law of cosines

Area of Triangles

• Oblique Triangles: Non-right triangles with varying side lengths.

  • Area can be calculated using two sides and the included angle (SAS).

• Area Formulas:

  • For two sides a, b, and included angle C:
    $Area = \frac{1}{2} a b \sin(C)$
  • Variants for sides and angles:
  • $Area = \frac{1}{2} b c \sin(A)$
  • $Area = \frac{1}{2} a c \sin(B)$

• Example: Area of triangle ABC where angle B = 92°, a = 12, c = 10:

  • $Area = \frac{1}{2} \times 12 \times 10 \times \sin(92°) = 59.962 \, ft^2$

• Regular Hexagon Area: Inscribed in circle of radius 5 m.

  • Hexagon has 6 triangles; each triangle's area via triangle formula:
  • $Area = \frac{1}{2} \times 5 \times 5 \times \sin(60°)$
  • Area of one triangle = 10.825 m²; hexagon area = $6 \times 10.825 = 64.95 \, m^2$

• Heron's Formula: For triangles with all three sides a, b, c:

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

• Example Using Heron's Formula: Triangle with a = 19, b = 19√2, c = 38:

  • Calculate s:
    $s = \frac{19 + 19\sqrt{2} + 38}{2} = 41.935$
  • Compute area:
    $Area = \sqrt{41.935 \cdot (41.935 - 19) \cdot (41.935 - 19\sqrt{2}) \cdot (41.935 - 38)}$
  • Resulting area ≈ 238.778 in².

• Final Example: Area of a sail with foot = 9 ft, luff = 7 ft, leech = 12.885 ft:

  • Semi-perimeter:
    $s = \frac{9 + 7 + 12.885}{2} = 14.443$
  • Area:
    $Area = \sqrt{14.443 \cdot (14.443 - 9) \cdot (14.443 - 7) \cdot (14.443 - 12.885)}$
  • Resulting area ≈ 30.193 ft².