Area, Surface Area, Volume

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  FUNDAMENTAL CONCEPTS 1.  Results on Triangles: i.Sum of the angles of a triangle is 180°.ii.The sum of any two sides of a triangle is greater than the third side.iii.  Pythagoras Theorem: In a right-angled triangle, (Hypotenuse !   (#ase !  $ (Height ! .i%.The line &oining the mid-point of a side of a triangle to the positi%e %erte' is alled the median .%.The point where the three medians of a triangle meet, is alled entroid!  The entroid di%ided eah of the medians in the ratio ! ) 1.%i.In an isoseles triangle, the altitude from the %erte' *isets the *ase.%ii.The median of a triangle di%ides it into two triangles of the same area.%iii.The area of the triangle formed *y &oining the mid-points of the sides of a gi%en triangle is one-fourth of the area of the gi%en triangle.!.  Results on uadrilaterals: i.The diagonals of a parallelogram *iset eah other.ii.+ah diagonal of a parallelogram di%ides it into triangles of the same area.iii.The diagonals of a retangle are eual and *iset eah other.i%.The diagonals of a suare are eual and *iset eah other at right angles.%.The diagonals of a rhom*us are uneual and *iset eah other at right angles.%i. parallelogram and a retangle on the same *ase and *etween the same parallels are eual in area.%ii.f all the parallelogram of gi%en sides, the parallelogram whih is a retangle hasthe greatest area. #MPORTANT FORMULAE I.1. rea of a retangle (/ength ' #readth. /ength  reaand #readth  rea.#readth/engthII.!. erimeter of a retangle !(/ength $ #readth.III.rea of a suare (side !   (diagonal ! .I.rea of 2 walls of a room ! (/ength $ #readth ' Height..1. rea of a triangle ' #ase ' Height.!. rea of a triangle s ( s - a ( s - b ( s - c   where a , b , c   are the sides of the triangle and s   ( a  $ b  $ c  .3. rea of an euilateral triangle 3' (side !. 22. 4adius of inirle of an euilateral triangle of side a   a .!35. 4adius of irumirle of an euilateral triangle of side a    a .36. 4adius of inirle of a triangle of area and semi-perimeter r    . s I.1. rea of parallelogram (#ase ' Height.!. rea of a rhom*us ' (rodut of diagonals.3. rea of a trape7ium ' (sum of parallel sides ' distane *etween them.II.1. rea of a irle 4 ! , where 4 is the radius.!. irumferene of a irle !4.3. /ength of an ar !4, where is the entral angle.3602. rea of a setor 1(ar ' 4 ! .  4!360III.1. irumferene of a semi-irle 4.!. rea of semi-irle 4 ! .!1.  CU$O#D /et length l  , *readth b  and height h  units. Theni.  %olume   ( l   ' b  ' h  u*i units.ii.  Sur&ae area   !( lb  $ bh  $ lh  s. units.iii.  Diagonal   l  !  $ b !  $ h !  units.!.  CU$E /et eah edge of a u*e *e of length a . Then,i.  %olume   a 3  u*i units.ii.  Sur&ae area   6 a !  s. units.iii.  Diagonal   3 a  units.3.  C'L#NDER /et radius of *ase r   and Height (or length h . Then,i.  %olume   (  r  ! h  u*i units.ii.  Cur(ed sur&ae area )  (!  rh  s. units.iii.  Total sur&ae area   !  r  ( h  $ r   s. units.2.  CONE /et radius of *ase r   and Height h . Then,i.  Slant height*   l    h !  $ r  !  units.ii.  %olume   r  ! h  u*i units.iii.  Cur(ed sur&ae area   (  rl   s. units.i%.  Total sur&ae area   (  rl   $ r  !  s. units.5.  SP+ERE /et the radius of the sphere *e r  . Then,i.  %olume   r  3  u*i units.ii.  Sur&ae area   (2  r  !  s. units.6.  +EM#SP+ERE /et the radius of a hemisphere *e r  . Then,i.  %olume   r  3  u*i units.ii.  Cur(ed sur&ae area   (!  r  !  s. units.iii.  Total sur&ae area   (3  r  !  s. units.9ote) 1 litre 1000 m 3 .1.  right triangle with sides 3 m, 2 m and 5 m is rotated the side of 3 m to form a one. The %olume of the one so formed is) A! 1! m 3 $! 15 m 3 C! 16 m 3 D! !0 m 3 Ans,er:  ption AE-.lanation:  learly, we ha%e r    3 m and h   2 m. olume 1 r  ! h   1' ' 3 !  ' 2m 3 1! m 3 .33iew nswer  :isuss in ;orum <or=spae 4eport !. In a shower, 5 m of rain falls. The %olume of water that falls on 1.5 hetares of ground is) A! >5 u. m $! >50 u. m C! >500 u. m D! >5000 u. m Ans,er:  ption $E-.lanation: 1 hetare 10,000 m ! So, rea (1.5 ' 10000 m !   15000 m ! .:epth 5m 1m.100!0 olume (rea ' :epth 15000 '1m 3 >50 m 3 .!0iew nswer  :isuss in ;orum <or=spae 4eport 3.  hall is 15 m long and 1! m *road. If the sum of the areas of the floor and the eiling is eual to the sum of the areas of four walls, the %olume of the hall is) A! >!0 $! ?00 C! 1!00 D! 1800 Ans,er:  ption CE-.lanation: !(15 $ 1! ' h   !(15 ' 1!  h   180m !0m.!>3   olume 15 ' 1! '!0m 3 1!00 m 3 .3iew nswer  :isuss in ;orum <or=spae 4eport 2. 66 u*i entimetres of sil%er is drawn into a wire 1 mm in diameter. The length of the wire in metres will *e) A! 82 $! ?0 C! 168 D! 336 Ans,er:  ption AE-.lanation: /et the length of the wire *e h .4adius 1mm 1m.Then,!!0!!'1'1' h   66.>!0!0  h   66 ' !0 ' !0 '> 8200 m 82m.!!iew nswer  :isuss in ;orum <or=spae 4eport 5.   hollow iron pipe is !1 m long and its e'ternal diameter is 8 m. If the thi=ness of the pipe is 1 m and iron weighs 8 g@m 3 , then the weight of the pipe is) A! 3.6 =g $! 3.6?6 =g C! 36 =g D! 36.? =g Ans,er:  ption $E-.lanation: +'ternal radius 2 m,Internal radius 3 m.olume of iron 26! m 3 . <eight of iron (26! ' 8gm 36?6 gm 3.6?6 =g. !!' A(2 !  - (3 ! B ' !1m 3 > !!' >' 1'!1m 3 >
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