To find the area of triangle $ABC$ with sides $AB = 11 \, \text{cm}$, $BC = 7 \, \text{cm}$, and $CA = 8 \, \text{cm}$, we can use Heron's formula.

Step 1: Calculate the semi-perimeter $s$

The semi-perimeter $s$ is given by: $s = \frac{AB + BC + CA}{2} = \frac{11 + 7 + 8}{2} = \frac{26}{2} = 13 \, \text{cm}$

Step 2: Use Heron's formula

The area $A$ of the triangle is calculated using the formula: $A = \sqrt{s(s - AB)(s - BC)(s - CA)}$ Substituting the values: $A = \sqrt{13(13 - 11)(13 - 7)(13 - 8)}$ $A = \sqrt{13 \times 2 \times 6 \times 5}$ $A = \sqrt{13 \times 60} = \sqrt{780}$

Step 3: Calculate the square root

$A \approx \sqrt{780} \approx 27.928$

Step 4: Round to 3 significant figures

$\text{Area} \approx 27.9 \, \text{cm}^2$

Final Answer

The area of triangle $ABC$ is $27.9 \, \text{cm}^2$ to 3 significant figures.

Would you like more details or have any questions?

Here are some related questions you might find useful:

1. How do you derive Heron's formula?
2. What are other methods to find the area of a triangle if angles are known?
3. Can you use trigonometric ratios to solve this triangle?
4. What is the significance of the semi-perimeter in Heron's formula?
5. How does Heron's formula apply to right-angled triangles?

Tip: When using Heron's formula, ensure you calculate the semi-perimeter accurately as it is crucial for determining the correct area.