Question

Ratio of two sides of polygon is 1:2 and ratio of their interior angle is 3:4. Find the number of sides of the polygon with more number of sides.

Answer 1

Let the number of sides of the polygon with fewer sides as n. Therefore, the number of sides of the polygon with more sides would be 2n, since the ratio of their sides is 1:2.
The interior angle of a regular polygon can be calculated using the formula:
Interior Angle\(=\frac {(n−2)×180}{n}\)
Now, given that the ratio of interior angles is 3:4, we can set up the following proportion:

\(\frac {\text {Interior\ Angle\ of\ n-sided\ polygon}}{\text {Interior\ Angle\ of\ 2n\ sided\ polygon}}=\frac 34\)

Using the formula for interior angles:
\(\frac {\frac {(n−2)×180}{n}}{\frac {(2n−2)×180}{2n}}=3:4\)

\(4×2n×(n−2)×180=3×n×(2n−2)×180\)
\(8n(n−2)=3n(2n−2)\)
\(8n^2−16n=6n^2−6n\)
\(2n^2−10n=0\)
\(2n(n−5)=0\)
This equation has two solutions: \(n=0\) or \(n=5\). 
Since the number of sides cannot be zero, the only valid solution is \(n=5\).

So, the polygon with more sides has \(2n=2×5=10\) sides.