Triangle Area Calculation Methods

Mastering various techniques for calculating the area of a triangle is crucial for effectively tackling geometry problems. This guide covers multiple methods, from fundamental formulae to trigonometric and coordinate geometry applications.

Introduction to Basic Area Formula

To determine the area of a triangle using the base and height, use the formula:

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

• Key Points:

  • Any side can be the base: You can choose any side of the triangle as the base, depending on the given context.
  • Height: Always refers to the line perpendicular to the base extending to the opposite vertex.

• Common Mistakes:

  • Misidentifying the line that functions as the perpendicular height.

Introduction to Trigonometric Formula

The trigonometric formula for calculating the area is:

$\text{Area} = \frac{1}{2} ab \sin(C)$

• Components:
  • Sides (a and b): The known lengths of two sides of the triangle.
  • Angle (C): Essential for determining the sine component.

Worked Examples

• Example 1: Given $a = 5$ cm, $b = 7$ cm, $C = 30^{\circ}$.

  • Calculate: $\frac{1}{2} \times 5 \times 7 \times 0.5 = 8.75$ cm²
  • Result: Area = 8.75 cm²

• Example 2: With $a = 8$ cm, $b = 10$ cm, $C = 120^{\circ}$.

  • Calculate: $\frac{1}{2} \times 8 \times 10 \times 0.866 \approx 34.64$ cm²
  • Result: Area ≈ 34.64 cm²

Heron's Formula

Heron's Formula serves as an alternative to find the area without utilising angles:

$A = \sqrt{s(s-a)(s-b)(s-c)}$

• Semi-Perimeter (s): $s = \frac{(a+b+c)}{2}$

Example: For sides $a = 5$, $b = 6$, $c = 7$:

• Step 1: Calculate $s$: $s = \frac{5+6+7}{2} = 9$.
• Step 2: Solve for $(s-a) = 4$, $(s-b) = 3$, $(s-c) = 2$.
• Step 3: Substitute into the formula: $A = \sqrt{9 \times 4 \times 3 \times 2} = \sqrt{216}$
• Result: $A = \sqrt{216} \approx 14.7$ square units.

Coordinate Geometry Application

When vertices are defined, apply:

$\text{Area} = \frac{1}{2} \left| x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2) \right|$

Example:

• With vertices $(2, 3), (5, 7), (8, 4)$
• Substitute values: $\frac{1}{2} |2(7-4) + 5(4-3) + 8(3-7)|$
• Simplify: $\frac{1}{2} |2(3) + 5(1) + 8(-4)| = \frac{1}{2} |6 + 5 - 32| = \frac{1}{2} |21|$
• Area = 10.5 square units

Exam Strategies and Tips

Selecting the Appropriate Formula

• Use formulae based on the given data: base-height, two sides & included angle, all three sides, or vertex coordinates.

Practice Exam Question

• Sides: 7 cm, 8 cm, angle: 60°
• Solution: Using the trigonometric formula: $\text{Area} = \frac{1}{2} \times 7 \times 8 \times \sin(60°)$ $\text{Area} = \frac{1}{2} \times 7 \times 8 \times 0.866$ $\text{Area} \approx 24.25$ cm²

Calculator Tips: Regularly check the mode setting to ensure accuracy.