ATRACTOR
ÁREAS E VOLUMES 2
!"#$%&'( &)( *+!( ,'"+*-!./'( 0)'+1#"$2!( !-#)"%!#$3!( &'( 4)'")+!( &)( $#50'"!67( 1( $%&$2!&!( *+!( 2'%6#"*./'( 8*)( !( 2!&!(
2'"'!(2$"2*-!"(!66'2$!(*+(&$62'(2'+(!(+)6+!(5")!9(:7(");)#$%&'()66)(;"'2)&$+)%#'(2'+(!6(2'"'!6(2$"2*-!")6(8*)(6)('<#=+(
6)22$'%!%&'(*+!()6,)"!()(*+(2$-$%&"'(2$"2*%62"$#'(;'"(*+!(
,!+>-$!(&)(;-!%'6(;)";)%&$2*-!")6(!'()$?'(&)6#)7(&)#)"+$%!")+'6('(3'-*+)(&!()6,)"!9
No âmbito de uma colaboração
entre a Gazeta e o Atractor, este
é um espaço da responsabilidade
do Atractor, relacionado com
conteúdos interactivos do seu
site www.atractor.pt .
Quaisquer reacções ou sugestões
serão bem-vindas para
[email protected]
'%6$&)")+'6(*+(#"$@%0*-'(")2#@%0*-'()(#"=6(8*!&"!&'6(2'%6#"*>&'6(6'<")(
'6(6)*6(#"=6(-!&'69(A!(6*2)66/'(&)($+!0)%6(!2$+!(B(&)(8*)(!($+!0)+(C%!-(1(
!($%&$2!&!(D()68*)"&!7(6);!"!&!(&!6('*#"!6(B(!6(;!"#)6(!(3)"+)-E'(#=+($0*!-(5")!(
)('(+)6+'(3!-)(;!"!(!6(2'-'"$&!6(!(!+!")-'9(F)(,!2#'7(;!"!(!(C0*"!(3)"+)-E!G(
)+(H7(I7(J()(K7(L7(M7(;'"8*)(!(5")!(&)(*+(;!"!-)-'0"!+'(6N(&);)%&)(&'(2'+;"$+)%#'(&)(*+(-!&'()(&!(!-#*"!(2'"")6;'%&)%#)(OHP7()()+(J7(Q7(K7(;'"8*)(*+!("'#!./'(2'%6)"3!(&$6#@%2$!6()7(;'"#!%#'7(#!+<1+(5")!69(R%!-'0!+)%#)(;!"!(!(C0*"!(
!+!")-!G(;!66'6(M7(S7(T()(HH7(HI7(C%!-(;)-!(;"$+)$"!("!U/'()(T7(HV7(HH(;)-!(6)0*%&!9(
W'%2-*6/'G(!(5")!(&'(8*!&"!&'(2$%U)%#'(2'%6#"*>&'(6'<")(!(E$;'#)%*6!(1($0*!-(D(
6'+!(&!6(5")!6(&'6(8*!&"!&'6(2'-'"$&'6(2'%6#"*>&'6(6'<")('6(2!#)#'6(X,'"+*-!./'(0)'+1#"$2!(&'(4)'")+!(&)( $#50'"!6Y9(
C
ATRACTORrÁreas e Volumes 2
03
W'+'( C0*"!6( 6)+)-E!%#)6( 2'+( "!U/'( &)( 6)+)-E!%.!( k( #=+( 5")!6( 8*)( )6#/'( %!(
"!U/'(kI(OHP7(;'&)+'6(2'%2-*$"(8*)7(;!"!(8*!$68*)"('*#"!6(C0*"!6(X6)+)-E!%#)6()%#")(6$Y7(2'%6#"*>&!6(X&)($0*!-(,'"+!Y(6'<")('6(2!#)#'6()(!(E$;'#)%*6!7(2'%#$%*!(!(6)"(
