Area of non right angled triangles using the sine rule
The sine rule for area is used to calculate the area of non right angled triangles. This is useful when the height of the triangle is unknown. The formula is derived from the basic triangle area formula you used in junior mathematics, that is,
Area = 1/2 x b x h and the Sine ratio - remember the triangle?
Area = 1/2 x b x h and the Sine ratio - remember the triangle?
However, the triangle below has a base that is labelled 'c' and vertical height 'h'. Hence
Area = 1/2 x c x h.
It is common practice to label the triangle in the manner below when exploring Trig formula. A vertical line (h) has been drawn extending from the apex to the base of triangle ABC creating two smaller triangles. The symbol θ (theta) is used to represent an unknown angle.
Area = 1/2 x c x h.
It is common practice to label the triangle in the manner below when exploring Trig formula. A vertical line (h) has been drawn extending from the apex to the base of triangle ABC creating two smaller triangles. The symbol θ (theta) is used to represent an unknown angle.
OK, from the Sine Ratio, the sine of the angle is equal to the opposite side (h) divided by the hypotenuse (b). Rearrange this so that h is the subject. h=bsinθ. Now substitute this in place of the 'h' in line 2 below.
We swap the c and the b around to put them in alphabetical order in the formula. This pleases the founding fathers. For the formula to work you need an angle in the centre of two known sides. So in the example below, we notice the angle θ of 74 degrees is at the vertex (centre) of two sides, 60m and 44m. We substitute these values into the formula to work out the area. Follow the steps below using your calculator. Make sure your calculator is in degrees, not radians or gradians!
And in the calculator enter in the numbers, rounding to 2 decimal places. Don't forget the unit of measurement.
NOTE: You do not have to derive or memorize the area formula. It is given in the assessment - you just need to know how to use it.
Your turn:
A triangle has the following side lengths which converge at an angle of...
1) 15m and 18m at 80 degrees.
2) 7.6cm and 7.6cm at 43 degrees.
3) 3m and 7m at 35 degrees.
4) 62m and 59m at 49 degrees.
A triangle has the following side lengths which converge at an angle of...
1) 15m and 18m at 80 degrees.
2) 7.6cm and 7.6cm at 43 degrees.
3) 3m and 7m at 35 degrees.
4) 62m and 59m at 49 degrees.
ANSWERS:
1) 132.95m² 2) 19.70cm² 3) 6.02m² 4) 1380.36m²
1) 132.95m² 2) 19.70cm² 3) 6.02m² 4) 1380.36m²