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Trigonometry

Area of Triangle

Introduction

In getting the area of a triangle we use the formula \(A = \dfrac{1}{2}bh\) where \(b\) is the base and \(h\) is the height. This would be applicable for a right triangle having one side as its base and another side as its height. But for other types of triangle given at least 2 sides and an angle, we could use the formula: \(\mathit{Area} = \dfrac{1}{2}bc\sin A = \dfrac{1}{2}ab\sin C = \dfrac{1}{2}ac\sin B\)

Figure 1.0 — Triangle ABC with vertex A at the top and vertices C (left) and B (right) on the base. Side a runs from C to B along the base, side b from C up to A, and side c from A down to B. A perpendicular of length x is drawn from vertex A to the base, meeting side a at a right angle (marked by a small square) near B; x is the height of the triangle on base a.

In Figure 1.0, triangle ABC with sides \(a\), \(b\) and \(c\) and angles \(A\), \(B\) and \(C\), base would be \(a\) and height is \(x\). To get the value of \(x\), we use \(x = b(\sin C)\) since \(\triangle abx\) is a right triangle and \(b\) is the hypotenuse. Therefore, \(\mathit{Area} = \dfrac{1}{2}a(b \sin C)\).

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