3)"&!&)(8*)(!(5")!(&!(C0*"!(2'%6#"*>&!(6'<")(!(E$;'#)%*6!(1($0*!-(D(6'+!(&!6(5")!6(
&!6(2'%6#"*>&!6(6'<")('6(2!#)#'69(:+(2!&!($+!0)+(!(C0*"!(2$%U)%#!(#)+(2'+'(5")!(
!(6'+!(&!6(5")!6(&!6(C0*"!6(3)"+)-E!()(!+!")-!9
R(C0*"!(!<!$?'( 6*0)")(*+!(,'"+*-!./'(0)'+1#"$2!(!-#)"%!#$3!( ;!"!('(#)'")+!(
&)( $#50'"!69(W'-'2!%&'(*+!(2N;$!(&'(8*!&"!&'(3)"+)-E'(%'(8*!&"!&'(2$%U)%#'7(
'<#1+B6)(*+!(2'"'!(8*!&"!&!7(2*^!(5")!(1()?!2#!+)%#)(!(&'(8*!&"!&'(!+!")-'(!66'2$!&'(!'('*#"'(2!#)#'9(
À esquerda, a coroa a cinzento e o quadrado amarelo têm mesma área
!"!(2!&!($+!0)+7(#=+(5")!($0*!-(!6(C0*"!6(2'+(!(+)6+!(2'"(
X3)"(D(&$")$#!Y9
Z#$-$U!%&'( &)3$&!+)%#)( )6#!6( ;"';"$)&!&)67( ;'&)")+'6( )%2'%#"!"( ,!2$-+)%#)( *+!( 6'-*./'( ;!"!( '( ;"'<-)+!( &)7( &!&!( *+!(
[2'"'!\(<!6)!&!(%*+!(C0*"!(&)(2)"#!(,'"+!7()%2'%#"!"+'6(*+!(
C0*"!( 2'+( !( +)6+!( ,'"+!( X$9)9( 6)+)-E!%#)Y( 2'+( !( +)6+!( 5")!9((
'"( )?)+;-'7( 2'+'( 2'%6#"*$"( *+( 8*!&"!&'( 2'+( 5")!( $0*!-( D( &!(
2'"'!( 8*!&"!&!( !+!")-!( %!( C0*"!( !'( -!&']( A'( 2!6'( 0)"!-7( <!6#!(
)62'-E)"(%!(C0*"!(+!$'"(&'$6(;'%#'67(2'%6#"*$"(*+(#"$@%0*-'(")2#@%0*-'(8*)(#)%E!(2'+'(+)&$&!(&!(E$;'#)%*6!(!(&$6#@%2$!()%#")(
)66)6(&'$6(;'%#'6(&!(C0*"!()(2'+'(+)&$&!(&)(*+(&'6(2!#)#'6(!(
&$6#@%2$!( )%#")( '6( ;'%#'6( 2'"")6;'%&)%#)6( &!( C0*"!( 6)+)-E!%#)(
04
GAZETA DE MATEMÁTICA r165
&)( 2$"2*%,)"=%2$!( *+( ;'%#'9( _( 6)0+)%#'( X!+!")-'Y( 3)"#$2!-(
&)6#!(")2#!(2'+()?#")+'(%)66)(;'%#'(1('(2!#)#'(&)6)^!&'9(_(
8*!&"!&'(;"'2*"!&'(#)+(2'+'(2'+;"$+)%#'(&'(-!&'('(&'<"'(
&'(&)6#)(2!#)#'9(
!"!(;"$6+!6(")2#'6(!66)%#)6(6'<")()6#!6(C0*"!6(;-!%!67(#)+'6(
")6*-#!&'6( !%5-'0'67( !0'"!( )%3'-3)%&'( '6( ")6;)2#$3'6( 3'-*+)69( '"()?)+;-'7(%!(C0*"!(!<!$?'7(;!"#$%&'(&!(2'"'!(;"$6-
+!$6(;)8*)%!7(%!(2'"'!9(W'%6#"*$%&'(!(C0*"!(6)+)-E!%#)()+(
