![Polygon Polygon](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg9qKyoOBzInUlYLr5Y72isyrpuiEYMaIK0cCnPs7zZMZWR6joy7a4blrm8Azc8hwjTeCZLxgB04F6ZBXR9B13WXCx15n5PO_5nykCNUvBXf5Nf1QQL2I6rJ6XT9T7PpK_9YLc0A6cSZVlgSFp_QWLQFyCF29anwb7mT1W99KGGejT8Lp_s8DtbJIzurg/s16000/polygon%20banner.png)
A closed bounded geometrical figure with three or more sides is called a polygon. There are specific names of the polygon according to the number of sides in the polygon. They are as follows:
![Number of sides, name of the polygons and the figure of the polygons.](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhp9ZvbDnHSMfPgwG2-AX9fXcqZfOnBEaRsbdzZibIrP2Ng593ud719CPI2n33b34OmNV-Hcl3o0MMk8v0mrJdpIfvoOc6XYnY986bvz1M389H1dTgnM-SJUGz3yZesA1Bcy8hOUPjqSP3iKV44ekho56qSA4zDZTRuuf0uQTXTnyG8GBR-zTqaiZ5P3g/s16000/number%20of%20sides-name%20of%20polygons-polygon%20figure.png)
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Interior and Exterior Angle of a Polygon
An angle that forms inside the polygon is called an interior angle. When a side of a polygon is produced, the angle that forms outside of the polygon is called an exterior angle.
![Interior and exterior angles of a polygon](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjQY2Rhu4BzATJwPIbX1YuAK3HENS7N-6epR93pAabUx6DuELId2mr4FwtW3dO9ffpSuWX-CvXc71_O0rJ7z6QzZNpCpZ_Cu-yXDzJlTLXLmIAuRXUUw45XIYB4az_2Zw9X-nhNddm8CGOlsX_m_N2-PboLFLTEyBMtve0DGjinp-OuxsBAsNINUUFhNw/s16000/interior%20angle%20-%20exterior%20angle%20in%20a%20polygon.png)
The sum of interior angles of a polygon is given by the formula (n – 2) × 180° where n is the number of
sides of the polygon.
i.e. Sum of interior
angles of a polygon = (n – 2) × 180°
Therefore, by using this formula,
Sum of interior angles of a triangle = 180°
Sum of interior angles of a quadrilateral = 360°
Sum of interior angles of a pentagon = 540°
Sum of interior angles of a hexagon = 720°
Sum of interior angles of a heptagon = 900°
Sum of interior angles of an octagon = 1080°
Sum of interior angles of a nonagon = 1260°
Sum of interior angles of a decagon = 1440°
Sum of interior angles of a hendecagon = 1620°
Sum of interior angles of a dodecagon = 1800°
Regular Polygon
A polygon having all the sides equal and each interior angle also equal is called a regular polygon. The Interior and exterior angles of a regular polygon is given by the formula,
![The formula of interior and exterior angle of a regular polygon.](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhbmcxfZtHz1hke3Yik65U7h1hxz5KeNcQAneqJwyCLXNDbexXLlyNNV_npktu_pKJisDLki2IkqjH3XwBKw7W0eBOiJGlQ7FGA_fI_q-WPpHuv2G7UBxTs-n40CN9iarBdG_kmacw7ScIKE4Ek86tgYS2ezTHBPeKtc7DGMUxgBF6IxVUhqQvxvjAccg/s16000/polygon%20interior%20exterior%20angle%20formula.png)
Construction of Regular Polygon
Example 1: Construct a regular pentagon of side 4 cm.
Solution: Here,
Length of side = 4 cm
No. of side (n) = 5
Interior angle = (n – 2)180°/n
= (5 – 2)180°/5
= 108°
Steps of construction:
Step 1: Draw a baseline XY and cut an arc of 4 cm on the line XY and
mark it AB.
Step 2: Draw 108° angle at points A and B using a protractor.
Step 3: Cut 4 cm arc AF and BG and mark as E and C.
Step 4: Make an angle of 108° at points E and C.
Step 5: Draw EH and CF and mark point D at the cross point.
![Construction of Pentagon](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjjwh2eElV-sDHzlO3cmwXtdHPSNpMK6hl4s98-qPF7gpxr9kep71GcdyFiT8G04eys2Lo4Cqm5FSf_Y49yVWJYN1qLvxpzmkrKgSK37Na_LzFofedjbe3gXkTsscYqHeSWg1wXAFG1ngifW8V_2nT95sSclPsB3EBKD00GsXbP_LG47PwCPHRA1yAzCw/s16000/construction%20of%20pentagon.png)
Hence, ABCDE is a regular pentagon of side 4 cm.
