Chapter 12 Area of Polygons

Transcription

1 88 Chapter 1 Area f Plygs Area f Plygs 1.1 Plyg A plae figure buded by a umber f straight lies is called plyg. A plyg is said t be regular whe all its sides ad agles are equal. A five sided plyg is called a petag. A six sided plyg is called a hexag. A seve sided plyg is called a heptag. A eight sided plyg is called a ctag. A ie sided plyg is called a ag. A te sided plyg is called a decag. A twelve sided plyg is called a ddecag. Iscribed Plyg If a circle passes thrugh the crers f a plygs the this plyg is called iscribed plyg ad the circle is called the circumscribed circle. If the iscribed plyg is a regular plyg the cetre f the circle is als the cetre f the plyg. I the figure is the cetre f the plyg ad OF is the radius f the circumscribed circle ad is deted by R. Fig. Circumscribed Plyg: Fig.1 A plyg which is draw utside a circle s that the sides tuch circumfereces f the circle. I this case circle is called iscribed circle. If the plyg is regular the the cetre f the circle is als the cetre f the plyg. The radius f the iscribed circle is deted by r i the figure OG r

2 89 Area f Plygs Iterir Agle: The iterir agle f a regular 4 Plyg f sides x 90 Fr example fr a hexag, 6 Its iterir agle x 6 4 x x x Agle at the cetre ppsite t each side f a regular. 360 Plyg 1. Methd fr Fidig Area f Regular Plyg A regular plyg ca be divided it equal issceles triagles by jiig all the crers t the cetre. Obviusly, the umber f triagle is the same as the umber f sides f the plyg, the the area f the plyg is equal t the area f ay e triagle multiplied by the umber f triagle. (i) Area f the Regular Plyg f sides whe legth f a side is give: Let, AB a be the legth f the side f regular plyg f sides, be the cetre f the plyg, we frmed similar triagles. Let, AOB be e f the triagles. 360 The <AOB, <AOG Area f the regular plyg x area f AOB AB x OG x Fig.3

3 90 Area f Plygs a x x OG Nw, i right triagle AOG OG Ct AG OG AG Ct a Ct Area f plyg x a x a Ct a Ct Sq. uit 4 Perimeter f plyg a Example 1: Fid the cst f carpetig a ctagal flr with sides measurig 1 meter if the carpet csts Rs. 1/- per square meter. Sluti: 8, a 1m a Area Ct 4 8 8x(1) Ct 4 8 8x144 Ct x.41 A Sq. m Cst f carpetig x 1 Rs (ii) Area f a regular plyg f side whe the radius f the iscribed circle r is give Area f plyg x area f AOB AB?, AB AG OG r

4 91 Area f Plygs AG ta OG AG OG ta AB r ta AB r ta Fig.5 Area f plyg x AB x OG r x r ta x r ta Sq. uit Perimeter r ta Example : A regular petag is circumscribed abut circle with a radius f 0 cm. Fid the area f the petag. Sluti: Radius r 0cm 5 (petag) Area f circumscribed plyg r ta 5(0) ta 5 5 x 400 x ta x (.76) 1453 sq. cm (iii)area f Regular Plyg f sides whe the radius f the circumscribed circle R is give Let, OA R, the radius f the circumscribed circle Area f the plyg area f AOB

5 9 Area f Plygs Area OA x OB 360 x Si R x R 360 x Si R 360 Si sq. uit Perimeter x side x AB AG But Si OA AG OA Si AB AG AB OA Si AB OA Si Perimeter x AB x OA Si x x R Si R Si

6 93 Area f Plygs R 360 A Si P R Si Example 3: A regular plyg f 10 sides is iscribed i a circle whse radius is 30 cm. Fid the area ad the perimeter f the plyg. Sluti: We have, R 30cm, 10 R 360 Area Si 10 x 30 x Si x Si x (.5078) 650 Sq. cm. (1) Perimeter R Si x 10 x 30 x Si 600 x Si x (.31) 186cm. () Example 4: Cmpare the areas f a equilateral triagle, a square ad a regular hexag f equal perimeter. Sluti: Let, perimeter f each figure x Oe side f equilateral triagle x 3 Oe side f square x 4

7 94 Area f Plygs Oe side f regular hexag x 6 Area f a equilateral triagle 3 a 4 3 x 4 3 3x 36 Area f a square a. x x. (ii) 3 16 a Area f regular hexag Ct 4 6(x/6) Ct x x Ct x ( 3).. (iii) 4 Cmparis f area is 3 x 3 x : : x Multiplyig by x : : Multiplyig by : 9 : 6 3 Example 5: A grassy plt has the shape f a regular hexag each side 100m. Withi the plt ad alg its sides a ft path is made 4m wide all arud. Fid the area f the grassy plt left withi. Sluti: Fr ier hexag (i)

