Pentagon Angle Calculation Using Ratios

The sum of the interior angles of a polygon with n sides is given by the formula:

$ \text{Sum} = (n-2) \times 180^\circ $

For a pentagon, n = 5. Therefore, the sum of its interior angles is:

$ (5-2) \times 180^\circ = 3 \times 180^\circ = 540^\circ $

Ratio Analysis for Pentagon Angles

The interior angles are in the ratio 1 : 3 : 5 : 7 : 11. Let the angles be represented as $1x$, $3x$, $5x$, $7x$, and $11x$, where x is a common multiplier.

The sum of these angles is:

$ 1x + 3x + 5x + 7x + 11x = 27x $

Determining the Smallest Angle

Equating the sum of the ratio parts to the total sum of the interior angles of the pentagon:

$ 27x = 540^\circ $

Now, solve for x:

$ x = \frac{540^\circ}{27} $

$ x = 20^\circ $

The smallest interior angle corresponds to the smallest part of the ratio, which is $1x$. Therefore, the smallest angle is:

$ 1x = 1 \times 20^\circ = 20^\circ $