SOLUTION: Two regular polygons of the same number of sides have sides 48m and 55m in length respectively. What is the length of the side of another regular polygon of the same number of side

Algebra ->  Polygons -> SOLUTION: Two regular polygons of the same number of sides have sides 48m and 55m in length respectively. What is the length of the side of another regular polygon of the same number of side      Log On


   

Question 251475: Two regular polygons of the same number of sides have sides 48m and 55m in length respectively. What is the length of the side of another regular polygon of the same number of sides, if its area is equal to the sum of the two ?
Answer by Theo(13342) About Me  (Show Source):
You can put this solution on YOUR website!
let n = the number of sides of each polygon.
let s = length of a side of each polygon.
let a = area of each polygon.

s[1] = 48
s[2] = 55
s[3] = ?????

formula for area of a polygon is:

a+=+%28s%5E2%2An%29+%2F+%284%2Atan%28pi%2Fn%29%29

a%5B1%5D+=+%2848%5E2%2An%29+%2F+%284%2Atan%28pi%2Fn%29%29

a%5B2%5D+=+%2855%5E2%2An%29+%2F+%284%2Atan%28pi%2Fn%29%29

a%5B3%5D+=+a%5B1%5D+%2B+a%5B2%5D

since the number of sides are equal, then the measurement of each of the angles of each of the polygons is also equal.



this is equivalent to:



factor out the n in the numerator to get:




simplify by performing the indicated operations to get:

a%5B3%5D+=+a%5B1%5D+%2B+a%5B2%5D+=+%28%285329%29%2An%29+%2F+%284%2Atan%28pi%2Fn%29%29

take the square root of 5329 and substitute in the equation to get:


a%5B3%5D+=+a%5B1%5D+%2B+a%5B2%5D+=+%28%2873%5E2%29%2An%29+%2F+%284%2Atan%28pi%2Fn%29%29

The side of the third polygon is equal to 73.

this will be true for any value of n because the denominator in the equation and the variable n both drop out of the equation.

to confirm, let:

a[1] + a[2] = a[3]

this becomes:



simplify the fraction on the left side of the equation to get:



multiply both sides of the equation by %284%2Atan%28pi%2Fn%29%29 and divide both sides of the equation by %28n%29 to get:

%28%2848%5E2%29+%2B+%2855%5E2%29%29+=+%2873%5E2%29

select the following link to see a reference where I got the equations from.

http://www.mathopenref.com/polygonregulararea.html