A small card is in the shape of an isosceles triangle with a semicircle on its base. If the diameter of the semicircle is 6 cm, and the hright of the triangle is 3 cm, find the area of the card.

Answer

To find the area of the card that is in the shape of an isosceles triangle with a semicircle on its base, we can follow these steps:

### Step 1: Understand the dimensions given
- The diameter of the semicircle is 6 cm.
- The height of the triangle is 3 cm.

### Step 2: Calculate the radius of the semicircle
The radius \( r \) of the semicircle is half of the diameter:
\[ r = \frac{\text{diameter}}{2} = \frac{6 \text{ cm}}{2} = 3 \text{ cm} \]

### Step 3: Calculate the area of the isosceles triangle
The area \( A_t \) of a triangle is given by the formula:
\[ A_t = \frac{1}{2} \times \text{base} \times \text{height} \]
Here, the base of the triangle is equal to the diameter of the semicircle, which is 6 cm, and the height is given as 3 cm. Thus, we can substitute these values into the formula:
\[ A_t = \frac{1}{2} \times 6 \text{ cm} \times 3 \text{ cm} = \frac{1}{2} \times 18 \text{ cm}^2 = 9 \text{ cm}^2 \]

### Step 4: Calculate the area of the semicircle
The area \( A_s \) of a semicircle is given by the formula:
\[ A_s = \frac{1}{2} \pi r^2 \]
Substituting the value of \( r \):
\[ A_s = \frac{1}{2} \times \pi \times (3 \text{ cm})^2 = \frac{1}{2} \times \pi \times 9 \text{ cm}^2 = \frac{9\pi}{2} \text{ cm}^2 \]
Using \( \pi \approx \frac{22}{7} \):
\[ A_s = \frac{9 \times \frac{22}{7}}{2} = \frac{198}{14} \text{ cm}^2 = \frac{99}{7} \text{ cm}^2 \approx 14.14 \text{ cm}^2 \]

### Step 5: Calculate the total area of the card
The total area \( A \) of the card is the sum of the area of the triangle and the area of the semicircle:
\[ A = A_t + A_s = 9 \text{ cm}^2 + \frac{99}{7} \text{ cm}^2 \]
To add these, convert \( 9 \text{ cm}^2 \) into a fraction with a denominator of 7:
\[ 9 \text{ cm}^2 = \frac{63}{7} \text{ cm}^2 \]
Now, add the two areas:
\[ A = \frac{63}{7} \text{ cm}^2 + \frac{99}{7} \text{ cm}^2 = \frac{162}{7} \text{ cm}^2 \]
Calculating this gives:
\[ A \approx 23.14 \text{ cm}^2 \]

### Final Answer
Therefore, the area of the card is approximately \( 23.14 \text{ cm}^2 \).

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