Sum of the Exterior Angles of an n-sided Polygon

Here we will discuss the theorem of the sum of all exterior angles of an n-sided polygon and sum related example problems.

If the sides of a convex polygon are produced in the same order, the sum of all the exterior angles so formed is equal to four right angles.

Given: Let ABCD .... N be a convex polygon of n sides, whose sides have been produced in the same order.

Sum of the Exterior Angles of an n-sided Polygon

To prove: The sum of the exterior angles is 4 right angles, i.e., ∠a’ + ∠b’ + ∠c’ + ..... + ∠n’ = 4 × 90° = 360°.

Proof:

Statement

Reason

1. ∠a + ∠a’ = 2 right angles. Similarly, ∠b + ∠b’ = 2 right angles, ...., ∠n + ∠n’ = 2 right angles.

1. They form a linear pair.

2. (∠a + ∠b + ∠c + ..... + ∠n) + (∠a’ + ∠b’ + ∠c’ + ..... + ∠n’) = 2n right angles.

2. The polygon has n sides, and using statement 1.

3. (2n – 4) right angles + (∠a’ + ∠b’ + ∠c’ + ..... + ∠n’) = 2n right angles.

3. ∠a + ∠b + ∠c + ..... + ∠n = (2n – 4) right angles

4. ∠a’ + ∠b’ + ∠c’ + ..... + ∠n’

                = [2n - (2n – 4)] right angles.

                = 4 right angles

                = 4 × 90°

                = 360°.        (Proved)

4. From statement 3.

Note:

1. In a regular polygon of n sides, each exterior angle = \(\frac{360°}{n}\).

2. If each exterior angle of a regular polygon is x°, the polygon has \(\frac{360}{x}\) sides.

3. The greater the number of sides of a regular polygon, the greater is the value of each interior angle and the smaller is the value of each exterior angle.

Solved examples on finding the sum of the interior angles of an n-sided polygon:

1. Find the measure of each exterior angle of a regular pentagon.

Solution:

Here, n = 5.

Each exterior angle = \(\frac{360°}{n}\)

                             = \(\frac{360°}{5}\)

                             = 72°

Therefore, the measure of each exterior angle of a regular pentagon is 72°.


2. Find the number of sides of a regular polygon if each of its exterior angles is (i) 30°, (ii) 14°.

Solution:

We know, total number of sides of a regular polygon is \(\frac{360}{x}\) where, each exterior angle is x°.

(i) Here, exterior angle x = 30°

Number of sides = \(\frac{360°}{30°}\)

                        = 12

Therefore, there are 12 sides of the regular polygon.


(ii) Here, exterior angle x = 14°

Number of sides = \(\frac{360°}{14°}\)

                        = 25\(\frac{5}{7}\), is not a natural number

Therefore, such a regular polygon does not exist.


3. Find the number of sides of a regular polygon if each of its interior angles is 160°.

Solution:

Each interior angle = 160°

Therefore, each exterior angle = 180° - 160° = 20°

We know, total number of sides of a regular polygon is \(\frac{360}{x}\) where, each exterior angle is x°.

Number of sides = \(\frac{360°}{20°}\) = 18

Therefore, there are 18 sides of a regular polygon.


4. Find the number of sides of a regular polygon if each interior angle is double the exterior angle.

Solution:

Let each exterior angle = x°

Therefore, each interior angle = 180° - x°

According to the problem, each interior angle is double the exterior angle i.e.,

180° - x° = 2x°

⟹ 180° = 3x°

⟹ x° = 60°

Therefore, the number of sides = \(\frac{360}{x}\)

                                             = \(\frac{360}{60}\)

                                             = 6

Therefore, there are 6 sides of a regular polygon when each interior angle is double the exterior angle.


5. Two alternate sides of a regular polygon, when produced, meet at right angles. Find:

(i) each exterior angle of the polygon,

(ii) the number of sides of the polygon

Solution:

(i) Let ABCD ...... N be a regular polygon of n sides and each interior angle = x°

Alternate Sides of a Regular Polygon

According to the problem, ∠CPD = 90°

∠PCD = ∠PDC = 180° - x°

Therefore, from ∆CPD,

180° - x° + 180° - x° + 90° = 180°

⟹ 2x° = 270°

⟹ x° = 135°

Therefore, each exterior angle of the polygon = 180° - 135° = 45°.

(ii) Number of sides = \(\frac{360°}{45°}\) = 8.


6. There are two regular polygons with number of sides equal to (n – 1) and (n + 2). Their exterior angles differ by 6°. Find the value of n.

Solution:

Each exterior angle of the first polygon = \(\frac{360°}{ n – 1}\).

Each exterior angle of the second polygon = \(\frac{360°}{ n + 2}\).

According to the problem, each exterior angle of the first polygon and the second polygon differs by 6° i.e., \(\frac{360°}{ n – 1}\) - \(\frac{360°}{ n + 2}\).

⟹ 360° (\(\frac{1}{ n – 1}\) - \(\frac{1}{ n + 2}\)) = 6°

⟹ \(\frac{1}{ n – 1}\) - \(\frac{1}{ n + 2}\) = \(\frac{6°}{360°}\)

⟹ \(\frac{(n + 2) – (n – 1)}{(n – 1)(n + 2)}\) = \(\frac{1}{60}\)

⟹ \(\frac{3}{n^{2} + n - 2}\) = \(\frac{1}{60}\)

⟹ n\(^{2}\) + n – 2 = 180

⟹ n\(^{2}\) + n – 182 = 0

 ⟹ n\(^{2}\) + 14n – 13n – 182 = 0

⟹ n(n + 14) – 13(n + 14) = 0

⟹ (n + 14)(n - 13) = 0

Therefore, n = 13 (since n ≠ -14).




9th Grade Math

From Sum of the Exterior Angles of an n-sided Polygon to HOME PAGE




Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.



New! Comments

Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.




Share this page: What’s this?

Recent Articles

  1. 5th Grade Circle Worksheet | Free Worksheet with Answer |Practice Math

    Jul 11, 25 02:14 PM

    Radii of the circRadii, Chords, Diameters, Semi-circles
    In 5th Grade Circle Worksheet you will get different types of questions on parts of a circle, relation between radius and diameter, interior of a circle, exterior of a circle and construction of circl…

    Read More

  2. Construction of a Circle | Working Rules | Step-by-step Explanation |

    Jul 09, 25 01:29 AM

    Parts of a Circle
    Construction of a Circle when the length of its Radius is given. Working Rules | Step I: Open the compass such that its pointer be put on initial point (i.e. O) of ruler / scale and the pencil-end be…

    Read More

  3. Combination of Addition and Subtraction | Mixed Addition & Subtraction

    Jul 08, 25 02:32 PM

    Add and Sub
    We will discuss here about the combination of addition and subtraction. The rules which can be used to solve the sums involving addition (+) and subtraction (-) together are: I: First add

    Read More

  4. Addition & Subtraction Together |Combination of addition & subtraction

    Jul 08, 25 02:23 PM

    Addition and Subtraction Together Problem
    We will solve the different types of problems involving addition and subtraction together. To show the problem involving both addition and subtraction, we first group all the numbers with ‘+’ and…

    Read More

  5. 5th Grade Circle | Radius, Interior and Exterior of a Circle|Worksheet

    Jul 08, 25 09:55 AM

    Semi-circular Region
    A circle is the set of all those point in a plane whose distance from a fixed point remains constant. The fixed point is called the centre of the circle and the constant distance is known

    Read More