Example 2: Construct a regular hexagon of side 3 cm.
Solution: Here,
Length of side = 3 cm
No. of side (n) = 6
Interior angle = (n – 2)180°/n
= (6 – 2)180°/6
= 120°
Steps of construction:
Step 1: Draw a baseline XY and cut an arc AB of 3 cm on XY.
Step 2: Make an angle 120° at each vertex and cut an arc of 3 cm in
each. Join the line segment.
![Construction of Hexagon](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiKRMAr8vVvZsfkZoYW6Z3Q0sKAqMXIPiLZUvr4iEJVgiUN62B5X4slSyIobjegKoBk02hsumaT60mY_aLxpX-o0Nxuq2Zj7lVI5hCJtGArnQutr0ryE1oZFdjbgLBAWRCnDDjl3jergbHAf6_SLqsyGh1lRSY-pEV8VY69nJAjJxZRQycTkRxJ-qpnyQ/s16000/construction%20of%20hexagon.png)
Hence, ABCDEF is a regular hexagon of side 3 cm.
Example 3: Construct a regular octagon of side 3.5 cm.
Solution: Here,
Length of side = 3.5 cm
No. of side (n) = 8
Interior angle = (n – 2)180°/n
= (8 – 2)180°/8
= 135°
Steps of construction:
Step 1: Draw a baseline XY and cut an arc AB of 3.5 cm on XY.
Step 2: Make an angle 135° at each vertex and cut an arc of 3.5 cm in
each. Join the line segment.
![Construction of Octagon](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi6oxHObrUBZR_oXC3WhNyiFIJ9v-h3xkhgJjic6huIcMtCeV0pkVRax4CqOYQpgjZMDRnDLKN9lSoQCbsT-ZEPhpll5vVB-u-poWE4j6bTTczGAMrwMJWrYjQGkYMCH3K1bjpCZT_eAjaJef5ogr2pSqZEg6LHwGLtXJI3nyBJX9AulGGwu1oCB9HDnw/s16000/construction%20of%20octagon.png)
Hence, ABCDEFGH is a regular octagon of side 3.5 cm.
Worked Out Examples
Example 4: Find the value of x.
![Example 4: Figure of Polygon](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiNYjF1MD6xR1fldPTEVJ-v_BAI2_huA1agcS771CQ7KN9fGUQpHdSDBUe_NSz5tdbk_gNG2oXXIEaZgzHakMsDcSJ6XF5pduNakj7qXYXam5oed6ol_DaEylXj8UXKuQ-k_ekngTmJqn_Ul0Oe2EkH0WiemS6wzN6AfyLQKQaAnL1U9OG63RuJ-z0-3w/s16000/example%204%20polygon.png)
Solution: Here,
From the figure,
No. of sides (n) = 5
Sum of interior angles = (n – 2)180°
i.e. 3x + 4x + 2x + 2x + x = (5 – 2)180°
or, 12x = 540°
or, x = 540°/12
or, x = 45° Ans.
Example 5: Find the value of x.
![Example 5: Figure of Polygon](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgN1wRgmSnN_GqMomazzUUiFZ1H7vLlB9xffKf7xOE_EbL8x-W039QD7RYlcfX8G_QkMJpjvYT6K4c2WGtA1AeBBn-A_o7ltpAxLIUAxlFoUz_H7GGt5Pus_rbbSYe94tql72xusJoL4aVftJKttw55skPPFNC_uCGXcaXpi2BKAEBU8syt7Smn9gsnVg/s16000/example%205%20polygon.png)
Solution: Here,
From the figure,
No. of sides (n) = 6
Sum of interior angles = (n – 2)180°
i.e. ∠A + ∠B + ∠C + ∠D + ∠E + ∠F = (6 – 2)180°
or, 90° + 80° + 240° + x + 95° + 130° = 720°
or, x + 635° = 720°
or, x = 720° - 635°
or, x = 85° Ans.
Example 6: Find the number of sides of a regular polygon where each interior
angle is 120°.
Solution: Here,
Each interior angle = 120°
i.e. (n – 2)180°/n = 120°
or, (n – 2)180° = 120°n
or, 180°n - 360° = 120°n
or, 180°n - 120°n = 360°
or, 60°n = 360°
or, n = 360°/60°
or, n = 6
∴ Number of side (n) = 6 Ans.
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