Find the area of a triangle whose sides are 13 cm, 14 cm and 15 cm, using Heron's formula.

Step 1: Calculate the semi-perimeter

Heron's formula requires the semi-perimeter of the triangle, denoted by $s$. The semi-perimeter is half the sum of the lengths of the three sides of the triangle. Here, the side lengths are $a=13 \text{ cm}$, $b=14 \text{ cm}$, and $c=15 \text{ cm}$.

Step 2: Substitute values and compute semi-perimeter

Substitute the given side lengths into the formula for the semi-perimeter and perform the addition and division to find its value. The sum of the sides is $13 + 14 + 15 = 42 \text{ cm}$, and half of this sum is $21 \text{ cm}$.

Step 3: Apply Heron's formula

Heron's formula provides a way to calculate the area of a triangle when only the lengths of its three sides are known. The formula involves the semi-perimeter $s$ and the lengths of the sides $a$, $b$, and $c$.

Step 4: Substitute values into Heron's formula

Substitute the calculated semi-perimeter $s=21 \text{ cm}$ and the given side lengths $a=13 \text{ cm}$, $b=14 \text{ cm}$, $c=15 \text{ cm}$ into Heron's formula. This sets up the expression for the area calculation.

Step 5: Simplify the expression

Perform the subtractions inside the parentheses: $21-13=8$, $21-14=7$, and $21-15=6$. This simplifies the expression under the square root.

Step 6: Calculate the product and square root

Multiply the numbers under the square root: $21 \times 8 \times 7 \times 6 = 7056$. Then, calculate the square root of $7056$, which is $84$. The area is expressed in square centimeters.

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