3.2-Sum of the Measures of the Exterior Angles of a Polygon

3.2-Sum of the Measures of the Exterior Angles of a Polygon Important Formulae

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3.2 - Sum of the Measures of the Exterior Angles of a Polygon
  • The sum of the exterior angles of any polygon is always 360°.
  • Exterior angles are formed when one side of a polygon is extended.
  • The exterior angle at each vertex is supplementary to the interior angle.
  • For a polygon with $n$ sides, the exterior angles add up to 360°.
  • The measure of each exterior angle of a regular polygon is $\frac{360^\circ}{n}$.
  • Exterior angles are positive and are measured in the direction of rotation.

3.2 - Sum of the Measures of the Exterior Angles of a Polygon

In this section, we will learn about the sum of the exterior angles of a polygon, a fundamental property that helps us understand the geometric relationships in polygons, especially quadrilaterals.

An exterior angle of a polygon is formed when one side of the polygon is extended. It is the angle between the extended side and the adjacent side. Every polygon has exterior angles, and we will explore how their measures are related to the number of sides in the polygon.

Key Concepts:
  • Exterior Angles: An exterior angle of a polygon is formed by one side of the polygon and the extension of an adjacent side.
  • Interior Angles: The interior angles are the angles inside the polygon formed by adjacent sides meeting at each vertex.
  • Exterior Angle Theorem: The sum of the measures of the exterior angles of any polygon is always constant.
Sum of the Exterior Angles of a Polygon:

For any polygon, irrespective of the number of sides, the sum of the exterior angles is always $360^{\circ}$. This is a key result in geometry that holds for any convex polygon, whether it is a triangle, quadrilateral, pentagon, etc.

This means that if we add up all the exterior angles of a polygon (one exterior angle at each vertex), the total will always be $360^{\circ}$. This rule is true for both regular and irregular polygons.

Formula:

The sum of the exterior angles of any polygon is:

$$ \text{Sum of exterior angles} = 360^{\circ} $$

Now, let us break down the reasoning behind this result:

Why the sum of exterior angles is always 360°?

Consider a polygon with $n$ sides. At each vertex of the polygon, there is an exterior angle formed by extending one side of the polygon. The interior angle at each vertex and the exterior angle at that same vertex are supplementary angles, meaning they add up to $180^{\circ}$. Therefore, the exterior angle at each vertex is:

$$ \text{Exterior angle} = 180^{\circ} - \text{Interior angle} $$

However, no matter the number of sides, if you add all the exterior angles around the entire polygon, they will always sum to $360^{\circ}$. This is because as you move around the polygon, the turns or rotations accumulate in a complete circle, which is $360^{\circ}$.

Example Calculation:

Consider a polygon with 4 sides, i.e., a quadrilateral. No matter whether it is a square, rectangle, or any irregular quadrilateral, the sum of its exterior angles will be:

$$ \text{Sum of exterior angles of quadrilateral} = 360^{\circ} $$

If we were to calculate the exterior angles at each vertex individually (such as for a regular quadrilateral like a square), each exterior angle would be:

$$ \text{Exterior angle of regular quadrilateral} = \frac{360^{\circ}}{4} = 90^{\circ} $$

Similarly, for any polygon with $n$ sides, if the polygon is regular (all sides and angles are equal), the measure of each exterior angle can be calculated by:

$$ \text{Exterior angle} = \frac{360^{\circ}}{n} $$

This is because the total of the exterior angles is $360^{\circ}$, and there are $n$ exterior angles in the polygon. Therefore, dividing $360^{\circ}$ by the number of sides gives the measure of each exterior angle for a regular polygon.

Important Notes:
  • The sum of the exterior angles of any polygon, whether regular or irregular, is always $360^{\circ}$.
  • The exterior angles at each vertex form supplementary angles with the interior angles of the polygon.
  • For a regular polygon, the measure of each exterior angle is given by $360^{\circ} / n$, where $n$ is the number of sides.
  • This rule holds true for both convex and concave polygons.

Find the measure of each exterior angle of a regular polygon of:
(i) 9 sides
(ii) 15 sides

Solution:

Find the measure of each exterior angle of a regular polygon

To find the measure of each exterior angle of a regular polygon, use the formula:

$\text{Exterior Angle} = \frac{360^\circ}{n}$

Where $n$ is the number of sides of the polygon.

