To calculate the area of a triangle, the fundamental formula is given by:
Area = \(\frac{1}{2} \times \text{base} \times \text{height}\)
This formula applies to all types of triangles, including non-right triangles, as long as the height is known. However, in cases where the height is not provided, it can be determined using trigonometric functions, specifically the sine function.
For a triangle with a known angle and the lengths of two sides, the height can be derived using the sine of the angle. If we denote the sides as \(a\), \(b\), and \(c\) (where \(c\) is the hypotenuse), and the angle opposite side \(a\) as \(A\), the relationship can be expressed as:
\(\sin(A) = \frac{h}{c}\)
Rearranging this gives us the height:
\(h = c \times \sin(A)\)
Substituting this back into the area formula, we can express the area in terms of the sides and angles. For example, if \(b\) is the base and \(c\) is the hypotenuse, the area can be calculated as:
Area = \(\frac{1}{2} \times b \times (c \times \sin(A))\)
This can be simplified to:
Area = \(\frac{1}{2} \times b \times c \times \sin(A)\)
There are variations of this formula depending on which sides and angles are known. The three common forms are:
1. Area = \(\frac{1}{2} \times b \times c \times \sin(A)\)
2. Area = \(\frac{1}{2} \times a \times c \times \sin(B)\)
3. Area = \(\frac{1}{2} \times a \times b \times \sin(C)\)
In each case, the angle corresponds to the third vertex of the triangle that is not associated with the two sides being multiplied. This approach allows for the calculation of the area of non-right triangles using the relationships between their sides and angles.