8*)('6(&'$6(;'%#'6(2'"")6;'%&)%#)6(!'6(!%#)"$'")6(&$6#!+()%#")(6$()?!2#!+)%#)('(2'+;"$+)%#'(&'(%'3'(2!#)#'(&'(#"$@%0*-'7(
#)+'6('(;"'<-)+!(")6'-3$&'(XC0*"!(!2$+!Y9
!"!(2'%6#"*$"('(#!-(#"$@%0*-'(")2#@%0*-'7(*+(;"'2)66'(6$+;-)6(1('(&)(&)6)%E!"(*+!(2$"2*%,)"=%2$!(2*^'("!$'(6)^!(!(E$;'#)%*6!(&)6)^!&!7(+!"2!"(%*+("!$'(*+(6)0+)%#'(2'+('"$0)+(
%'(2)%#"'()(2'+;"$+)%#'($0*!-(!'(&'(2!#)#'(&)6)^!&'()(#$"!"(
*+!( ;)";)%&$2*-!"( ;)-'( '*#"'( )?#")+'9( '"( )?)+;-'7( ;!"#$%&'(&!(2'"'!(8*!&"!&!(!+!")-!(!%#)"$'"7(!(2'%6#"*./'(0)'+1#"$2!($%&$2!&!(%!(C0*"!(!2$+!(-)3!(!(*+(8*!&"!&'(2'+(!(
+)6+!(5")!(&!(2'"'!9(_(!"2'(&)(2$"2*%,)"=%2$!($%&$2!&'(#)+(
2'+'("!$'(!(+)&$&!(&)(+)$'(-!&'(&'(8*!&"!&'(0"!%&)(&!(2'"'!( )( '( 6)0+)%#'( 3)"+)-E'( E'"$U'%#!-( 2'+( *+( )?#")+'( %'(
2)%#"'(&!(2$"2*%,)"=%2$!(#)+(2'+;"$+)%#'($0*!-(!'(&)(+)#!&)(
&'(-!&'(&'(8*!&"!&'(3)"+)-E'9(R(")2#!(;)";)%&$2*-!"(!()6#)(
6)0+)%#'(;!66!%&'(;)-'(6)*('*#"'()?#")+'(&)#)"+$%!(%'(!"2'(
+5#$2!(&!()68*)"&!()(!;-$2!%&'(!(2'%6#"*./'(!%#)"$'"(D(6)2./'(
")2#!( XE'"$U'%#!-Y7( 8*)( 1( *+!( 2'"'!( 8*!&"!&!7( '<#)+'6( *+(
;"$6+!(&)(6)2./'(8*!&"!&!(2'+('(+)6+'(3'-*+)(&!([2'"'!(
;"$6+5#$2!\(&!()68*)"&!9(
`!6( )6#!+'6( ;!"#$2*-!"+)%#)( $%#)")66!&'6( %*+!( 6$#*!./'(
+!$6( 2'+;-)?!7( )+( 8*)( !( ;!"#)( )?#)"$'"( 1( *+( ;"$6+!( X'*(
2$-$%&"'Y()(!($%#)"$'"(#)+(6)2.a)6(3!"$53)$6(2'+(!(!-#*"!9(b)7(
)+(2!&!(%>3)-7(!;-$2!"+'6(!(2'%6#"*./'(!%#)"$'"7('<#)+'6(&'(
-!&'()68*)"&'(2'"'!6()+(8*)(!6(C0*"!6(&)(,'"!(6/'(#'&!6(2'%0"*)%#)67(+!6(%/'(!6(&)(&)%#"'9(_(")6*-#!&'(1('(8*)(!6(C0*"!6(
!<!$?'($-*6#"!+9( !"!(2!&!(;!"7('(3'-*+)(&!(")0$/'(X$+!0)+(
&!()68*)"&!Y(2'+;"))%&$&!()%#")('(;"$6+!('*(2$-$%&"'()?#)"$'"( )( '( 6N-$&'( 8*)( )6#5( &)%#"'( 1( $0*!-( !'( 3'-*+)( &'( 6N-$&'(
X3)"&)Y9(c6#'(")6*-#!(&'(,!2#'(&)(!6(6)2.a)6(;'"(2!&!(;-!%'(E'B
