, 90° + 80° + 240° + x + 95° + 130° = 720°

or, x + 635° = 720°

or, x = 720° - 635°

or, x = 85° Ans.

**Example 6:** Find the number of sides of a regular polygon where each interior
angle is 120°.

**Solution:** Here,

Each interior angle = 120°

i.e. (n – 2)180°/n = 120°

or, (n – 2)180° = 120°n

or, 180°n - 360° = 120°n

or, 180°n - 120°n = 360°

or, 60°n = 360°

or, n = 360°/60°

or, n = 6

∴ Number of side (n) = 6 Ans.

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