Question

What is the formula for the area of a non-right angled triangle?

Answer 1

Hint: We can find area of non-right angled triangle in two ways:
1. General formula: $$\mathrm{\Delta }Area={\displaystyle \frac{1}{2}}b\times h$$, where b is base length of triangle and h is height of triangle.
2. Heron’s formula: $$\mathrm{\Delta }Area=\sqrt{s\left(s-a\right)\left(s-b\right)\left(s-c\right)}$$ , where $$s={\displaystyle \frac{1}{2}}\left(a+b+c\right)$$ and a,b and c are the sides of triangle. We can use any one.

Complete step by step solution:
Using general formula:
The area of ΔABC can be expressed as:
$$\mathrm{\Delta }Area={\displaystyle \frac{1}{2}}c\times h$$
Where a represents the side (base) and h represents the height drawn to that side.

Using trigonometry, In right angled triangle CDA, we can state that
$$\sin C={\displaystyle \frac{h}{b}}$$ (and multiplying by b gives) $$b\sin C=h$$
The height, h of the triangle can be expressed as $$b\sin C$$
Substituting this new expression for the height, h, into the general formula for the area of a triangle gives:
$$\mathrm{\Delta }ABC={\displaystyle \frac{1}{2}}ab\sin C$$
Where a and b be two sides and C is the included angle
With this new formula, we no longer have to rely on finding the altitude (height) of a triangle in order to find its area. Now, if we know two sides and the included angle of a triangle, we can find the area of the triangle. This is a valuable new formula!
Example:
Find the area triangle whose sides are $$12$$ and $$18$$ and the angle between them is $${55}^{\circ }$$.