Trigonometric Area of Triangles and Polygons

Find the area of a triangle with sides of length 6 cm and 9 cm and an included angle of 30 degrees.

13.5 sq cm. (Area = 0.5 * 6 * 9 * sin(30°) = 27 * 0.5 = 13.5)

A triangle has side lengths of 7 m and 12 m with an included angle of 60 degrees. What is its exact area in radical form, and its rounded value?

21√3 sq m, or approximately 36.4 sq m. (Area = 0.5 * 7 * 12 * (√3 / 2) = 21√3)

A triangle has an area of 28 square feet. Two of its sides measure 8 feet and 14 feet. Find the measure of the acute included angle.

30 degrees. (sin(C) = (2 * 28) / (8 * 14) = 56 / 112 = 0.5; arcsin(0.5) = 30°)

A parallelogram has adjacent sides of length 10 cm and 15 cm with an interior angle of 45 degrees. Calculate its area.

Approximately 106.1 sq cm. (Area = 10 * 15 * sin(45°) = 150 * 0.7071 ≈ 106.1)

A triangle has sides of 5 cm and 8 cm with an area of 18 square centimeters. Find the two possible values for the included angle rounded to the nearest tenth of a degree.

64.2 degrees and 115.8 degrees. (sin(C) = 36 / 40 = 0.9; arcsin(0.9) ≈ 64.2°, and supplementary angle is 180° - 64.2° = 115.8°)

Find the area of a regular hexagon with a side length of 6 cm by dividing it into six equilateral triangles.

Approximately 93.5 sq cm. (Each triangle has sides 6 cm, 6 cm and angle 60°: Area = 6 * [0.5 * 6 * 6 * sin(60°)] = 6 * [18 * √3 / 2] = 54√3 ≈ 93.5)