Area of Triangles: Trigonometric Methods and Applications

9.4 Area of Triangles

Objective 1: Determining the Area of Oblique Triangles

The area of a triangle can be determined using several formulas, especially when the triangle is not a right triangle (oblique triangle). These formulas utilize trigonometric relationships and are essential in solving problems involving non-right triangles.

• Standard Area Formula: The area of any triangle is given by: $\text{Area} = \frac{1}{2}bh$ where b is the length of the base and h is the height (altitude) drawn to that base.

• Area Using Sine: If the measures of two sides and the included angle are known, the area can be found using: $\text{Area} = \frac{1}{2}ab\sin C$ $\text{Area} = \frac{1}{2}bc\sin A$ $\text{Area} = \frac{1}{2}ac\sin B$ where a, b, and c are the lengths of the sides, and A, B, and C are the measures of the corresponding angles.

• Application: These formulas are particularly useful for finding the area of oblique triangles, where the height is not easily measured.

• Example: Given triangle ABC with sides a = 7, b = 10, and included angle C = 45°, the area is: $\text{Area} = \frac{1}{2} \times 7 \times 10 \times \sin 45^\circ$

Objective 2: Using Heron's Formula to Determine the Area of a SSS Triangle

Heron's Formula allows the calculation of the area of a triangle when all three sides are known (SSS case). This is especially useful when no height or angle is given.

• Heron's Formula: For a triangle with sides a, b, and c:

  • Semi-perimeter: $s = \frac{1}{2}(a + b + c)$

  • Area: $\text{Area} = \sqrt{s(s-a)(s-b)(s-c)}$

• Application: Heron's Formula is widely used in geometry and trigonometry for triangles where only the side lengths are known.

• Example: For a triangle with sides a = 5, b = 6, c = 7: $s = \frac{1}{2}(5 + 6 + 7) = 9$ $\text{Area} = \sqrt{9(9-5)(9-6)(9-7)} = \sqrt{9 \times 4 \times 3 \times 2}$

Objective 3: Solving Applied Problems Involving the Area of Triangles

Trigonometric area formulas are essential for solving real-world problems involving triangles, such as land surveying, architecture, and physics.

• Key Steps:

  1. Identify the known sides and angles.

  2. Select the appropriate area formula (standard, sine, or Heron's).

  3. Substitute the known values and solve for the area.

• Example Application: Calculating the area of a plot of land shaped as a triangle when only the side lengths are known.

Formula	When to Use	Required Information
$\frac{1}{2}bh$	Any triangle with known base and height	Base and height
$\frac{1}{2}ab\sin C$	Oblique triangle (SAS)	Two sides and included angle
$\sqrt{s(s-a)(s-b)(s-c)}$	SSS triangle	All three sides

Additional info: These formulas are foundational in trigonometry and are directly related to Chapter 9: Applications of Trigonometry, specifically in the context of solving for areas of triangles using trigonometric relationships.