Question

The ratio of the number of sides of two regular polygons is 1 : 2, and the ratio of the

sum of their interior angles is 3 : 8. Find the number of sides in each polygon​

Answer 1

Answer:

sides of the given polygon are 5 and 10

Step-by-step explanation:

$\text{Let }n_1\text{ and }n_2\text{ be the number of sides the given two polygons}$

$\text{Given:}$

$n_1:n_2=1:2$

$\implies\,n_1=k\text{ and }n_2=2k$

$\textit{We know that, the sum of interior angles of a regular polygon is }\bf\,(n-2)\times\,180^{\circ}$

$\text{As per given data, }$

$(n_1-2)\times\,180^{\circ}:(n_2-2)\times\,180^{\circ}=3:8$

$\implies\frac{(n_1-2)\times\,180^{\circ}}{(n_2-2)\times\,180^{\circ}}=\frac{3}{8}$

$\implies\frac{n_1-2}{n_2-2}=\frac{3}{8}$

$\implies\frac{k-2}{2k-2}=\frac{3}{8}$

$\implies\frac{k-2}{2(k-1)}=\frac{3}{8}$

$\implies\frac{k-2}{k-1}=\frac{3}{4}$

$\implies\,4k-8=3k-3$

$\implies\,k=5$

$\therefore\,n_1=5\text{ and }n_2=10$

Answer 2

Answer:

The number of sides of each polygon is 3 and 6.

Step-by-step explanation:

Since, it is being given that ;

The ratio of the number of sides of two regular polygons is 1 : 2;

Let,     n1 : n2  =  1 : 2

∴ $\frac{n1}{n2}  =  \frac{1}{2}$

the ratio of the  sum of their interior angles is 3 : 8;

Let , the sum of the interior angles be A1 and A2 respectively;

So,       A1 : A2 =  3 : 8

∴     $\frac{A1}{A2}  =  \frac{3}{8}$

Since , the formula for sum of the interior angle of the polygon is $\frac{(n - 2)  \times  180}{n}$

∴  A1  =  $\frac{(n1 - 2)  \times  180}{n1}$...........(1)

&

   A2 =  $\frac{(n2 - 2)  \times  180}{n2}$,,,,,,,,,,(2)

Dividing  (1) and (2)

∴  $\frac{A1}{A2} = \frac{(n1 - 2) \times 180 }{n1} \times\frac{n2}{(n2 -2) \times 180}$

∴ $\frac{3}{8}  = \frac{(n1-2) \times n2}{(n1-2) ]\times n1}$

∴$\frac{3}{8}  = 2 \times \frac{n1 - 2}{n2 -2}$

$3 \times n2  - 6 = 16 \times (n1-2)$

Putting,    n2 = 2n1 ;

We get;

$6 \times n1 - 16 \times n1 = -26$

∴ We get n1 = 2.6 ≅ 3

∴ Number of sides of first polygon is 3.

Since$n2 = 2 \times n1$

∴$n2 = 2 \times  3$

n2 = 6

∴ Number of sides of second polygon is 6.