ATRACTORrÁreas e Volumes 2
05
"$U'%#!-(#)")+(!(+)6+!(5")!9(_(#!+!%E'(&)66!6(6)2.a)6(&);)%&)(&!(!-#*"!(6)0*%&'(*+!(,*%./'(8*)7(%)6#)()?)+;-'7(1(!(
+)6+!(;!"!('6(&'$6(;"$+)$"'6(;!")6()(&$,)")%#)(;!"!('(d-#$+'(
X3)"(C0*"!Y9(
b)(!(2*"3!(&)#)"+$%!%&'(!(3!"$!./'(&'(#!+!%E'(&!6(6)2.a)6(
&'(6N-$&'(3)"+)-E'(,'"(*+!(2$"2*%,)"=%2$!7('<#)+'6(*+(2!6'(
;!"#$2*-!"+)%#)($%#)")66!%#)e(%'#)B6)(%!(C0*"!(D(&$")$#!(X+)$'Y7(
!( 2*"3!( 2'"")6;'%&)%#)( !+!")-!9( b)( !6( ,'"+!6( *6!&!6( ,'")+(
2$"2*-!")67('(6N-$&'(3)"+)-E'(&!()68*)"&!(&)%#"'(&'(2$-$%&"'(1(
*+!()6,)"!()('(6N-$&'(3)"&)(&!(&$")$#!(*+(X&*;-'Y(2'%)9
f*!-(!("!U/'](b)(3'-#!"+'6(!'(;"'2)66'(!%#)"$'"(&)(2'%6#"*./'7(&)62'<"$+'6(8*)(%)6#)(2!6'(!(!-#*"!(&!(6)2./'(1($0*!-(
!'( 2'+;"$+)%#'( &'( 2!#)#'( 2'%6#"*>&'7( 8*)7( ;'"( 6*!( 3)U7( 3!$(
6)"('("!$'(&!(6)2./'(&!(C0*"!(
3)"&)(&!(&$")$#!9(c6#'(17('("!$'(
&)6#!( 6)2./'( 1( ;")2$6!+)%#)(
$0*!-(D(!-#*"!(&!(6)2./'9( '"#!%#'7('<#)+'6(0)"!#"$U)6(")2#$->%)!6(&)(*+(&*;-'(2'%)(&)(
")3'-*./'( 2'+( *+!( $%2-$%!./'( &)( QKg( ")-!#$3!+)%#)( !'(
)$?'9( R-$567( %'( 2!6'( )+( !%5-$6)7(E5(*+!(2'%6#"*./'()8*$3!-)%#)()(+!$6(&$")2#!(&'(2!#)#'(&)6)^!&'(X8*)(6)"5('("!$'(
&'( 2>"2*-'( 2'+( 5")!( $0*!-( D(
&!(2'"'!(2$"2*-!"(&)(;!"#$&!Y9(
W'+'(!(C0*"!(D()68*)"&!($%&$2!7(%)66)(2!6'(;!"#$2*-!"(&'(
2>"2*-'7(<!6#!(+!"2!"(*+("!$'(E'"$U'%#!-(&'(2>"2*-'(3)"+)-E'G(
'(6)0+)%#'(!+!")-'(3)"#$2!-(;!"#$%&'(&'()?#")+'(&)66)("!$'(
)(#)"+$%!%&'(%!(2$"2*%,)"=%2$!(!+!")-!(1('(2!#)#'(;"'2*"!&'9(
_(#"$@%0*-'(")2#@%0*-'(!66$+('<#$&'(1(2-!"!+)%#)(2'%0"*)%#)(
2'+('(8*)(6)('<#1+(;)-!(2'%6#"*./'(0)"!-7(35-$&!(;!"!(#'&!6(
!6(,'"+!69(W'+'('(3'-*+)(&'(&*;-'(2'%)(1(*+(#)".'(&'(3'-*+)(&'(2$-$%&"'(2'+(!(+)6+!(<!6)()(!-#*"!(X'(&!()68*)"&!Y(OHP7(
'(3'-*+)(&!()6,)"!(6)"5(!(&$,)")%.!()%#")('6(&'$67('*(6)^!7(&'$6(
