Law of cosines calculator
Sides of 7 and 10 with 60° between them give a third side of 8.8882 in the law of cosines calculator, since c² = 49 + 100 - 70 = 79.
Third side
- Third side, by the law of cosines c² = a² + b² - 2ab cos C = 49 + 100 - 2 × 7 × 10 × cos 60° = 79, so c = 8.8882.
- 8.8882
- Sides
- a = 7, b = 10, c = 8.8882
- Angles Summing to 180 degrees, as every plane triangle must.
- A = 43deg, B = 77deg, C = 60deg
- Area
- 30.3109
- Perimeter
- 25.8882
- Type
- scalene, acute
- Heights The perpendicular from each side to the opposite vertex. Area is half the base times the height for any of the three.
- to a: 8.6603, to b: 6.0622, to c: 6.8205
- Circumradius The radius of the circle through all three vertices.
- 5.1316
- Inradius The radius of the largest circle that fits inside.
- 2.3417
| a | b | c | Multiples |
|---|---|---|---|
| 3 | 4 | 5 | 6-8-10, 9-12-15, 30-40-50 |
| 5 | 12 | 13 | 10-24-26, 15-36-39 |
| 8 | 15 | 17 | 16-30-34 |
| 7 | 24 | 25 | 14-48-50 |
| 20 | 21 | 29 | 40-42-58 |
| 9 | 40 | 41 | 18-80-82 |
- The Pythagorean theorem holds only for right triangles. For any other triangle the law of cosines generalizes it: c squared equals a squared plus b squared minus 2ab cos(C), which reduces to Pythagoras when C is 90 degrees and cos(C) is zero.
- Any two sides must add to more than the third, or the shape does not close. That triangle inequality is checked here before anything else is computed.
How the Triangle Calculator works
The law of cosines relates all three sides of any triangle to one of its angles: c² = a² + b² - 2ab cos C, where C is the angle between sides a and b, opposite side c. It solves the two cases the law of sines cannot start from: two sides and the angle between them, and three sides.
For a = 7, b = 10 and C = 60°: cos 60° = 0.5, so c² = 49 + 100 - 2 × 7 × 10 × 0.5 = 149 - 70 = 79, and c = √79 = 8.8882.
Rearranged, it gives an angle from three sides: cos A = (b² + c² - a²) ÷ 2bc. Here cos A = (100 + 79 - 49) ÷ (2 × 10 × 8.8882) = 130 ÷ 177.76 = 0.7313, so A = 43.0°, and B = 180° - 60° - 43.0° = 77.0°.
When C is 90°, cos C is 0 and the last term disappears, which leaves the Pythagorean theorem. An angle over 90° has a negative cosine, so the third side comes out longer than Pythagoras would give; an angle under 90° makes it shorter.
The area follows from the same two sides and angle: ½ab sin C = ½ × 7 × 10 × sin 60° = 30.31.
Triangle Calculator FAQ
When do I use the law of cosines instead of the law of sines?
When you know two sides and the angle between them, or all three sides. The law of sines needs an angle and the side opposite it, which neither case gives you.
What is the law of cosines for angle A?
a² = b² + c² - 2bc cos A, or, solved for the angle, A = arccos((b² + c² - a²) ÷ 2bc). The formula is the same for each angle with the letters rotated.
Does it work when the angle is obtuse?
Yes. With sides 7 and 10 and 120° between them, cos 120° = -0.5, so c² = 149 + 70 = 219 and c = 14.80.
More from the Triangle Calculator
- Right triangle calculator
- Law of sines calculator
- Equilateral triangle calculator
- Isosceles triangle solver
- Pythagorean theorem calculator
- Triangle area calculator