Polygon Area Ratio Basics
We are considering two regular polygons that share the same number of sides. The ratio comparing their side lengths is given as $3:5$. Our task is to find the ratio of their corresponding areas.
Area Formula and Ratio Derivation
The area ($A$) of a regular polygon is calculated using the formula $A = \frac{1}{4} n s^2 \cot\left(\frac{\pi}{n}\right)$, where $n$ is the number of sides and $s$ is the side length.
Let the two polygons have side lengths $s_1$ and $s_2$, and areas $A_1$ and $A_2$. Since both polygons have the same number of sides ($n$), the ratio of their areas is:
$ \frac{A_1}{A_2} = \frac{\frac{1}{4} n s_1^2 \cot\left(\frac{\pi}{n}\right)}{\frac{1}{4} n s_2^2 \cot\left(\frac{\pi}{n}\right)} $The terms $\frac{1}{4}$, $n$, and $\cot\left(\frac{\pi}{n}\right)$ are identical for both polygons and thus cancel out. This simplifies the relationship to:
$ \frac{A_1}{A_2} = \frac{s_1^2}{s_2^2} = \left(\frac{s_1}{s_2}\right)^2 $Side Length Ratio to Area Ratio Calculation
We are provided with the ratio of the side lengths: $ \frac{s_1}{s_2} = \frac{3}{5} $ Substitute this ratio into the derived formula for the areas:
$ \frac{A_1}{A_2} = \left(\frac{3}{5}\right)^2 $ $ \frac{A_1}{A_2} = \frac{3^2}{5^2} = \frac{9}{25} $Therefore, the ratio of the areas of the two regular polygons is $9:25$.
Similar Questions
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Important Questions from 2-D Mensuration
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