Sine and cosine rule calculator (triangle solver)
Enter any valid combination of three measurements — three sides, two sides and the angle between them, or two angles and a side — and this solver completes the triangle. It tells you which rule applies at each step, gives the missing sides and angles to one decimal place, and works out the area using ½ab sin C.
Solve a triangle — enter any three values
Side a is opposite angle A, side b opposite B, side c opposite C. Leave the unknowns blank.
a = 8, b = 11, c = 6.666
A = 46.2°, B = 96.8°, C = 37°
Area: 26.48 · Perimeter: 25.666
Working
- Cosine rule: missing side² = 8² + 11² − 2 × 8 × 11 × cos(37°)
- Remaining angles from the cosine rule rearranged for cos of each angle.
- Area = ½ × 8 × 11 × sin(37°)
The method behind it
- Sine rule: a / sin A = b / sin B = c / sin C.
- Cosine rule: a² = b² + c² − 2bc cos A, and cos A = (b² + c² − a²) / 2bc.
- Area = ½ab sin C, using the angle between the two known sides.
How to use this tool
1. Label your triangle
Side a is opposite angle A, side b is opposite angle B and side c is opposite angle C. Getting the labelling right is the step most marks are lost on.
2. Fill in what you know
Type the three values you have and leave the rest blank. Angles are in degrees.
3. Check the rule used
The solver names the rule for each step, so you can copy the correct method into your working rather than just the answer.
Frequently asked questions
When do you use the sine rule and when the cosine rule?
Use the sine rule (a/sin A = b/sin B) when you have a matching side and opposite angle pair. Use the cosine rule (a² = b² + c² − 2bc cos A) when you have three sides, or two sides and the angle between them, and no matching pair.
What is the cosine rule formula?
a² = b² + c² − 2bc cos A for a side, rearranged to cos A = (b² + c² − a²) / 2bc when you are finding an angle from three sides.
How do you find the area of a non-right-angled triangle?
Use Area = ½ab sin C, where C is the angle between the two sides a and b. This calculator applies it automatically once the triangle is solved.
What is the ambiguous case of the sine rule?
When you know two sides and an angle that is not between them, the sine rule can give two possible triangles — an acute angle and its obtuse partner (180° − θ). Check the diagram or the question wording to decide which is valid; this solver flags when a second triangle exists.
Practise this properly, not just once
A solver gives you the answer; exam marks come from the working. MathsGradeUp pairs exam-board questions with step-by-step feedback and revision notes on the exact topics you keep dropping marks on.