Area of triangle with 3 sides formula

General
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area of triangle with 3 sides formula

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Area of a Triangle with 3 Sides Formula

To find the area of a triangle when you know the lengths of all three sides, you can use Heron’s Formula. This formula is particularly useful because it does not require knowledge of the height of the triangle, only the side lengths.

Heron’s Formula

Heron’s Formula states that the area of a triangle whose sides have lengths (a), (b), and (c) is given by:

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

where (s) is the semi-perimeter of the triangle, calculated as:

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

Step-by-Step Calculation

  1. Calculate the Semi-Perimeter (s):

    Given the side lengths (a), (b), and (c), first find the semi-perimeter (s):

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

  2. Apply Heron’s Formula:

    Plug the values of (s), (a), (b), and (c) into Heron’s Formula to find the area:

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

Example Calculation

Let’s go through an example to illustrate the use of Heron’s Formula.

Example:

Suppose you have a triangle with sides (a = 7), (b = 8), and (c = 9).

  1. Calculate the Semi-Perimeter (s):

    s = \frac{7 + 8 + 9}{2} = \frac{24}{2} = 12

  2. Apply Heron’s Formula:

    \text{Area} = \sqrt{12(12-7)(12-8)(12-9)} = \sqrt{12 \cdot 5 \cdot 4 \cdot 3}
    \text{Area} = \sqrt{12 \cdot 60} = \sqrt{720} \approx 26.83

Therefore, the area of the triangle is approximately (26.83) square units.

Conclusion

Heron’s Formula is a powerful tool for calculating the area of a triangle when only the side lengths are known. By following the steps outlined above, you can easily determine the area of any triangle given its three sides. This method is especially useful in various fields of science, engineering, and mathematics where direct measurement of the height may not be feasible.