To calculate the area of the triangle, we can use Heron's formula since we know the lengths of all three sides of the triangle. The sides of the triangle are given as:

• $a = 7 \, \text{cm}$
• $b = 8 \, \text{cm}$
• $c = 10 \, \text{cm}$

Step 1: Calculate the semi-perimeter ($s$):

The semi-perimeter is half of the triangle's perimeter, calculated as:

$s = \frac{a + b + c}{2} = \frac{7 + 8 + 10}{2} = 12.5 \, \text{cm}$

Step 2: Apply Heron's formula:

Heron's formula for the area $A$ of a triangle is:

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

Substitute the values of $s$, $a$, $b$, and $c$:

$A = \sqrt{12.5(12.5 - 7)(12.5 - 8)(12.5 - 10)}$ $A = \sqrt{12.5(5.5)(4.5)(2.5)}$

I'll now calculate the exact area.The area of the triangle is approximately 27.81 cm².

Would you like more details or have any further questions? Here are five related questions you may find interesting:

1. How does Heron's formula apply to different types of triangles?
2. What are other methods for calculating the area of a triangle besides Heron's formula?
3. Can we always use Heron's formula, or are there cases where it's not applicable?
4. How do we calculate the area of a right triangle using simpler formulas?
5. What is the relationship between the semi-perimeter and the perimeter of any polygon?

Tip: Always check if a triangle is right-angled. If so, using simpler formulas like $\frac{1}{2} \times \text{base} \times \text{height}$ can save time!