Lesson Solved problems on angles of a regular polygon

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Solved problems on angles of a regular polygon


This lesson is a small collection of typical comparatively simple problems on finding angles of a regular polygon.
The goal of this text is to teach you by examples to make first steps in this area.

Problem 1

Find an interior angle of a regular pentagon.  Find its exterior angle.

Solution

The sum of interior angles of any polygon with n sides/vertices is  (n-2)*180°.

For a pentagon it is   (5-2)*180° = 3*180° = 540°.
Since all interior angles of a regular pentagon are congruent,  a single interior angle is  540°/5 = 108°.

Then an exterior angle of a regular pentagon is  180° - 108° = 72°.

Answer.  An interior angles of a regular pentagon are of  108°.  An interior angles are of  72°.


Problem 2

Find an interior angle of a regular hexagon.  Find its exterior angle.

Solution

The sum of interior angles of any polygon with n sides/vertices is  (n-2)*180°.

For a hexagon it is   (6-2)*180° = 4*180° = 720°.
Since all interior angles of a regular pentagon are congruent,  a single interior angle is  720°/6 = 120°.

Then an exterior angle of a regular pentagon is  180° - 120° = 60°.

Answer.  An interior angles of a regular hexagon are of  120°.  An interior angles are of  60°.


Problem 3

Find an interior and an exterior angles of a regular octagon;  of a regular decagon;  of a regular 12-gon.

Solution

For a regular octagon: an interior angle is   %28%28n-2%29%2A180%5Eo%29%2Fn = %28%288-2%29%2A180%5Eo%29%2F8 = %286%2A180%5Eo%29%2F8 = 135°;     an exterior angle is 180° - 135° = 45°.

For a regular decagon: an interior angle is   %28%28n-2%29%2A180%5Eo%29%2Fn = %28%2810-2%29%2A180%5Eo%29%2F10 = %288%2A180%5Eo%29%2F10 = 144°;     an exterior angle is 180° - 144° = 36°.

For a regular 12-gon: an interior angle is   %28%28n-2%29%2A180%5Eo%29%2Fn = %28%2812-2%29%2A180%5Eo%29%2F12 = %2810%2A180%5Eo%29%2F12 = 150°;     an exterior angle is 180° - 150° = 30°.


Problem 4

The measure of an interior angle of a regular polygon is  160°.  What is the number of sides/vertices in the polygon?

Solution

To solve the problem,  you have to solve this equation

%28%28n-2%29%2A180%29%2Fn = 160.

for  n.  To do it,  multiply both sides by  n  and simplify:

(n-2)*180 = 160n,
180n - 360 = 160n,
180n - 160n = 360,
20n = 360,

n = 360%2F20 = 18.

Answer.  The number of sides/verices in the polygon is  18.


Problem 5

The exterior angle of a regular polygon is  15°.  Find  n,  the number of sides in the polygon.

Solution

From the condition,  the interior angle has the measure  180° - 15° = 165°.

Therefore,  to solve the problem,  you have to solve this equation

%28%28n-2%29%2A180%29%2Fn = 165.

for  n.  To do it,  multiply both sides by  n  and simplify:

(n-2)*180 = 165n,
180n - 360 = 165n,
180n - 165n = 360,
15n = 360,

n = 360%2F15 = 24.

Answer.  The number of sides/verices in the polygon is  24.


Problem 6

The ratio of exterior angles and the interior angles of a regular polygon is  1:17.
What is the number of sides/vertices in the polygon?

Solution

Let x be the measure of the exterior angle,  in degrees.
Then the measure of the interior angle is  17x,  according to the condition.
Since the sum of the interior and the exterior angles is  180 degrees,  you have an equation:

x + 17x = 180.   Hence,  18x = 180.   Then   x = 180%2F18 = 10.
Thus the exterior angle is 10 degrees.   (Then the interior angle is 17*10 = 170).

Now,  the sum of exterior angles of any polygon is  360 degrees.

Hence,  the number of vertices in the given polygon is  360%2F18 = 20.

Answer.  The number of sides/vertices in the given polygon is  20.


Problem 7

The difference between an interior angle and an exterior angle of a regular polygon is  140°.
What is the number of sides/vertices in the polygon?

Solution

Let  x  be the measure of the interior angle,  in degrees.
Let  y  be the measure of the exterior angle.

Since the sum of an interior and the exterior angle is  180°,  you have an equation:

x + y = 180.

From the condition,  you have the second equation

x - y = 140.

Add the two equations.  You will get

2x = 180 + 140 = 320.

Hence,  x = 320%2F2 = 160.

Thus the interior angle is of  160°.  Then the exterior angle is  180° - 160° = 20°.

Now,  the sum of exterior angles of any polygon is  360°.

Hence,  the number of vertices is  360%2F18 = 20.

Answer.  The number of sides/vertices in the polygon is  20.


My other introductory lessons on finding angles of triangles, parallelograms, quadrilaterals and polygons in this site are
    - Solved problems on supplementary and complementary angles
    - Solved problems on angles of a triangle
    - Solved problems on angles of a parallelogram
    - Solved problems on angles of a quadrilateral
    - Solved problems on angles of a polygon
    - Solved problems on missed angle of a polygon
    - OVERVIEW of solved problems on angles of triangles, parallelograms, quadrilaterals and polygons

Use this file/link  ALGEBRA-I - YOUR ONLINE TEXTBOOK  to navigate over all topics and lessons of the online textbook  ALGEBRA-I.

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