(i) For a regular polygon with 9 sides:

Exterior Angle = $\frac{360^\circ}{9}$ = $40^\circ$

(ii) For a regular polygon with 15 sides:

Exterior Angle = $\frac{360^\circ}{15}$ = $24^\circ$

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How many sides does a regular polygon have if the measure of an exterior angle is 24°?

Solution:

How many sides does a regular polygon have if the measure of an exterior angle is 24°?

The exterior angle of a regular polygon is given by the formula:

$\text{Exterior angle} = \frac{360^\circ}{n}$

Where $n$ is the number of sides of the polygon.

We are given that the exterior angle is $24^\circ$. So, we can set up the equation:

$24 = \frac{360}{n}$

To solve for $n$, multiply both sides of the equation by $n$:

$24n = 360$

Now, divide both sides by 24:

$n = \frac{360}{24}$

$n = 15$

So, the polygon has 15 sides.

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How many sides does a regular polygon have if each of its interior angles is 165°?

Solution:

How many sides does a regular polygon have if each of its interior angles is 165°?

We know that the sum of the interior angles of a regular polygon with $n$ sides is given by:

Sum of interior angles = $(n - 2) \times 180°$

Each interior angle of a regular polygon is the same, so we can write:

Each interior angle = $\frac{(n - 2) \times 180°}{n}$

We are given that each interior angle is 165°. So, we set up the equation:

$$\frac{(n - 2) \times 180°}{n} = 165°$$

Multiplying both sides by $n$:

$(n - 2) \times 180° = 165n$

Expanding the left-hand side:

$180n - 360° = 165n$

Now, subtract $165n$ from both sides:

$180n - 165n = 360°$

$15n = 360°$

Dividing both sides by 15:

$n = \frac{360°}{15} = 24$

Thus, the regular polygon has 24 sides.

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(a) Is it possible to have a regular polygon with measure of each exterior angle as 22°?
(b) Can it be an interior angle of a regular polygon? Why?

Solution:

3. (a) Is it possible to have a regular polygon with measure of each exterior angle as 22°?

The measure of each exterior angle of a regular polygon is given by the formula: $ \frac{360^\circ}{n} $, where $n$ is the number of sides. For the exterior angle to be $22^\circ$, we set up the equation: $ \frac{360^\circ}{n} = 22^\circ $. Solving for $n$: $ n = \frac{360^\circ}{22^\circ} = 16.36 $. Since $n$ must be a whole number, it is not possible to have a regular polygon with each exterior angle as $22^\circ$.

3. (b) Can it be an interior angle of a regular polygon? Why?

The interior angle of a regular polygon is related to the exterior angle by: $ \text{Interior angle} = 180^\circ - \text{Exterior angle} $. If the exterior angle is $22^\circ$, the interior angle would be: $ 180^\circ - 22^\circ = 158^\circ $. For a polygon with an interior angle of $158^\circ$, we can check if it’s possible by solving: $ \frac{(n-2) \times 180^\circ}{n} = 158^\circ $. This yields $n \approx 7.9$, which is not a whole number, meaning $158^\circ$ cannot be the interior angle of a regular polygon.

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(a) What is the minimum interior angle possible for a regular polygon? Why?
(b) What is the maximum exterior angle possible for a regular polygon?

Solution:

(a) What is the minimum interior angle possible for a regular polygon? Why?

The minimum interior angle for a regular polygon occurs when the number of sides is maximized. In the case of a regular polygon, all interior angles are equal. The formula for the interior angle of a regular polygon with $n$ sides is:

Interior angle = $ \frac{(n-2) \times 180^\circ}{n} $

As $n$ increases, the interior angle approaches $180^\circ$. The minimum interior angle occurs when the polygon is a regular triangle, where $n = 3$. For a regular triangle, the interior angle is:

Interior angle = $ \frac{(3-2) \times 180^\circ}{3} = 60^\circ $

Thus, the minimum interior angle is $60^\circ$.

(b) What is the maximum exterior angle possible for a regular polygon?

The exterior angle of a regular polygon with $n$ sides is given by the formula:

Exterior angle = $ \frac{360^\circ}{n} $

The maximum exterior angle occurs when the number of sides, $n$, is minimized. The polygon with the minimum number of sides is a regular triangle ($n = 3$), which gives the maximum exterior angle. For a regular triangle:

Exterior angle = $ \frac{360^\circ}{3} = 120^\circ $

Thus, the maximum exterior angle is $120^\circ$.

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