Which polgon has 35 diagonals?

Answer 1

The number of diagonals in a polygon with n sides is given by n(n-3)/2 (see related question). So, 35 = n(n-3)/2 is what we have to solve. 70 = n(n-3) 70 = n2 - 3n n2 - 3n - 70 = 0 Factorizing the quadratic: (n-10)(n+7) = 0 This gives solutions n = 10 and n = -7. Clearly only the positive answer makes any sense (a shape cannot have minus seven sides!) so our shape has 10 sides: it is a decagon.