Area of a Triangle

Overview

The area of a triangle can be determined using several methods, depending on the information given. This note focuses on the key approaches specified in the syllabus, including the base-height formula and the application of trigonometric methods.

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Base-Height Formula

Formula:

$$\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$$

Key Idea:

• Select any side of the triangle as the base.
• The corresponding height is the perpendicular distance from the opposite vertex to the base.

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Using Trigonometry (Sine Rule)

Formula:

$$\text{Area} = \frac{1}{2} \times a \times b \times \sin(C)$$

• $a$ and $b$: Two sides of the triangle.
• $C$: The angle between the two sides.

Key Idea:

• This formula is particularly useful when two sides and the included angle are known.

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Worked Examples

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Summary

• Base-Height Formula: $\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$
• Trigonometric Formula: $\text{Area} = \frac{1}{2} \times a \times b \times \sin(C)$
• Use the base-height method when the height is given or easily determined.
• Use the trigonometric formula for triangles where two sides and the included angle are known. Understanding and applying these formulas ensures accurate calculations for different types of triangles.