(例題対比)扇形の面積,円錐の表面積

【円の面積】
 円周率をπで表わすとき,半径 r の円の面積 S
S=π r2
になる.右参照→
 r は円の直径ではなく,半径であることに注意

[例題1]
(1) 半径の長さが 3 の円の面積を求めなさい.
(答案)
____S=π×32=9π …(答)
(2) 右図の水色で示した図形の面積を求めなさい.
(答案)
____水色で示した大きい円の半径は 4,白で示した小さい円の半径は 2 だから,
S=π×42−π×22=16π−4π=12π …(答)

(3) 右図のような競技場の面積を求めなさい.
(答案)
____正方形の面積は 10000(m2),半円2つ(=円)の面積は π×502=2500π(m2) だから,S=2500π+10000(m2) …(答)
注意書き
○ π3.141592 ··· のような無限に続く小数になる.
 数学の答案では,正確な値 π を使って表わすことが多い.近似値(およその値)が必要になったときは π として小数第2位までの値 3.14 を使えば,ほとんどの場合十分なので長々と覚える必要はない.
○ たとえば,左の公式で 半径 2 のとき,円の面積 S は,正確な値で S=4π と答えるのがよく,S=12.56 は近似値に過ぎないので S=12.56 と答えるのはよくない.
○ 「半径の長さが 2 の円の面積と1辺の長さが 3 の正方形の面積では,どちらが大きいか」というような問題では, S1=4π のままでは S2=9 と比べられないので, S1=12.56S2=9 を比べて,円の方が大きいと答える.

[問題1]
(1) 右図の水色で示した輪の面積を求めなさい.
__________π

(2) 右図の水色で示した部分の面積を求めなさい.__________π

__________採点する やり直す 解説
【半円,4分円,6分円の面積】
○ 半円の面積は円の面積の半分に等しい.
 半径 r の円の半円の面積は .πr22nn

○ 4分円の面積は円の面積の4分の1に等しい.
 半径 r の円の4分円(右図のように中心角が 90° の扇形)の面積は .πr24nn

○ 半径 r の円の中心角が 60° の扇形の面積は .πr26nn

[例題2]
 1辺の長さが 2 の正方形において右図のように弧を描くとき,水色で示した部分の面積を求めなさい.
(答案)
__正方形の面積は 4
__4分円の面積は .π224nnn
_だから,4−π …(答)
[問題2]
(1) 右図の水色で示した部分の面積を求めなさい.
__________π−


(2) 右図の水色で示した部分の面積を求めなさい.
__________−π
__________採点する やり直す 解説


【扇形の面積】 半径 r, 中心角 x° の扇形の面積は

πr2×.x360nnn
(解説)
 ミカンやグレープフルーツを横に切って見ると,中心角が2倍,3倍,・・・になると扇形の面積も2倍,3倍,・・・となることが分かる.

 すなわち,扇形の面積は中心角に比例する.
 そこで,扇形の面積を S,円の面積を πr2 とおくと,
_________S : πr2=x : 360
_________360S=πr2x
_________S=πr2×.x360nnn


 上で述べた半円は,中心角 180° だから

S=πr2×.180360nnn=.πr22nn

に等しい.
[例題3]
 半径 4,中心角 72° の扇形の面積を求めなさい.
(答案)
_______S=π×42×.72360nnn=.16π5nnn …(答)
[問題3]
(1) 半径 6,中心角 120° の扇形の面積を求めなさい.
__________π

(2) 半径 4,中心角 45° の扇形の面積を求めなさい.
__________π

__________採点する やり直す 解説


○ 中学校で「円錐」といえば「直円錐」のことをいう.
 直円錐では,頂点から底面にひいた垂線は円の中心で交わる.また,母線の長さはどこで測っても等しい.
 以下において「直円錐」のことを単に「円錐」という.
【円錐の表面積】
○ 底面の半径が r,母線の長さが R の円錐の表面積を求めるには,右図のように展開図で考え,底面積=円と側面積=扇形の面積を各々求めて加えるとよい.
○ 底面は半径 r の円だから,その面積は πr2(1)
○ 側面の扇形の面積を求めるためには,その中心角を求めることが重要になる.
 円錐の展開図において,扇形の「弧の長さ」が底面の「円周の長さ」と等しいことから扇形の中心角が求められる.
 弧は,半径 R の円の一部分で,その長さは中心角に比例する.中心角が 360° のときは円になって 2πR
 中心角が x° のとき,弧の長さは 2πR×.x360nnn
 これが底面の円周の長さ 2πr に等しいことから,
_________2πR×.x360nnn=2πr

_________x=.360rRnnnn

 半径 R,中心角 .360rRnnnn の扇形の面積を求めると

_________πR2×.x360nnn=πR2×.rRn=πRr(2)

(1)(2)を加えて,πr2+πRr
[例題4]
  右図の円錐の表面積を求めなさい.
(答案)
 底面積は π×22=4π(1)
 展開図における扇形の中心角を x° とおくと
_________×6×.x360nnn=2π×2
_________x=120°
 側面積は π×62×.120360nnn=12π(2)
(1)(2)より 16π …(答)
[問題4]
(1) 右の円錐において,
____底面積は π
____展開図における扇形の中心角は °
____側面積は π
____表面積は π
となる.

(2) 右の円錐において,
____底面積は π
____展開図における扇形の中心角は °
____側面積は π
____表面積は π
となる.

(3) 底面の半径が 6,母線の長さが 9 である円錐の表面積は,
____π
____採点する やり直す 解説

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