To find the area of a triangle with three known sides, we can use Heron's formula. Heron's formula is given by:

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

where $s$ is the semi-perimeter of the triangle, calculated as:

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

Step 1: Calculate the semi-perimeter $s$

Given:

• $a = 9$
• $b = 13$
• $c = 13$

$s = \frac{9 + 13 + 13}{2} = \frac{35}{2} = 17.5$

Step 2: Apply Heron's formula

Substitute the values of $s$, $a$, $b$, and $c$ into the formula:

$A = \sqrt{17.5(17.5 - 9)(17.5 - 13)(17.5 - 13)}$

$A = \sqrt{17.5 \times 8.5 \times 4.5 \times 4.5}$

Now, calculate this expression.

$A = \sqrt{17.5 \times 8.5 \times 4.5^2}$ $A \approx \sqrt{17.5 \times 8.5 \times 20.25}$ $A \approx \sqrt{3021.1875} \approx 54.96$

Final Answer:

The area of the triangle is approximately 54.96 square units.

Would you like more details or have any questions about this process?

Here are some related questions:

1. How do you calculate the semi-perimeter for a triangle with different side lengths?
2. What is the significance of Heron's formula in geometry?
3. How do you find the area of an equilateral triangle using Heron's formula?
4. Can Heron's formula be used for non-triangular shapes?
5. How does Heron's formula differ from the basic formula for area using base and height?

Tip: Heron's formula is a powerful tool for triangles when the height is not given explicitly.