Acute Triangle Calculator
The Acute Triangle Calculator uses the law of cosines to find the angle opposite side A from the three side lengths, then tells you whether that angle is acute. An acute triangle has all three angles below 90°. Enter the three sides to check angle A.
Formula
Angle A = arccos((b²+c²−a²)/2bc); repeat for B and C; acute if all cosines > 0
- Each angle is found with the law of cosines: cos A = (b² + c² − a²) / (2bc).
- A triangle is acute when ALL three angles are less than 90°, which is equivalent to all cosines being positive.
- Check the angle opposite the longest side first: if that angle is less than 90°, all angles are.
- If any angle equals 90° the triangle is right-angled; if any exceeds 90° it is obtuse.
Example Calculation
Inputs
- Side A: 5
- Side B: 6
- Side C: 7
With sides 5, 6, and 7, angle A = arccos((6² + 7² − 5²) / (2×6×7)) = arccos(60/84) ≈ 44.4°. Since it is below 90°, this corner is acute.
Frequently asked questions
What is an acute triangle?
An acute triangle is one where all three interior angles are less than 90°. If any angle equals 90° it is right-angled, and if any exceeds 90° it is obtuse.
How is angle A found from the sides?
By the law of cosines: cos A = (b² + c² − a²) / (2bc), where a is the side opposite angle A. Taking the inverse cosine gives the angle.
How do I confirm the whole triangle is acute?
Check the angle opposite the longest side. If that largest angle is below 90°, every angle is, so the triangle is acute.
What if the sides cannot form a triangle?
The three sides must satisfy the triangle inequality (each side less than the sum of the other two). Otherwise no valid triangle exists.
Does the unit matter?
No — angles depend only on the ratios of the sides, so any consistent unit (cm, m, in) gives the same angle.