How do I find the length of a side of a triangle using the cosine rule?

Suppose we have a triangle with 2 sides of length 4 and 5 (in any units), and the angle between those sides is 60 degrees. We want to find the length of the side opposite the 60 degree angle.

Write the cosine rule, a2=b2+c2-2bcCos(A). Here a, b and c are the side lengths, and A is the angle opposite side a.

Given our example we have a2=42+52-2*4*5*Cos(60)

so a2=16+25-20 = 21, a = sqrt(21) which is irrational (approximately 4.58).

We can also use the cosine rule if we know two side lengths and an angle opposite one of those sides. Suppose we have a triangle with side of length 2 and 5, and we know that the angle opposite the side of length 5 is 120 degrees. We want to work out the 3rd side length.

Again we write out a2=b2+c2-2bcCos(A), but this time we want a to be the side of length 5, since this side is opposite the known angle, giving us




Solving with the quadratic formula we get

c = (-5 +/- sqrt(25+84)) / 2 which is again irrational.

Notice this time we get two different values for c, since in this case there are two possible triangles that meet the criteria described in the question.

Angus H. GCSE Maths tutor, A Level Maths tutor, GCSE Physics tutor, A...

2 years ago

Answered by Angus, a GCSE Maths tutor with MyTutor

Still stuck? Get one-to-one help from a personally interviewed subject specialist


£18 /hr

Rhianna C.

Degree: Mathematics (Bachelors) - Bristol University

Subjects offered:Maths, Chemistry+ 1 more


“I am currently studying Maths at University and absolutely love it! I can't wait to share my enthusiasm with you!”

£18 /hr

Suraiyya J.

Degree: Medicine (Bachelors) - Nottingham University

Subjects offered:Maths, Science+ 5 more

-Personal Statements-
-Medical School Preparation-

“Friendly Patient Good listener Chemistry and Biology Medical student Creative Flexible Hard working Clear and concise Confident Reliable”

MyTutor guarantee

£22 /hr

Ed W.

Degree: Medicine (Bachelors) - Edinburgh University

Subjects offered:Maths, Further Mathematics + 3 more

Further Mathematics
-Medical School Preparation-

“Hi! I'm Ed and I'm currently studying Medicine at the University of Edinburgh. I tutor at GCSE and A Level forBiology, Chemistry and Maths, and at GCSE level forFurther Maths. These are subjects I'm very comfortable in and enjoy. With ...”

About the author

Angus H.

Currently unavailable: for new students

Degree: Computer Science (Bachelors) - Cambridge University

Subjects offered:Maths, Physics+ 2 more

Further Mathematics

“Top tutor from the renowned Russell university group, ready to help you improve your grades.”

MyTutor guarantee

|  14 completed tutorials

You may also like...

Other GCSE Maths questions

Make n the subject of the formula: m = 5n - 21

What is Pythagorus' Theorem?

Solve these simultaneous equations. 5x + 2y = 20 and x + 4y = 13.

Factorise and solve x^2 + 3x - 4 = 0

View GCSE Maths tutors

We use cookies to improve your site experience. By continuing to use this website, we'll assume that you're OK with this. Dismiss