Question

1. Select each of the constructible regular n-gons listed below.

a) 204-gon

b) 13-gon

c) 100-gon

2. Select each of the contructible angles below.

a) (3/20)°

b) 9°

c) (3/8)°

d) 1°

e) 17°

Answer 1

1. A regular n-gon is constructible if

(i) n can be expressed as a power of two (like ) or a Fermat prime () or

(ii) If it satisfies Gauss theorem:

If n can be expressed as:

where are distinct odd primes such that are all powers of 2.

a) 204-gon

n=204 which can be factorized as:

here and

So 204-gon satisfies Gauss theorem.

Hence a 204-gon is constructible.

b) 13-gon:

n=13

13 itself is a prime number.

Gauss theorem is not satisfied.

So a 13-gon is not constructible.

c) 100-gon:

25 is not a prime number. So Gauss theorem is not satisfied

So a 100-gon is not constructible.

2. Constructible angles:

An angle is constructible if n is constructible according to Gauss theorem.

a) :
We equate the given angle to to find the value of n.

n does not satisfy Gauss theorem.
Hence is not constructible.

b)

5 is a distinct prime.

So Gauss theorem is satisfied.

Hence is constructible.

c)

3 and 5 are distinct primes.

Hence Gauss theorem is satisfied.

So is constructible.

d)

9 is not a distinct prime. So Gauss theorem is not satisfied.
So is not constructible.

e)

n must be a whole number since the number of sides of a polygon cannot be fractional.

Hence is not constructible.