Let n represent the number of sides of the polygon with fewer sides. Consequently, the polygon with more sides will have 2n sides, as their side ratio is 1:2.
The formula for calculating the interior angle of a regular polygon is:
Interior Angle\(=\frac {(n−2)×180}{n}\)
Given that the ratio of their interior angles is 3:4, we can establish the following proportion:
\(\frac {\text {Interior\ Angle\ of\ n-sided\ polygon}}{\text {Interior\ Angle\ of\ 2n\ sided\ polygon}}=\frac 34\)
Applying the interior angle formula:
\(\frac {\frac {(n−2)×180}{n}}{\frac {(2n−2)×180}{2n}}=3:4\)
\(4×2n×(n−2)×180=3×n×(2n−2)×180\)
\(8n(n−2)=3n(2n−2)\)
\(8n^2−16n=6n^2−6n\)
\(2n^2−10n=0\)
\(2n(n−5)=0\)
This equation yields two solutions: \(n=0\) or \(n=5\).
Since a polygon cannot have zero sides, the only valid solution is \(n=5\).
Therefore, the polygon with more sides has \(2n=2×5=10\) sides.