Find the area of a triangle whose three sides are 6 cm, 8 cm, and 11 cm and circumradius is 5 cm

Options

A. ` 10 5 . 6 cm^(2)`
B. ` 528 cm ^(2)`
C. ` 26.4 cm^(2)`
D. ` 23.41 cm^(2)`

Answer

To find the area of a triangle with sides of lengths 6 cm, 8 cm, and 11 cm, we can use Heron's formula. Here are the steps to solve the problem:

### Step 1: Identify the sides of the triangle
Let the sides of the triangle be:
- \( A = 6 \, \text{cm} \)
- \( B = 8 \, \text{cm} \)
- \( C = 11 \, \text{cm} \)

### Step 2: Calculate the semi-perimeter (S)
The semi-perimeter \( S \) is calculated using the formula:
\[ S = \frac{A + B + C}{2} \]
Substituting the values:
\[ S = \frac{6 + 8 + 11}{2} = \frac{25}{2} = 12.5 \, \text{cm} \]

### Step 3: Calculate \( S - A \), \( S - B \), and \( S - C \)
Now we calculate:
- \( S - A = 12.5 - 6 = 6.5 \, \text{cm} \)
- \( S - B = 12.5 - 8 = 4.5 \, \text{cm} \)
- \( S - C = 12.5 - 11 = 1.5 \, \text{cm} \)

### Step 4: Apply Heron's formula to find the area
Heron's formula for the area \( A \) of the triangle is:
\[ \text{Area} = \sqrt{S \times (S - A) \times (S - B) \times (S - C)} \]
Substituting the values we calculated:
\[ \text{Area} = \sqrt{12.5 \times 6.5 \times 4.5 \times 1.5} \]

### Step 5: Calculate the area
Now we compute the product inside the square root:
1. Calculate \( 12.5 \times 6.5 = 81.25 \)
2. Calculate \( 4.5 \times 1.5 = 6.75 \)
3. Now multiply these two results:
\[ 81.25 \times 6.75 = 548.4375 \]
4. Finally, take the square root:
\[ \text{Area} = \sqrt{548.4375} \approx 23.4 \, \text{cm}^2 \]

### Final Answer
The area of the triangle is approximately \( 23.4 \, \text{cm}^2 \).

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