8 95 Area f Plygs Area f the regular plyg a Whe side is give Ct 4 T fid the legth (side) CD, draw perpedicular CE ad DF AB. I right agle triagle ACE CE ta 60 AE CE AE ta AE.30 FB FB.30 similarly Als, CD EF AB AE FB CD m Nw, 6, a 95.40m a A Ct 4 6 6(95.4) x A Sq. meter Exercise 1 Q1. The area f a regular ctagal rm is 51 sq. m Fid the legth f its side. Q. A regular decag is iscribed i a circle the radius f which is 10cm. Fid the area f the decag. Q3. Fid the crss-sectial area f the largest hexagal shak that ca be cut frm a rud bar f cpper cm i diameter. Q4. Regular plygs f 15 sides are iscribed i ad circumscribed abut a circle whse radius is 1cm shw that the differece f their areas is early 0 square cm. Q5 A regular ctag circumscribes a circle f cm radius. Fid the area f the ctag.

9 96 Area f Plygs Q6. The distace betwee the crers f a hexagal ut is.8 cm. Fid the distace betwee the jaws f the wrech eeded t fit this ut. Q7. Fid the area f a regular hexag whse perimeter is 48cm. What is radius f its iscribed circle. Q8. Fid the area f a regular hexag whse perimeter is 48cm, fid als the perimeter ad area f circumscribed circle f hexag. Q9. What is the legth f the side ad area f the largest hexag that ca be cut frm 8cm. rud bar. Aswers 1 Q Q sq. cm Q3..59 sq. cm Q sq.cm Q cm Q7. A sq. cm, r 6.93cm Q sq. cm, 50.8 cm, sq. cm. Q sq.cm Summary 1. The iterir agle f a regular Plyg f sides 4 x 90. Area f the regular plyg f sides whe legth f a side a is a give A Ct Sq. Uit Perimeter a 3. Area f a regular plyg f side whe the radius f the iscribed aide r is give A r ta sq. uit P r ta 4. Area f regular plyg f sides whe the radius f the circumscribed circle R is give R 360 A Si Sq. uit P R Si

10 97 Area f Plygs Q.1: Q.: Q.3: Q.4: Q.5: Q.6: Q.7: Q.8: Q.9: Defie a plygs. Shrt Questis Defie a regular plyg. Defie iscribed plyg (circumscribed circle). Defie circumscribed plyg (iscribed circle) Write the frmula t fid the agle f a regular plyg f sides. Fid the iterir agle f hexag. Write the area f the regular plyg f sides whe the legth f a side is give. Write the frmula f area f a regular plyg f sides whe the radius f iscribed circle r is give. Write the area f regular plyg f sides whe the radius f the circumscribed circle R is give. Q.10: The perimeter f a regular hexag is 1 cm, fid its area Q.11: Fid the area f regular hexag circumscribed abut a circle f radius cm. Aswers Q5. The agle f a regular plyg f sides is Q6. 10 Q sq. cm. Q sq. cm.

11 98 Area f Plygs Objective Type Questis Q.1 Each questis has fur pssible aswers. Chse the crrect aswer ad ecircle it. 1. A seve sided figure is called (a) Square (b) ctag (c) heptag (d) petag. If a is the side f plyg f sides, the its area is a (a) 4 Ct a (b) 3 a (c) Si 360 (d) Ne f these 3. Area f a regular plyg f sides whse radius f circumscribed circle is R is R (a) Si 360 R (b) Si 360 R (c) Ct (d) Ne f these 4. Decag have umber f sides (a) 8 (b) 9 (c) 10 (d) 1 _5. Area f a regular hexag f side a is r a 3 3 a (a) (b) (c) (d) 3a The perimeter f a regular hexag is 1 cm, its area is (a) sq. cm (b) cm (c) sq (d) cm 7. Each agle f a ctag is (a) 60 (b) 10 (c) 135 (d) Area f regular ctag f side a is (a) a (b) a (c) a (d) a 9. A regular plyg havig ifiite umber f sides is called (a) Octag (b) hexag (c) circle (d) heptag 10. Area f regular hexag circumscribed abut a circle f radius cm is (a) (b) (c) (d) Aswers Q.1 (1) c () a (3) a (4) c (5) c (6) b (7) c (8) c (9) c (10) a

12 99 Area f Plygs