Curriculum Hub → Maths Lessons → Geometry → Trigonometry → Solving Problems with Non-Right-Angled Triangles
Once a triangle loses its right angle, SOH CAH TOA runs out of road. That’s where the Sine Rule, Cosine Rule and Area Rule take over.
This is how I teach the topic at GCSE: what each rule is for, how to prove them from first principles, and how to help students decide which one to reach for.
GCSE papers rarely test one rule in isolation any more. Students need to see how the Sine Rule, Cosine Rule and Area Rule connect to Pythagoras’ Theorem, bearings and algebraic lengths, not just how to apply each one on its own.
Download the free PDF worksheet or work through the video tutorials below with your class.
These three formulae work on any triangle. If a right angle is present, SOH CAH TOA is still the quickest method. For everything else, reach for one of the rules below.
a ÷ sin(A) = b ÷ sin(B) = c ÷ sin(C)
Use it when you know a matching pair: an angle and the side directly opposite it, plus one more side or angle.
a² = b² + c² − 2bc × cos(A)
Rearrange it to find an angle instead: cos(A) = (b² + c² − a²) ÷ 2bc. Use the length version when two sides and the angle between them are known. Use the angle version when all three sides are known.
Area = ½ × a × b × sin(C)
Use it when two sides and the angle between them are known. It also works backwards: give students the area and ask for a missing side or angle.
Want more depth on the Sine Rule alone, including the ambiguous case (SSA) and the misconceptions students bring with them? I’ve written a dedicated Sine Rule lesson, worksheet and video tutorial guide.
Every lesson in this sequence starts with a derivation, not a statement of the formula. That’s how it connects to what students already know.
Here are three ways I get students deriving the Sine, Cosine and Area rules for themselves.
Students struggle most when a problem needs more than one formula, usually because they haven’t stopped to plan. I ask mine to spend a minute sketching a flowchart first, breaking the problem into smaller steps before they pick up a calculator.
If a question has no diagram, get one on paper straight away. Without a sketch, students find it almost impossible to see what’s actually being asked.
That decision-making is the problem solving. A student who can only apply a rule once you’ve told them which one to use hasn’t really learned to solve these triangles. They’ve only learned to substitute into a formula.
I begin the sequence with the Sine Rule.
The lesson starts with a jumbled-up derivation using right-angled trigonometry. It connects the new rule straight back to what students already know.
In the main teaching phase, we work through a series of problems involving missing angles and lengths.
The plenary is tougher. Students need angle facts (angles in a triangle, angles on a straight line, bearings) just to find a matching angle and side before they can even use the rule.
After the Sine Rule, we progress to deriving and using the Cosine Rule to calculate unknown lengths. The lesson starts with another jumbled-up proof for students to complete.
When teaching the derivation, most students spot that a² = b² + c² − 2bc cos(A) reduces to Pythagoras’ Theorem when A = 90°, since cos(90°) = 0. Making this connection explicit is one of the clearest ways to show that right-angled trigonometry is a special case, not a separate topic.
Next, students practise substituting known values into the formula. As they progress, questions bring in bearings, algebraic lengths, and diagrams that need both the Sine and Cosine Rules together.
In this lesson, students find an unknown angle when all three lengths are known, rearranging the Cosine Rule to cos(A) = (b² + c² − a²) ÷ 2bc. We start with another jumbled-up proof, then move quickly on to problems given only as written descriptions.
Mini-whiteboards are brilliant here. Sketching the diagram first helps students see what’s actually known, especially in bearings problems where the diagram decides which angle you’re really being asked for.
This is the final lesson in the sequence. Students bring together both the Sine and Cosine Rules from the previous two lessons to solve area problems.
They start by finding the area itself. Then, as they progress, they work backwards from a given area to find a missing angle or length, combining the Area Rule with the Sine Rule or Cosine Rule in the same question.
Exam questions often include the area of a triangle on the non-calculator paper, since sin(30°) and similar values can be worked out exactly without a calculator.
That’s really the point of the whole sequence. Each lesson derives from right-angled trigonometry, applies the new rule, then combines it with the one before.
No rule is ever taught in isolation. Every lesson connects back to Pythagoras’ Theorem, bearings, algebraic lengths, or whatever came before it. That’s what teaches problem solving, not a separate “problem solving lesson” bolted on at the end.
Download the free GCSE worksheet on non-right-angled trigonometry. It brings the Sine Rule, Cosine Rule and Area Rule together in one set of exam-style questions, with space for full working.
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Members get all four lessons: editable slides, jumbled-proof starters, differentiated worksheets and answers, plus hundreds more KS3, GCSE and A-Level resources.
Try these 4 GCSE-style questions, then reveal the solutions to check your working. Similar questions are in the free PDF worksheet.
These two questions are designed to challenge the most able GCSE students. Both are worked through in full as video tutorials.
This one asks students to derive a geometrical proof using the Area Rule. It’s a step beyond applying the formula: they need to reason about why the rule holds, not just use it.
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Yes. The Sine Rule (a ÷ sin A = b ÷ sin B = c ÷ sin C) works on any triangle, including right-angled ones, but it is specifically needed for non-right-angled triangles where SOH CAH TOA cannot be used directly. Use it when you know a matching pair of an angle and its opposite side.
Yes, the Cosine Rule works on any triangle. Use a² = b² + c² − 2bc cos(A) to find a missing length when two sides and the included angle are known, or rearrange it to cos(A) = (b² + c² − a²) ÷ 2bc to find a missing angle when all three sides are known.
The Area Rule states that Area = ½ × a × b × sin(C), where a and b are two sides of the triangle and C is the angle between them. This works for any triangle, not just right-angled ones, and is derived from the standard ½ × base × height formula using right-angled trigonometry.
Sketch the triangle and label what you know. If you have a matching angle and opposite side, use the Sine Rule. If you know two sides and the angle between them, or all three sides, use the Cosine Rule. If you need the area, or are given the area to find a length or angle, use the Area Rule. Many GCSE problems require more than one rule in sequence.
Yes. The Sine Rule, Cosine Rule and Area Rule are all trigonometric formulae designed specifically for triangles without a right angle. Standard SOH CAH TOA only works when a right angle is present.
Use the Sine Rule when you have a matching angle-and-side pair. Use the Cosine Rule to find a length when two sides and the included angle are known, or to find an angle when all three sides are known. Use the Area Rule when the area is given or asked for.
Download the free PDF worksheet from this page, or work through the four GCSE exam-style questions with worked solutions above. Two additional Grade 9 extension questions with video solutions are also included for the most able students.
Build a coherent non-right-angled trigonometry sequence with these related resources. For the full curriculum, visit the Trigonometry hub.
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