What is the perimeter of the star?

Question

The sides of a regular pentagon, side length 20cm, are extended to form a 5-pointed star. What is the length of its perimeter?

Answer 1

yu need tu draw the pentagon,,,draw outside serkel thru the 5 points,

tgeb draw lines from senter av serkel tu poly points=kut polygon intu 5 identikal

triangels with base=1 side av original polygon.   Senter angel=360/5=108 deg.

Eech isosaleez triangel hav base angels=(180-72)/2=54 deg.

pentagon=5 av the identikal isos triangels.   Side av isos=radius av serkel.  Base =pentagon side.

The star have 5 identikal isos triangels with base=1 side av original pentagon.

Split in 2...2 identikal rite triangels, base=(side/2), 90 deg, 54 deg at base, 28 deg at top.

sine(28 deg)=0.4694715627859...base/diagonal, but base=half leng av 1 side or original penta.

1/sine(28 deg)=2.1300544681894706=raesheeo av 2 outsides tu original (1 side av pentagon)

So yu hav 5 such points, so sum av length av the 5 pointed thangs=1.1300544681894*

rim-leng av original pentagon..

If origina side=20, 5 sides=100 & sum av pointed thangs=100*1.130054468

=213.00544681894706

Comments

Line 3 should be centre angle=72 degrees as shown in line 4.
Line 7: the angles of the right-angled triangle are 54 and 36 degrees. Where do you get 28 degrees from? The extended sides of the pentagon form the points of the star. These are isosceles triangles with a base of 20cm (side of pentagon) and equal base angles of 72 degrees (180-108). The equal sides of the isosceles triangle have length 10sec72=32.36068cm approx. The perimeter of the star has 10 of these lengths=323.6068cm. Since the pentagon is completely enclosed by the star, the perimeter of the pentagon is not included in the perimeter of the star.