Question 193049
good  start,,somewhat  complex  solution,,your  geometry  book  probably  has  a  good  ref

lets  start  with  the  sketch,,,regular  hexagon  has  6  equal  sides,12  for  this  problem.
If  we  find  the  center  of  the  hexagon,  and  then  draw  lines  to  each  vertex,  we  have  6 identical  triangles.

Area  of  a  triangle  =1/2 base  * height
ok,  but  we  only  know  base,  we  need  to  find  height

We  also  know  the  central  angle,,,,,remember  sum  of  angles  of  polygon  =(n-2)(180)
in  our  case  (6-2)(180)=720,,,,,with  6  angles,,,720/6=120,,,(central  angle  of  triangle).

note  the  120  deg  on  one  triangle.
also ,  please,  draw  an  altitude on  that triangle,  from  center  to  base.
this  is  the  height  of  that  triangle.
Note  the  altitude  bisects  the  120 angle  into  2  equal  60  deg  angles.
we  are  almost  there,
in  our  triangle  we  have  the  60  degree  angle  at  the  center  vertex,  90 deg  where  altitude  intersects  the  base,  therefore  the  other  vertex angle  is  30 degrees,,,(sum  of  triangle =180 degrees)

lets  mark  the  30  degree  vertex,  and  the  90 degree  angle.

we  now  have  a  30-60-90  degree  triangle,  with  one  side (opposite  the  60 angle) equal  to  6
which  is  half  of  the  full  side  12.

we  know  from  typical  triangles  that  a  30-60-90  has  sides  of  1,2,sqrt3.
in  our  case  we  need  height,h,  which  is  opposite  the  30 deg  angle.
if  the  height  is  proportional  to (1),  hyp  is  (2),  and  the  base  side (sqrt3),,,but  base  side  is  6

sqrt3/6=1/h
cross  multiply,,
(sqrt3)(h)=6
h=6/sqrt3=3.464

now,,,area  of  triangle  is  1/2 (base) (height) = 1/2 (12)(3.464)=20.78,,,,(full base=12)

and  remember  we  have  6  triangles  in  hexagon
area  if  hexagon=6(20.78)=124.7=125

this  is  generalized  to  Area  of  any  polygon  =1/2 (perimeter)(apothem)
in  our  case,,,per  =6*size  of  side,  apothem  is  height
Area=1/2 *(12*6)*3.464=  125  ok

note,  apothem  is  denoted  (a),  and  is  usually  given,,just  remember  1/2 * per*apothem