# Sum of the Measures of the Exterior Angles of a Polygon

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The knowledge of exterior angles may throw light on the **nature of interior angles and sides**.

A polygon say, a **pentagon ABCDE is drawn on the floor**, using a piece of chalk.

We stand at A and start walking along the side AB. On reaching B, we need to *turn through an angle (equal to the exterior angle subtended by side BC wrt x-axis)*, to walk along BC.

When you reach at C, we need to *turn through an angle (equal to the exterior angle subtended by side CD wrt y-axis)* to walk along CD. We **continue to move in this manner**, until we return to side AB.

We would have in fact made **one complete turn** i.e.

Therefore,

**The sum of the measures of the external angles of any polygon is 360°**

**This is true whatever be the number of sides of the polygon, as this same logic can be used for others as well**.

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**Example 1: Find measure of x.**

**Solution:** x + 90° + 50° + 110° =

x +

x =

**Thus, x is equal to 110°**

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### Try These

**1.What is the sum of the measures of its exterior angles x, y, z, p, q, r?**

For the given regular hexagon,

The measure of ∠a = ^{o}

The ∠x, ∠y, ∠z, ∠p, ∠q, ∠r in the given figure, are

x ,y ,z ,p ,q ,r are all exterior angles.

Hence, x + y + z + p + q + r = ^{o}

**2. Is x = y = z = p = q = r? Why?**

It is a hexagon with all sides

a + r = a + x = a + y = a + z = a + p = a + q = ^{o}

Hence, x = y = z = p = q = r (after

**3. What is the measure of each?**

**(i) exterior angle**

**(ii) interior angle**

**Solution:**

(i) The sum of the exterior angles of any polygon is always

For a regular polygon, all exterior angles are equal.

Therefore, the measure of each exterior angle is given by:

Exterior Angle =

For a hexagon, the number of sides is

Thus,

Exterior Angle =

(ii) The sum of the interior angles of a polygon with n sides is given by:

Sum of Interior Angles = (n−2) × 180°

For a hexagon, n = 6.

Sum of Interior Angles = (6−2) × 180° =

Since the hexagon is regular, all interior angles are equal.

Interior Angle =

**Thus, for a regular hexagon: the measure of each exterior angle is 60° while measure of each interior angle is 120°.**

**4. Repeat this activity for the cases of:**

**(i) a regular octagon**

**(ii) a regular 20-gon**

**Solution:**

(i)

**Exterior Angle:**

Exterior Angle =

**Interior Angle:**

Sum of Interior Angles = (n−2) × 180° = (8−2) × 180° =

Interior Angle =

**Thus, for a regular octagon: the measure of each exterior angle is 45° while measure of each interior angle is 135°.**

(ii)

**Exterior Angle:**

Exterior Angle =

**Interior Angle:**

Sum of Interior Angles = (n−2) × 180° = (20−2) × 180° =

Interior Angle =

**Thus, for a regular 20-gon: the measure of each exterior angle is 18° while measure of each interior angle is 162°.**

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**Example 2: Find the number of sides of a regular polygon whose each exterior angle has a measure of 45°.**

**Solution:** Total measure of all exterior angles =

Measure of each exterior angle =

Therefore, the number of exterior angles =

**The polygon has 8 sides.**