Acute Triangle Calculator

Enter the three side lengths of a triangle to check whether it's acute, right, or obtuse, and get its angles, area, and perimeter using the Law of Cosines and Heron's formula.

Quick Facts

Classification rule
Compare the square of the longest side to the sum of squares of the other two
c² < a² + b² is acute, c² = a² + b² is right, c² > a² + b² is obtuse (c = longest side).

Your Results

Calculated
Classification
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Acute, right, or obtuse
Largest angle
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Opposite the longest side
Area
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Heron's formula
Perimeter
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Sum of all three sides

Ready

Enter three side lengths and press Calculate.

Formula and method for classifying a triangle

Every triangle is acute, right, or obtuse, and the category is determined entirely by its three side lengths — no protractor required. This calculator uses the Law of Cosines relationship between the sides to classify the triangle, then reports its angles, area, and perimeter.

How the calculation works

Label the sides so that c is the longest. The Law of Cosines states c² = a² + b² − 2ab·cos(C), where C is the angle opposite c. Since a, b, and c are all positive, the sign of cos(C) — and therefore whether C is acute, right, or obtuse — depends only on comparing c² to a² + b²:

  • c² < a² + b²: angle C is less than 90°, and since C is the largest angle in the triangle, all three angles are under 90° — the triangle is acute.
  • c² = a² + b²: angle C is exactly 90° — the triangle is right (this is the Pythagorean theorem).
  • c² > a² + b²: angle C is greater than 90° — the triangle is obtuse.

Before classifying, the calculator checks the triangle inequality: the sum of any two sides must exceed the third (a + b > c, a + c > b, and b + c > a). If it fails, the three lengths cannot form a closed triangle at all. All three angles are computed exactly with the Law of Cosines (A = arccos((b² + c² − a²) / 2bc), and similarly for B and C, with the third angle found from 180° − A − B), the area comes from Heron's formula — area = √(s(s−a)(s−b)(s−c)) where s = (a + b + c) / 2 is the semi-perimeter — and the perimeter is simply a + b + c.

Common mistakes

  • Checking the wrong side: the c² vs. a² + b² test only works when c is the longest side. Comparing a shorter side gives a meaningless result.
  • Ignoring the triangle inequality: not every trio of positive numbers forms a triangle — 2, 3, and 10 do not, since 2 + 3 is less than 10.
  • Mixed units: enter all three sides in the same unit (all inches, all meters, etc.); the calculator has no way to convert between them.
  • Near-right triangles: when c² is very close to a² + b², small measurement errors can flip the classification between acute, right, and obtuse.

Real-world applications

  • Carpentry and construction use the 3-4-5 right-triangle rule to square corners, and the same math confirms whether a frame is exactly square or slightly off (acute/obtuse)
  • Trigonometry and geometry coursework rely on this classification as a first step before applying angle-specific formulas
  • Land surveying and CAD drafting use side-length checks to validate triangular plots and components
  • Structural engineering uses triangle geometry (trusses, bracing) where angle behavior under load depends on whether a triangle is acute, right, or obtuse

Frequently Asked Questions

How do you tell if a triangle is acute?
Find the longest side (call it c) and the other two sides (a and b). If c squared is less than a squared plus b squared, every angle is under 90 degrees and the triangle is acute. If c squared equals a squared plus b squared, it's a right triangle. If c squared is greater, it's obtuse. This works because the angle opposite the longest side is always the largest angle in the triangle.
What three side lengths always make a valid triangle?
Any three positive lengths where the sum of the two shorter sides is strictly greater than the longest side. This is the triangle inequality. Sides like 2, 3, and 10 fail it (2 + 3 = 5, which is less than 10) and cannot form a triangle at all — the two short sides can't reach each other.
How is the area calculated from three side lengths?
This calculator uses Heron's formula. First compute the semi-perimeter s = (a + b + c) / 2, then the area equals the square root of s(s−a)(s−b)(s−c). It works for any valid triangle without needing to know an angle or the height directly.
Can an equilateral or isosceles triangle be acute?
Every equilateral triangle is acute, since all three angles equal exactly 60 degrees. An isosceles triangle can be acute, right, or obtuse depending on the size of its apex angle — for example, a tall thin isosceles triangle with a small apex angle is acute, while a short, wide one is obtuse.

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