To find the area of a triangle when you have the lengths of all three sides (11 cm, 8 cm, and 9 cm), you can use Heron's formula instead of relying on base and height, because, from the given information, we don't yet know which sides represent the base and height.
Step 1: Calculate the semi-perimeter (s)
First, calculate the semi-perimeter \(s\) of the triangle:
\[ s = \frac{a + b + c}{2} = \frac{11 + 8 + 9}{2} = \frac{28}{2} = 14 \text{ cm} \]
Step 2: Apply Heron's formula
Now, use Heron's formula to find the area \(A\):
\[ A = \sqrt{s(s-a)(s-b)(s-c)} \]
Where \(a\), \(b\), and \(c\) are the lengths of the sides of the triangle.
Plugging in the values:
- \(a = 11\)
- \(b = 8\)
- \(c = 9\)
We get:
\[ A = \sqrt{14(14 - 11)(14 - 8)(14 - 9)} \]
Calculating each part:
\[ A = \sqrt{14 \times 3 \times 6 \times 5} \]
Calculating further:
\[ 3 \times 6 = 18 \] \[ 14 \times 18 = 252 \] \[ 252 \times 5 = 1260 \]
Now take the square root:
\[ A = \sqrt{1260} \]
Step 3: Simplifying \(\sqrt{1260}\)
You can simplify \(1260\):
\[ 1260 = 36 \times 35 = 6^2 \times 35 \] \[ A = 6 \times \sqrt{35} \]
Calculating \(\sqrt{35} \approx 5.916\):
\[ A \approx 6 \times 5.916 \approx 35.496 \text{ cm²} \]
Conclusion:
The area of the triangle is approximately:
\[ A \approx 35.5 \text{ cm}^2 \]
You can round if needed based on context to either:
\[ A \approx 35.5 \text{ cm}^2 \quad \text{or} \quad A \approx 36 \text{ cm}^2 \text{ (if rounding)} \]