What's the area of a triangle with sides 18 cm, 24 cm, and 30 cm?

Steps to Solve

1. Calculate the semi-perimeter

To apply Heron's formula, we first calculate the semi-perimeter $s$ of the triangle using the formula:

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

where $a$, $b$, and $c$ are the lengths of the sides of the triangle. For our triangle:

$$ s = \frac{18 , \text{cm} + 24 , \text{cm} + 30 , \text{cm}}{2} $$

2. Plug in the values to find semi-perimeter

Calculating this gives:

$$ s = \frac{72 , \text{cm}}{2} = 36 , \text{cm} $$

3. Apply Heron's formula to find the area

Now we apply Heron's formula to find the area $A$ of the triangle. The formula is:

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

Substituting the values:

$$ A = \sqrt{36 , \text{cm} \cdot (36 - 18)(36 - 24)(36 - 30)} $$

4. Calculate each term inside the square root

First, we calculate the values:

• $s - a = 36 - 18 = 18 , \text{cm}$
• $s - b = 36 - 24 = 12 , \text{cm}$
• $s - c = 36 - 30 = 6 , \text{cm}$

5. Final area calculation

Putting these into the equation gives:

$$ A = \sqrt{36 , \text{cm} \cdot 18 , \text{cm} \cdot 12 , \text{cm} \cdot 6 , \text{cm}} $$

Calculating the product:

$$ A = \sqrt{36 \cdot 18 \cdot 12 \cdot 6} $$

Now, we find:

$$ A = \sqrt{46656} \approx 216 , \text{cm}^2 $$