Acute Triangle Calculator
Type three side lengths and the calculator checks whether they form an acute triangle (every angle strictly less than 90°), reports all three angles via the law of cosines, and gives the area from Heron's formula.
Triangle classification
Acute ✓
- Angle A (opposite side a)
- 44.42°
- Angle B (opposite side b)
- 57.12°
- Angle C (opposite side c)
- 78.46°
- Largest angle
- 78.46°
- Area (Heron's formula)
- 14.6969 square units
- Perimeter
- 18
A triangle is acute when every interior angle is strictly less than 90°. Equivalently, for sides a ≤ b ≤ c, it is acute iff a² + b² > c² (the law of cosines applied to the largest angle). Angles come from the law of cosines; area from Heron's formula.
How to use this calculator
Enter the three side lengths a, b, and c in any single unit (cm, m, in, ft — anything, as long as you use the same one for all three). The calculator first checks the triangle inequality, then computes each interior angle using the law of cosines and the area using Heron's formula. The headline result is the classification: Acute if every angle is strictly less than 90°, Right if the largest angle is exactly 90°, Obtuse if any angle exceeds 90°. The breakdown shows each angle separately so you can see by how much the triangle clears or fails the acute test.
How the calculation works
A triangle is acute when all three interior angles measure less than 90°. Equivalently — and much faster to check by hand — if you order the sides so that c is the longest, the triangle is acute iff a² + b² > c² (a sharpened version of the Pythagorean theorem). If a² + b² = c² the triangle is right; if a² + b² < c² it is obtuse. To get the actual angles, the calculator uses the law of cosines: cos(A) = (b² + c² − a²) / (2bc), and similarly for B and C. The remaining angle is found from A + B + C = 180°. Area uses Heron's formula: with the semi-perimeter s = (a + b + c)/2, area = √(s(s − a)(s − b)(s − c)).
Worked example
Take sides a = 5, b = 6, c = 7. Triangle inequality holds (5 + 6 > 7, 5 + 7 > 6, 6 + 7 > 5). Apply the law of cosines: cos(C) = (5² + 6² − 7²) / (2·5·6) = 12/60 = 0.2, so C ≈ 78.46°. Similarly cos(A) = (6² + 7² − 5²) / (2·6·7) = 60/84 ≈ 0.7143, so A ≈ 44.42°, and B = 180° − A − C ≈ 57.12°. Every angle is below 90°, so the triangle is acute. Heron's formula gives s = 9 and area = √(9·4·3·2) = √216 ≈ 14.697 square units.
Frequently asked questions
What is an acute triangle?
An acute triangle is a triangle in which all three interior angles measure strictly less than 90°. Because the three angles must add to exactly 180°, the angles of an acute triangle each lie between 0° and 90°, with no angle reaching the right angle. An equilateral triangle (60°, 60°, 60°) is the most symmetric example; a 50°/60°/70° triangle is a less symmetric one. If even one angle reaches or exceeds 90°, the triangle stops being acute — it becomes right (one 90° angle) or obtuse (one angle bigger than 90°).
How can I tell if a triangle is acute from just the side lengths?
Order the sides so that c is the longest. Then compute a² + b² and compare with c². If a² + b² > c², the triangle is acute. If a² + b² = c², it is right (Pythagoras). If a² + b² < c², it is obtuse. This is the sharpened Pythagorean rule: the largest angle of any triangle sits opposite the longest side, so checking the largest angle against 90° is the same as comparing a² + b² with c². You do not need a calculator for this check — only squaring and addition.
Is an equilateral triangle acute?
Yes. An equilateral triangle has all three sides equal, and by symmetry all three angles are equal too. Since they must sum to 180°, each angle is exactly 60°, which is strictly below 90°. So every equilateral triangle is acute. More than that, the equilateral triangle is the "most acute" triangle in the sense that 60° is as close as the largest angle of an acute triangle can be made to the smallest (because pushing any angle above 60° forces another below it, and the largest can rise all the way to almost 90° without ever reaching it).
Can an acute triangle be isoceles?
Yes — most isoceles triangles are acute. An isoceles triangle has two equal sides and therefore two equal "base" angles. As long as both base angles are less than 90°, the triangle is acute; and because the base angles must each be less than 90° for the angles to sum to 180° with a positive apex angle, an isoceles triangle is acute whenever the apex angle is greater than 0° and the base angles are above 45°. An isoceles triangle becomes right when the apex is 90° (a 90°/45°/45° triangle) and obtuse when the apex exceeds 90°.
What is the area formula for an acute triangle?
There is no special area formula for acute triangles — every triangle, acute or otherwise, has the same set of area formulas. The two most useful ones are Heron's formula, area = √(s(s−a)(s−b)(s−c)) with s = (a+b+c)/2, which takes the three sides directly; and the "half base times height" formula, area = ½·b·h, where h is the perpendicular height from the opposite vertex. For SAS data, area = ½·a·b·sin(C), where C is the included angle. This calculator uses Heron's formula because the inputs here are the three sides.
Why does the calculator reject some inputs?
Three positive numbers only form a triangle if each is strictly less than the sum of the other two — the triangle inequality. For example, sides 1, 2, and 5 cannot form a triangle, because a side of length 5 would need the other two ends to reach 5, but they total only 3. The calculator checks this before computing anything else; if it fails, you get the inequality message. The other reason for rejection is non-positive inputs: a side length must be a positive number.