Formulas for Area of a Triangle.

There are a lot of formulas and techniques to find the area of a triangle. We can use many different formulas to calculate area of a triangle according to the given conditions. Here we shall derive some of the main formulas used to calculate area of a triangle.

Formulas for Area of a Triangle:

The area of a triangle is denoted by the symbol delta ( $\Delta$ )

We shall appeal to the formula:

$\Delta = \frac{1}{2} bc \sin A = \frac{1}{2} ac \sin B = \frac{1}{2} ab \sin C$

And the half angle formula:

$\sin \frac{1}{2} A = \sqrt{\dfrac{(s-b)(s-c)}{bc}} \, \, \, , \, \, \, \cos \frac{1}{2} A = \sqrt{\dfrac{s(s-a)}{bc}}$ etc.

Where “s” is the semi circumference of the triangle or, $s = \frac{a+b+c}{2}$

We shall now derive different formula for the area of triangle:

First formula for the area of a triangle:

$\Delta = \frac{1}{2} bc \sin A = \frac{1}{2} bc . 2 \sin \frac{A}{2} \cos \frac{A}{2} \\ \\ \\ or , \Delta = bc . \sqrt{\dfrac{s(s-a)(s-b)(s-c)}{bc . bc}} \\ \\ \\ or , \Delta = \sqrt{s(s-a)(s-b)(s-c)}$

Second formula for the area of a triangle:

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

Now , as 2s = (a+b+c)

$\Delta = \frac{1}{4} \sqrt{(a+b-c)(b+c-a)(c+a-b)(a+b-c)} \\ \\ So , \Delta = \frac{1}{4} \sqrt{2b^2 c^2 + 2c^2 a^2 + 2a^2 b^2 - a^4 - b^4 - c^4}$

Third formula for the area of a triangle:

$\Delta = \frac{1}{2} bc \sin A$

Now using sine law:

$\Delta = \frac{1}{2} bc \frac{a}{2R} \\ \\ So , \Delta = \dfrac{abc}{4R}$

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