The problem asks to find the area of a regular pentagon, given that the apothem (the distance from the center to the midpoint of a side) is 10 mm. The answer should be rounded to the nearest tenth.

1. Problem Description

The problem asks to find the area of a regular pentagon, given that the apothem (the distance from the center to the midpoint of a side) is 10 mm. The answer should be rounded to the nearest tenth.

2. Solution Steps

Let $a$ be the apothem, $n$ be the number of sides, and $s$ be the length of a side. The area $A$ of a regular polygon is given by the formula:

$A = \frac{1}{2}ans$

We are given that $a = 10$ mm and $n = 5$. We need to find $s$.

Consider a right triangle formed by the apothem, half of a side, and a radius of the circumscribed circle. The angle at the center of the polygon for each of the $n$ sides is $\frac{2\pi}{n}$. The angle in the right triangle opposite half of the side is half of this angle, or $\frac{\pi}{n}$.

In our case, this angle is $\frac{\pi}{5}$. We have:

$\tan(\frac{\pi}{n}) = \frac{s/2}{a}$

$s = 2a \tan(\frac{\pi}{n})$

$s = 2a \tan(\frac{\pi}{5})$

$s = 2(10) \tan(\frac{\pi}{5})$

$s = 20 \tan(\frac{\pi}{5})$

$s \approx 20(0.7265) \approx 14.53$

Now we can find the area:

$A = \frac{1}{2}ans = \frac{1}{2}(10)(5)(20\tan(\frac{\pi}{5}))$

$A = 500 \tan(\frac{\pi}{5})$

$A = 500(0.7265) = 363.25$

Rounded to the nearest tenth, $A \approx 363.3$ mm$^2$.

Alternatively, the formula for the area of a regular polygon is given by:

$A = \frac{1}{4} n s^2 \cot(\frac{\pi}{n})$

We have $a = 10$, $n = 5$, and $s = 2 a \tan(\frac{\pi}{n})$. Substituting $s$:

$A = \frac{1}{4} n (2 a \tan(\frac{\pi}{n}))^2 \cot(\frac{\pi}{n}) = n a^2 \tan(\frac{\pi}{n})$

$A = 5 \cdot (10)^2 \tan(\frac{\pi}{5}) = 500 \tan(\frac{\pi}{5})$

$A = 500 \tan(\frac{\pi}{5}) \approx 500(0.7265) = 363.25$

Rounded to the nearest tenth, $A \approx 363.3$ mm$^2$.

3. Final Answer

363.3