Find the ratio of the area of the semicircle formed on the diagonal of a square of side 14 cm to that of the semicircle forned on the side of the square.

Answer

To find the ratio of the area of the semicircle formed on the diagonal of a square of side 14 cm to that of the semicircle formed on the side of the square, we can follow these steps:

### Step 1: Calculate the area of the semicircle on the side of the square.
- The side of the square is given as 14 cm.
- The diameter of the semicircle on the side is equal to the side of the square, which is 14 cm.
- The radius \( r \) of the semicircle is half of the diameter:
\[ r = \frac{14}{2} = 7 \text{ cm} \]
- The area \( A \) of a semicircle is given by the formula:
\[ A = \frac{1}{2} \pi r^2 \]
- Substituting the radius:
\[ A = \frac{1}{2} \pi (7^2) = \frac{1}{2} \pi (49) = \frac{49\pi}{2} \text{ cm}^2 \]

### Step 2: Calculate the length of the diagonal of the square.
- The diagonal \( D \) of the square can be calculated using the Pythagorean theorem:
\[ D = \sqrt{(14^2 + 14^2)} = \sqrt{2 \times 14^2} = 14\sqrt{2} \text{ cm} \]

### Step 3: Calculate the area of the semicircle on the diagonal.
- The diameter of the semicircle on the diagonal is equal to the length of the diagonal, which is \( 14\sqrt{2} \) cm.
- The radius \( r \) of this semicircle is:
\[ r = \frac{14\sqrt{2}}{2} = 7\sqrt{2} \text{ cm} \]
- The area \( A \) of the semicircle on the diagonal is:
\[ A = \frac{1}{2} \pi r^2 = \frac{1}{2} \pi (7\sqrt{2})^2 = \frac{1}{2} \pi (49 \times 2) = \frac{98\pi}{2} = 49\pi \text{ cm}^2 \]

### Step 4: Find the ratio of the areas of the two semicircles.
- The area of the semicircle on the diagonal is \( 49\pi \) cm² and the area of the semicircle on the side is \( \frac{49\pi}{2} \) cm².
- The ratio of the area of the semicircle on the diagonal to the area of the semicircle on the side is:
\[ \text{Ratio} = \frac{49\pi}{\frac{49\pi}{2}} = \frac{49\pi \times 2}{49\pi} = 2:1 \]

### Final Answer:
The ratio of the area of the semicircle formed on the diagonal of the square to that of the semicircle formed on the side of the square is \( 2:1 \).

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