1. Deriving the Sine Area Formula

From basic geometry, \(\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}\). When the perpendicular height \(h\) is not given, use trigonometric ratios:

\(h = a \sin B \quad \text{or} \quad h = b \sin A\)

Substituting \(h\) into the basic formula gives:

\(\text{Area} = \frac{1}{2}ab \sin C = \frac{1}{2}ac \sin B = \frac{1}{2}bc \sin A\)

Condition: The angle used must be the included angle between the two given side lengths.

2. Finding Unknown Values from the Area

Given the area and two sides, the included angle \(\theta\) can be found by rearranging:

\(\sin \theta = \dfrac{2 \times \text{Area}}{a \times b}\)

Given the area, one side, and an included angle, the other side length can be calculated algebraically.

3. Heron's Formula

When only the lengths of all three sides \((a, b, c)\) are given, Heron's formula is used without needing to calculate an angle or the perpendicular height.

Historical background: Named after Hero of Alexandria (Heron), a Greek mathematician who recorded it in his work Metrica.

Step 1: Calculate the semi-perimeter, \(s = \frac{a + b + c}{2}\).

Step 2: Substitute into Heron's formula: \(\text{Area} = \sqrt{s(s - a)(s - b)(s - c)}\).

4. Solving Problems Involving Areas of Triangles

Combine Heron's formula, the sine area formula, the Cosine Rule, and the Sine Rule to solve composite geometric shapes, land plots, and 3D solids.

For multi-part figures, find the total area of the larger triangle or composite region, then subtract unshaded or non-target regions to find the required area.