已知正三角形的邊長為a,那么它的內(nèi)切圓與外接圓組成的圓環(huán)的面積S= .
【答案】
分析:根據(jù)題意畫出圖形,分別求出兩圓的半徑,再分別求出兩圓的面積,兩圓的面積之差即為內(nèi)切圓與外接圓組成的圓環(huán)的面積.
解答:解:如圖所示,BC=a,
連接OB、OC,過O作OD⊥BC;
∵△ABC是正三角形,
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101002757978133/SYS201311031010027579781017_DA/images0.png)
∴∠BOC=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101002757978133/SYS201311031010027579781017_DA/0.png)
=120°,
∵OB=OC,OD⊥BC,
∴∠BOD=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101002757978133/SYS201311031010027579781017_DA/1.png)
∠BOC=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101002757978133/SYS201311031010027579781017_DA/2.png)
×120°=60°,BD=CD=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101002757978133/SYS201311031010027579781017_DA/3.png)
BC=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101002757978133/SYS201311031010027579781017_DA/4.png)
,
∴OB=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101002757978133/SYS201311031010027579781017_DA/5.png)
=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101002757978133/SYS201311031010027579781017_DA/6.png)
=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101002757978133/SYS201311031010027579781017_DA/7.png)
;
∵∠BOD=60°,
∴∠DOB=90°-60°=30°,
∴OD=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101002757978133/SYS201311031010027579781017_DA/8.png)
×
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101002757978133/SYS201311031010027579781017_DA/9.png)
=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101002757978133/SYS201311031010027579781017_DA/10.png)
,
∴S
大圓=π(OB)
2=π(
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101002757978133/SYS201311031010027579781017_DA/11.png)
)
2=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101002757978133/SYS201311031010027579781017_DA/12.png)
,
S
小圓=π(OD)
2=π(
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101002757978133/SYS201311031010027579781017_DA/13.png)
)
2=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101002757978133/SYS201311031010027579781017_DA/14.png)
,
∴S
圓環(huán)=S
大圓-S
小圓=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101002757978133/SYS201311031010027579781017_DA/15.png)
-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101002757978133/SYS201311031010027579781017_DA/16.png)
=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101002757978133/SYS201311031010027579781017_DA/17.png)
.
點評:此題比較復(fù)雜,解答此題的關(guān)鍵是根據(jù)題意畫出圖形,根據(jù)正三角形的性質(zhì)分別求出兩圓的半徑及面積即可解答.