To determine the area of a triangle given the lengths of its three sides, you can use Heron's formula. Here's the process:

Step 1: Calculate the semi-perimeter

The semi-perimeter $s$ is half of the perimeter of the triangle.

$s = \frac{a + b + c}{2}$

For the given sides $a = 9$, $b = 14$, and $c = 11$:

$s = \frac{9 + 14 + 11}{2} = \frac{34}{2} = 17$

Step 2: Apply Heron's formula

Heron's formula for the area $A$ of the triangle is:

$A = \sqrt{s(s - a)(s - b)(s - c)}$

Substitute the values:

$A = \sqrt{17(17 - 9)(17 - 14)(17 - 11)}$ $A = \sqrt{17(8)(3)(6)}$ $A = \sqrt{17 \times 144}$ $A = \sqrt{2448}$

Step 3: Simplify the square root

Now, let's simplify $\sqrt{2448}$:

$A \approx 49.48 \, \text{square units}$

So, the area of the triangle is approximately 49.48 square units.

Would you like more details on any part of this solution, or do you have any other questions?

Here are five related questions you might find interesting:

1. How do you derive Heron's formula?
2. What are the other methods to find the area of a triangle?
3. How does the formula change if one angle of the triangle is known?
4. What is the importance of knowing the semi-perimeter in triangle calculations?
5. How can you apply Heron's formula to different types of triangles (isosceles, scalene, etc.)?

Tip: Always double-check your calculations, especially when dealing with square roots and areas, to avoid any small errors in large calculations.