are there pattern in area of polygon and inner-polygon formed when midpoints of all sides connected

Question

investigate the area of the polygon and inner-polygon formed when the midpoints of all sides are connected

Answer 1

For REGULAR polygon, kan get sum jenral ansers

konnekt midpoints av outside sides av reg poly, yu get smaller polygon,

upside down

around it will be n identikal isoseles triangels

n=3...equalateral triangel: yu get 4 identikal equalteral tris, so inside area=1/4 original triangel

n=4...square, inside square hav half the area av original square

n=5...yu get 5 identikal outside triangels, all=isosalees, with 2 sides=(1/2) av

1 side av original pentagon.  Pentagon inside angel=108 deg, so other 2

angels=36 deg.  half leng av inside triangel=(leng/2)*kosine(36 deg)

thus, leng av 1 side av inside polygon=kosine(36 deg)*leng av original side

polygon area=(n/4)*side*side*tan(beta), weer beta=(180-alfa)/2, & alfa=360 deg/n

for n=5, beta=54deg

portant thang is inside side leng=(original side leng)*kosine(36 deg)

(note...36deg=alfa / 2, alfa=360/5=72 deg)

sins area is proposhunal tu side*side, (inside area) / (outside area)=kosine^2(36 deg)

kosine(36deg)=0.809016994; kosine^2(36)=0.654508497...pentagon oenlee

n=6 (hexagon), alfa=360/6=60deg, so take kosine(30 deg) & square=0.75