To find the area of a triangle with side lengths 6, 9, and 12 using Heron's formula, follow these steps:

Step 1: Calculate the Semi-perimeter

The semi-perimeter $s$ is given by: $s = \frac{a + b + c}{2}$ where $a = 6$, $b = 9$, and $c = 12$.

$s = \frac{6 + 9 + 12}{2} = \frac{27}{2} = 13.5$

Step 2: Apply Heron's Formula

Heron's formula states: $\text{Area} = \sqrt{s(s - a)(s - b)(s - c)}$

Substitute $s = 13.5$, $a = 6$, $b = 9$, and $c = 12$: $\text{Area} = \sqrt{13.5 \times (13.5 - 6) \times (13.5 - 9) \times (13.5 - 12)}$

Step 3: Simplify Inside the Square Root

Calculate each term inside the square root: $13.5 - 6 = 7.5$ $13.5 - 9 = 4.5$ $13.5 - 12 = 1.5$

Substitute these values back: $\text{Area} = \sqrt{13.5 \times 7.5 \times 4.5 \times 1.5}$

Step 4: Multiply and Take the Square Root

$\text{Area} = \sqrt{682.875} \approx 26.13$

Final Answer

The area of the triangle is approximately $26.13$ square units.

Would you like further details on Heron's formula or another example?

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Here are five related questions to extend your understanding:

1. How does Heron's formula work with different side lengths?
2. What are the limitations of Heron's formula?
3. Can you find the area of an isosceles triangle with specific side lengths?
4. How do you calculate the height of a triangle given only side lengths?
5. What other formulas can be used to find the area of a triangle?

Tip: Heron's formula is particularly useful when you know all three sides but not the height of the triangle.