Question

The interior angle of heptagon
are : 2r, 3r, 2r,5r and 2r. find
the value of r.​

Answer 1

Step-by-step explanation:

Number of sides, all equal length a: n

Number of interior angles, all equal measure beta: n

Central angle subtending one side: alpha

Perimeter: P

Area: K

Radius of circumscribed circle: R

Radius of inscribed circle: r

beta = Pi(n-2)/n radians = 180o(n-2)/n

alpha = 2 Pi/n radians = 360o/n

alpha + beta = Pi radians = 180o

P = na = 2nR sin(alpha/2)

K = na2 cot(alpha/2)/4

= nR2 sin(alpha)/2

= nr2 tan(alpha/2)

= na sqrt(4R2-a2)/4

R = a csc(alpha/2)/2

r = a cot(alpha/2)/2

a = 2r tan(alpha/2) = 2R sin(alpha/2)