Lesson Solved problems on angles of a regular polygon

Source code of 'Solved problems on angles of a regular polygon'

This Lesson (Solved problems on angles of a regular polygon) was created by by ikleyn(52781)

: View Source, Show
About ikleyn:

<H2>Solved problems on angles of a regular polygon</H2>

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.

<H3>Problem 1</H3>Find an interior angle of a regular pentagon. &nbsp;Find its exterior angle.

<B>Solution</B>

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

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

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

<B>Answer</B>. &nbsp;An interior angles of a regular pentagon are of &nbsp;108°. &nbsp;An interior angles are of &nbsp;72°.

<H3>Problem 2</H3>Find an interior angle of a regular hexagon. &nbsp;Find its exterior angle.

<B>Solution</B>

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

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

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

<B>Answer</B>. &nbsp;An interior angles of a regular hexagon are of &nbsp;120°. &nbsp;An interior angles are of &nbsp;60°.

<H3>Problem 3</H3>Find an interior and an exterior angles of a regular octagon; &nbsp;of a regular decagon; &nbsp;of a regular 12-gon.

<B>Solution</B>

For a regular octagon: an interior angle is &nbsp;&nbsp;{{{((n-2)*180^o)/n}}} = {{{((8-2)*180^o)/8}}} = {{{(6*180^o)/8}}} = 135°; &nbsp;&nbsp;&nbsp;&nbsp;an exterior angle is 180° - 135° = 45°.

For a regular decagon: an interior angle is &nbsp;&nbsp;{{{((n-2)*180^o)/n}}} = {{{((10-2)*180^o)/10}}} = {{{(8*180^o)/10}}} = 144°; &nbsp;&nbsp;&nbsp;&nbsp;an exterior angle is 180° - 144° = 36°.

For a regular 12-gon: an interior angle is &nbsp;&nbsp;{{{((n-2)*180^o)/n}}} = {{{((12-2)*180^o)/12}}} = {{{(10*180^o)/12}}} = 150°; &nbsp;&nbsp;&nbsp;&nbsp;an exterior angle is 180° - 150° = 30°.

<H3>Problem 4</H3>The measure of an interior angle of a regular polygon is &nbsp;160°. &nbsp;What is the number of sides/vertices in the polygon?

<B>Solution</B>

To solve the problem, &nbsp;you have to solve this equation

{{{((n-2)*180)/n}}} = {{{160}}}.

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

(n-2)*180 = 160n,

180n - 360 = 160n,

180n - 160n = 360,

20n = 360,

n = {{{360/20}}} = 18.

<B>Answer</B>. &nbsp;The number of sides/verices in the polygon is &nbsp;18.

<H3>Problem 5</H3>The exterior angle of a regular polygon is &nbsp;15°. &nbsp;Find &nbsp;n, &nbsp;the number of sides in the polygon.

<B>Solution</B>

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

Therefore, &nbsp;to solve the problem, &nbsp;you have to solve this equation

{{{((n-2)*180)/n}}} = {{{165}}}.

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

(n-2)*180 = 165n,

180n - 360 = 165n,

180n - 165n = 360,

15n = 360,

n = {{{360/15}}} = 24.

<B>Answer</B>. &nbsp;The number of sides/verices in the polygon is &nbsp;24.

<H3>Problem 6</H3>The ratio of exterior angles and the interior angles of a regular polygon is &nbsp;1:17. 
What is the number of sides/vertices in the polygon?

<B>Solution</B>

Let x be the measure of the exterior angle, &nbsp;in degrees.
Then the measure of the interior angle is  &nbsp;17x,  &nbsp;according to the condition.

Since the sum of the interior and the exterior angles is  &nbsp;180 degrees, &nbsp;you have an equation:

x + 17x = 180. &nbsp;&nbsp;Hence, &nbsp;18x = 180. &nbsp;&nbsp;Then &nbsp;&nbsp;x = {{{180/18}}} = 10.

Thus the exterior angle is 10 degrees. &nbsp;&nbsp;(Then the interior angle is 17*10 = 170).

Now, &nbsp;the sum of exterior angles of any polygon is  &nbsp;{{{360}}} degrees.

Hence, &nbsp;the number of vertices in the given polygon is &nbsp;{{{360/18}}} = 20.

<B>Answer</B>. &nbsp;The number of sides/vertices in the given polygon is &nbsp;20.

<H3>Problem 7</H3>The difference between an interior angle and an exterior angle of a regular polygon is &nbsp;140°. 
What is the number of sides/vertices in the polygon?

<B>Solution</B>

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

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

x + y = 180.

From the condition, &nbsp;you have the second equation

x - y = 140.

Add the two equations. &nbsp;You will get

2x = 180 + 140 = 320.

Hence, &nbsp;x = {{{320/2}}} = 160.

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

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

Hence, &nbsp;the number of vertices is &nbsp;{{{360/18}}} = 20.

<B>Answer</B>. &nbsp;The number of sides/vertices in the polygon is &nbsp;20.

My other introductory lessons on finding angles of triangles, parallelograms, quadrilaterals and polygons in this site are

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Points-lines-and-rays/Solved-problems-on-supplementary-and-complementary-angles-and-pairs-of-angles.lesson>Solved problems on supplementary and complementary angles</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Points-lines-and-rays/Solved-problems-on-angles-of-a-triangle.lesson>Solved problems on angles of a triangle</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Points-lines-and-rays/Solved-problems-on-angles-of-a-parallelogram.lesson>Solved problems on angles of a parallelogram</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Points-lines-and-rays/Solved-problems-on-angles-of-a-quadrilateral.lesson>Solved problems on angles of a quadrilateral</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Points-lines-and-rays/Solved-problems-on-angles-of-a-polygon.lesson>Solved problems on angles of a polygon</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Points-lines-and-rays/Solved-problems-on-missed-angle-of-a-polygon.lesson>Solved problems on missed angle of a polygon</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Points-lines-and-rays/OVERVIEW-of-solved-problems-on-angles-of-triangles-quadrilaterals-and-polygons.lesson>OVERVIEW of solved problems on angles of triangles, parallelograms, quadrilaterals and polygons</A>

Use this file/link &nbsp;<A HREF=https://www.algebra.com/algebra/homework/quadratic/lessons/ALGEBRA-I-YOUR-ONLINE-TEXTBOOK.lesson>ALGEBRA-I - YOUR ONLINE TEXTBOOK</A>&nbsp; to navigate over all topics and lessons of the online textbook &nbsp;ALGEBRA-I.