To find the area of a triangle when you have the lengths of all three sides, you can use Heron's formula. First, let's summarize the given side lengths:
- \( a = 4 \) cm
- \( b = 3 \) cm
- \( c = 5 \) cm
Step 1: Calculate the semi-perimeter
The semi-perimeter \( s \) of the triangle is given by:
\[ s = \frac{a + b + c}{2} = \frac{4 + 3 + 5}{2} = \frac{12}{2} = 6 , \text{cm} \]
Step 2: Apply Heron's formula
Heron's formula for the area \( A \) of a triangle is:
\[ A = \sqrt{s(s-a)(s-b)(s-c)} \]
Substituting the values we have:
\[ A = \sqrt{6(6-4)(6-3)(6-5)} \]
Calculating each term inside the square root:
- \( s - a = 6 - 4 = 2 \)
- \( s - b = 6 - 3 = 3 \)
- \( s - c = 6 - 5 = 1 \)
Now substitute these values into the formula:
\[ A = \sqrt{6 \times 2 \times 3 \times 1} \]
Step 3: Calculate the area
Now calculate the product under the square root:
\[ 6 \times 2 = 12 \] \[ 12 \times 3 = 36 \] \[ 36 \times 1 = 36 \]
So, the area is:
\[ A = \sqrt{36} = 6 , \text{cm}^2 \]
Conclusion
The area of the triangle is:
\[ A = 6 , \text{cm}^2 \]