Acute Triangle Calculator — Area, Angles & Properties
Enter the three side lengths of a triangle to check whether it is acute (all angles under 90°) and compute its area, all interior angles, perimeter, inradius, and circumradius using the law of cosines and Heron's formula.
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All angles are acute (< 90°) — this is a valid acute triangle
- 1
Semi-perimeter
s = (5 + 6 + 7) ÷ 2 = 9 - 2
Product under the radical
s(s−a)(s−b)(s−c) = 9 × 4 × 3 × 2 = 216 - 3
Area (Heron's formula)
√216 = 14.6969All angles are acute, confirming this is a valid acute triangle.
How does this calculator work?
A triangle with sides a, b, c is acute if a²+b²>c², b²+c²>a², a²+c²>b² all hold. Area = √(s(s−a)(s−b)(s−c)) with s=(a+b+c)/2 (Heron). Angles from the law of cosines. Inradius r = Area/s; circumradius R = abc/(4·Area).
Formula
How this is calculated
A triangle is acute when all three interior angles are strictly less than 90°. In terms of side lengths, this translates to three simultaneous conditions: a² < b² + c², b² < a² + c², and c² < a² + b². If any one fails the triangle is obtuse (or right). The calculator checks all three and reports the type.
Once the sides are validated as a legal triangle (triangle inequality: each side must be less than the sum of the other two), the law of cosines finds every angle: for example angle A = arccos((b² + c² − a²) / (2bc)). The third angle is 180° − A − B to avoid any rounding drift. Heron's formula then gives the area: with semi-perimeter s = (a+b+c)/2, Area = √(s(s−a)(s−b)(s−c)) — no height required.
The inradius r = Area/s gives the radius of the largest circle that fits inside the triangle (incircle). The circumradius R = abc/(4·Area) gives the radius of the circumscribed circle that passes through all three vertices. Both are common in geometry and competition problems. All results are in the same linear unit as the input sides (area in unit²).
Frequently asked questions
Square all three sides. For each side, check that its square is less than the sum of the squares of the other two. If all three checks pass, every angle is below 90° and the triangle is acute. If one check fails, the angle opposite the longest side is ≥ 90° and the triangle is right or obtuse.
Yes — an equilateral triangle (all sides equal) has all angles exactly 60°, making it the most "acute" triangle possible. Any equilateral triangle is automatically acute.
The inradius r is the radius of the incircle — the largest circle that fits inside the triangle, tangent to all three sides. The circumradius R is the radius of the circumscribed circle that passes through all three vertices. For an equilateral triangle R = 2r, and for a very flat triangle R becomes very large while r shrinks toward zero.
Also known as
TG we-Calculate Editorial Team. (2026). Acute Triangle Calculator — Area, Angles & Properties [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/acute-triangle-calculator
TG we-Calculate Editorial Team. "Acute Triangle Calculator — Area, Angles & Properties." TG we-Calculate. 2026. https://we-calculate.com/calculator/acute-triangle-calculator.
TG we-Calculate Editorial Team, "Acute Triangle Calculator — Area, Angles & Properties," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/acute-triangle-calculator
@misc{wecalculate_acute_triangle_calculator, title = {Acute Triangle Calculator — Area, Angles & Properties}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/acute-triangle-calculator}}, year = {2026}, note = {TG we-Calculate} }
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