#)".'6(&'(3'-*+)(&)66)(2$-$%&"'9
:+(OHP(,'"!+(&!&!6(^*6#$C2!.a)6(&!6(!C"+!.a)6G($Y(!(5")!(
&)(*+(;!"!-)-'0"!+'(6N(&);)%&)(&'(2'+;"$+)%#'(&!(<!6)e(
$$Y( '( 3'-*+)( &)( *+( ;"$6+!( X2$-$%&"'Y( '*( &)( *+!( ;$"@+$&)(
X*+( 2'%)Y( 6N( &);)%&)( &!( !-#*"!( )( &!( 5")!( &!( <!6)( X%/'( &!(
,'"+!(&)6#!Ye($$$Y(*+!(;$"@+$&)(X*+(2'%)Y(#)+(2'+'(3'-*+)(
*+(#)".'(&'(3'-*+)(&'(;"$6+!(X2$-$%&"'Y(2'+(!(+)6+!(<!6)()(
!-#*"!9(:(&)2'"")(&)(;"';"$)&!&)6(^*6#$C2!&!6(%)66)(#)?#'(8*)(
!(5")!(&)(*+(#"$@%0*-'(6N(&);)%&)(&'(2'+;"$+)%#'(&!(<!6)()(
&!(!-#*"!7(;)-'(8*)(1(+)#!&)(&!(&'(;!"!-)-'0"!+'(2'%6#"*>&'(
6'<")(&'$6(&'6(6)*6(-!&'69(h$?!%&'(2'+'(*%$&!&)(&)(3'-*+)(
06
GAZETA DE MATEMÁTICA r165
'(3!-'"(&'(3'-*+)(&)(*+(2*<'(&)(!")6#!(*%$#5"$!()(*#$-$U!%&'(&)3$&!+)%#)(0")-E!6(#"$&$+)%6$'%!$6(2!&!(3)U(+!$6(C%!67(
'<#$&!6(!(;!"#$"(&!8*)-)(2*<'7(2'%2-*$B6)(8*)(8*!-8*)"(;"$6+!(
")2#!%0*-!"( ")2#'( #)+( 2'+'( 3'-*+)( '( ;"'&*#'( &'6( 2'+;"$+)%#'6(&)(#"=6(!")6#!67(&*!6(%!(<!6)()(*+!(3)"#$2!-9( '"('*#"'(
-!&'7(i(1(&)C%$&'(2'+'(!("!U/'(X8*)(6)(;"'3!(6)"(2'%6#!%#)Y(
)%#")('(;)">+)#"'()('(&$@+)#"'(&)(*+!(2$"2*%,)"=%2$!7('*(6)^!(
'(;)">+)#"'(17(;'"(&)C%$./'(&)(i7($0*!-(!(I(i(r7(6)%&'("('("!$'(
&!(2$"2*%,)"=%2$!9(c%62")3)%&'(*+(;'->0'%'(")0*-!"(&) n(-!&'6(%!(2$"2*%,)"=%2$!()(*%$%&'('6(31"#$2)6(!'(2)%#"'7('<#=+B6)(
%(#"$@%0*-'67(2*^!(5")!(#'#!-(1(+)#!&)(&'(;"'&*#'(&'(;)">+)#"'(
;)-'(!;N#)+!9(F!8*$(2'%2-*$B6)(8*)(!(5")!(&'(2>"2*-'(1(+)#!&)(
&'(;"'&*#'(&'("!$'(;)-'(;)">+)#"'7('*(6)^!(i(rjI(9(R66$+7('(3'-*+)(&'(2$-$%&"'(&)(")3'-*./'()%3'-3)%#)(&!()6,)"!(1(i(rjI(k(I(r
)('(&!()6,)"!(1(&'$6(#)".'6(&'(3'-*+)(&'(2$-$%&"'7('*(6)^!7(XIlJY(
i(rjI(9XIrYmXQlJY(i(rjJ9
REFERÊNCIAS
OHP(R#"!2#'"7([n")!6()(o'-*+)6\7( !"#$!%&#%'!$#()$*+!(HLQ7(!%'(
MI7(p*-(IVHH7(;;9(QBM