Calculating Triangle Area with Sides 8, 9, 13 cm

This problem asks us to find the area of a triangle given the lengths of its three sides: 8 cm, 9 cm, and 13 cm. We can solve this using Heron's formula, which is a useful method when all three side lengths are known.

Understanding Heron's Formula

Heron's formula allows us to calculate the area of a triangle when we know the lengths of all three sides. Let the sides of the triangle be $a$, $b$, and $c$.

First, we need to calculate the semi-perimeter ($s$) of the triangle, which is half of the perimeter. The formula for the semi-perimeter is:

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

Once we have the semi-perimeter, we can use Heron's formula for the area (A):

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

Step-by-Step Calculation

Let's apply Heron's formula to the given triangle with sides $a=8$ cm, $b=9$ cm, and $c=13$ cm.

Step 1: Calculate the Semi-Perimeter (s)

Using the formula $s = \frac{a+b+c}{2}$:

$s = \frac{8 \text{ cm} + 9 \text{ cm} + 13 \text{ cm}}{2}$

$s = \frac{30 \text{ cm}}{2}$

$s = 15 \text{ cm}$

Step 2: Apply Heron's Formula

Now, substitute the value of $s$ and the side lengths into Heron's formula:

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

Area $= \sqrt{15(15-8)(15-9)(15-13)}$

Area $= \sqrt{15(7)(6)(2)}$

Step 3: Calculate the Area

Let's multiply the numbers inside the square root:

Area $= \sqrt{15 \times 7 \times 6 \times 2}$

Area $= \sqrt{105 \times 12}$

Area $= \sqrt{1260}$

Step 4: Find the Square Root and Round

Now, we calculate the square root of 1260 and round it to two decimal places as requested:

Area $\approx 35.50 \text{ cm}^2$

Final Answer

The area of the triangle with sides 8 cm, 9 cm, and 13 cm is approximately 35.50 cm².