Regular Polygon, Apothem, Distance

In a regular n-sided polygon, the sum of the perpendicular distances from an interior point to the n sides being n times the apothem of the polygon.

See complete Problem 225 at:

gogeometry.com/problem/p225_viviani_theorem_regular_polygon.htm

Level: High School, SAT Prep, College geometry

## Wednesday, January 14, 2009

### Elearn Geometry Problem 225: Viviani Theorem Extension

Labels:
apothem,
distance,
interior point,
perpendicular,
regular polygon,
Viviani theorem

Subscribe to:
Post Comments (Atom)

One can prove that, in a equiangled polygon, the sum of the distances of an arbitrary inner point to the sides is constant =]

ReplyDelete(1)let the no. of sides be n.

ReplyDelete(2)let the length of each side be y.

(3)therefore the area of the given polygon is 1/2ap.y.n

(4)from point P, we can do tringulations (on the sides)

(5)We note that sum of area of those triangls=area by 1/2 ap.y.n method =(h1+h2+h3+....+hn).y.1/2

(6)then, 1/2& y get cancelled & we get ap.n = h1+h2+h3+...+hn

(7)hence the proof!!