34: A Trig Formula for the Area of a Triangle “Teach A Level Maths” Vol. 1: AS Core Modules 34: A Trig Formula for the Area of a Triangle © Christine Crisp

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3 Trig Ratios: A reminder In a right angled triangle, the 3 trig ratios for an angle x are defined as follows: x hypotenuse opposite

3 Trig Ratios: A reminder In a right angled triangle, the 3 trig ratios for an angle x are defined as follows: x hypotenuse adjacent

3 Trig Ratios: A reminder In a right angled triangle, the 3 trig ratios for an angle x are defined as follows: x opposite adjacent

3 Trig Ratios: A reminder Using the trig ratios we can find unknown angles and sides of a right angled triangle, provided that, as well as the right angle, we know the following: either 1 side and 1 angle or 2 sides

3 Trig Ratios: A reminder 7 y e.g. 1 Tip: Always start with the trig ratio, whether or not you know the angle. e.g. 2 10 8 (3 s.f.)

Scalene Triangles We will now find a formula for the area of a triangle that is not right angled, using 2 sides and 1 angle.

Area of a Triangle ABC is a non-right angled triangle. a, b and c are the sides opposite angles A, B and C respectively. ( This is a conventional way of labelling a triangle ). A B C b a c

C b a c A B h N Area of a Triangle ABC is a non-right angled triangle. Draw the perpendicular, h, from C to BA. C b a c A B h N

C b a c A B h N Area of a Triangle ABC is a non-right angled triangle. Draw the perpendicular, h, from C to BA. C b a c A B - - - - - (1) In h N

C b a c A B h N Area of a Triangle ABC is a non-right angled triangle. Draw the perpendicular, h, from C to BA. C b a c A B - - - - - (1) In h N

h b a c C N A B Area of a Triangle ABC is a non-right angled triangle. Draw the perpendicular, h, from C to BA. h b a c C N A B - - - - - (1) In

h b a c C N B A Area of a Triangle ABC is a non-right angled triangle. Draw the perpendicular, h, from C to BA. h b a c C N B - - - - - (1) In Substituting for h in (1) A

C b a a B A c Area of a Triangle ABC is a non-right angled triangle. Draw the perpendicular, h, from C to BA. C - - - - - (1) b a a In Substituting for h in (1) A B c

Area of a Triangle Any side can be used as the base, so Area = = = The formula always uses 2 sides and the angle formed by those sides

C b a A B c Area of a Triangle Any side can be used as the base, so The formula always uses 2 sides and the angle formed by those sides

C b a A B c Area of a Triangle Any side can be used as the base, so The formula always uses 2 sides and the angle formed by those sides

C b a A B c Area of a Triangle Any side can be used as the base, so The formula always uses 2 sides and the angle formed by those sides

R P Q Example 1. Find the area of the triangle PQR. 7 cm 8 cm R Q P Solution: We must use the angle formed by the 2 sides with the given lengths.

R P Q Example 1. Find the area of the triangle PQR. 7 cm 8 cm R Q P Solution: We must use the angle formed by the 2 sides with the given lengths. We know PQ and RQ so use angle Q

R P Q Example 1. Find the area of the triangle PQR. 7 cm 8 cm R Q P Solution: We must use the angle formed by the 2 sides with the given lengths. We know PQ and RQ so use angle Q cm2 (3 s.f.)

A r C r B Area of a Triangle A useful application of this formula occurs when we have a triangle formed by 2 radii and a chord of a circle. C A r r B

SUMMARY The area of triangle ABC is given by or The area of a triangle formed by 2 radii of length r of a circle and the chord joining them is given by where is the angle between the radii.

Exercises 1. Find the areas of the triangles shown in the diagrams. Y 12 cm 9 cm B A C (b) (a) O X radius = 4 cm., Ans: (a) cm2 (3 s.f.) (b) cm2 (3 s.f.)

The following slides contain repeats of information on earlier slides, shown without colour, so that they can be printed and photocopied. For most purposes the slides can be printed as “Handouts” with up to 6 slides per sheet.

C b a a A B c Area of a Triangle Any side can be used as the base, so The formula always uses 2 sides and the angle formed by those sides b a a A B c

R P Q e.g. Find the area of the triangle PQR. 7 cm 8 cm R Q P Solution: We must use the angle formed by the 2 sides with the given lengths. We know PQ and RQ so use angle Q cm2 (3 s.f.)

SUMMARY The area of triangle ABC is given by or or The area of a triangle formed by 2 radii of length r of a circle and the chord joining them is given by where is the angle between the radii.