1. Trigonometry Sine Area Rule By Mr Porter A B C a b c

2. Definition: The area of any triangle, ABC, can be found by using the formula: A B C a b c The formula is cyclic:

3. Example 1: Calculate the area of triangle ABC, correct to 3 sig. fig. Write the the formula for this triangle. Substitute the values for a, b and angle C. 62 m A B C 55 m 42°15’ c b a Label the sides of the triangle a, b and c.. Evaluate and round to 3 sig. fig. Example 2: Calculate the area of triangle ABC, correct to 1 dec. pl.. P Q R 12.5 cm 13.8 cm 108°40’ r p q Label the side of the triangle p, q and r. Write the the formula for this triangle. Substitute the values for p, r and angle Q. Evaluate and round to 1 dec. pl.

4. Example 3: Calculate the area of quadrilateral, correct to 3 sig. fig. 20 m 15 m 35 m 55°30’ Note: To find the area of the quadrilateral, you need to calculate the area of the 2 triangles. Label Quadrilateral (Triangles) A B C D Area of triangle ABD, A ABD = 1 / 2 x B x H A ABD = ½ x 20 x 15 A ABD = 150 m 2 Area of ∆BDC, sine area formula Side c is missing! (Hypotenuse of ∆ABD)  Pythagoras’ Thm. c 2 = a 2 + b 2 c 2 = 20 2 + 15 2 c 2 = 625 c =√625 c = 25 Side BD is 25 m. Area of ∆BDC, sine area formula Hence, the total area = 150 + 360.555 = 510.555 = 511 m 2

5. Example 4: Calculate the area of regular hexagon of radius 12 m, correct to 3 sig. fig. Draw a diagram with radial diagonals. 12 m Angle at the centre for a regular polygon is θ = 360÷ number of sides. θ = 60° 12 m 60° Area is 6 times the area of 1 triangle. Area of ∆ABC, sine area formula Substitute the values for a, b and angle C. Example 5: Calculate the area of regular pentagon of radius 15 cm, correct to 3 sig. fig. Draw a diagram with all radii from the centre drawn. Angle at the centre for a regular polygon is θ = 360÷ number of sides. 15 cm θ = 72° 15 cm 72° Area is 5 times the area of 1 triangle. Area of ∆ABC, sine area formula Substitute the values for a, b and angle C.

6. Example 6: The results of a radial survey are shown in the diagram (all measurements in metres). Calculate the total area of ∆XYZ (nearest m 2 ). X (045°T, 68m) Y (155°T, 102m) Z (275°T, 92 m) O North (0°T) Need to calculate the area of all 3 triangles. Must calculate the angles around ‘O’. From their bearing: Angle XOY = 155 – 45 = 110° Angle YOZ = 275 – 155 = 120° Now, Angle XOZ = (360 – 275) + 45 = 130° OR, Angle XOZ = 360 – (110 + 120) = 130° Now apply the sine area rule 3 times. Area of ∆XOZ: Likewise, areas for ∆XOY and ∆YOZ Area of ∆XOY= 3258.854 m 2 Area of ∆YOZ= 4063.391 m 2 Hence, total area = 2396.187 + 3258.854 + 4063.391 = 9718.432 m 2 = 9718 m 2