Question

Find the area.
A = [?]cm²
11 cm, 8 cm and 9 cm
Shape: Triangle
Area of a triangle: A = Base x Height/2
Which one is the base and height? 11, 8 or 9?

Answer 1

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:

\[
A \approx 35.5 \text{ cm}^2 \quad \text{or} \quad A \approx 36 \text{ cm}^2 \text{ (if rounding)}
\]