If area of a regular pentagon is `125sqrt(3)` sq cm, how long is its each side?

Options

A. 10 cm
B. 15 cm
C. 16 cm
D. 25 cm

Answer

To find the length of each side of a regular pentagon given its area, we can follow these steps:

### Step 1: Write down the formula for the area of a regular pentagon.
The formula for the area \( A \) of a regular pentagon with side length \( a \) is given by:
\[ A = \frac{5a^2 \sqrt{3}}{4} \]

### Step 2: Set the formula equal to the given area.
We know the area of the pentagon is \( 125\sqrt{3} \) sq cm. Therefore, we can set up the equation:
\[ \frac{5a^2 \sqrt{3}}{4} = 125\sqrt{3} \]

### Step 3: Eliminate \( \sqrt{3} \) from both sides.
Since \( \sqrt{3} \) is present on both sides of the equation, we can cancel it out:
\[ \frac{5a^2}{4} = 125 \]

### Step 4: Multiply both sides by 4 to eliminate the fraction.
To isolate \( a^2 \), multiply both sides by 4:
\[ 5a^2 = 500 \]

### Step 5: Divide both sides by 5.
Now, divide both sides by 5 to solve for \( a^2 \):
\[ a^2 = 100 \]

### Step 6: Take the square root of both sides to find \( a \).
Taking the square root of both sides gives us:
\[ a = \sqrt{100} = 10 \text{ cm} \]

### Conclusion:
Each side of the regular pentagon is \( 10 \) cm.

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