Team
math

Acute Triangle Calculator

An acute triangle has all three interior angles measuring less than 90°. Whether you're verifying a triangle's classification from its angles or testing side lengths against the Pythagorean inequality, this calculator handles multiple input modes. Enter three angles, three sides, two sides with an included angle, or two angles with a side—and instantly discover whether your triangle is acute, right, or obtuse, plus compute missing dimensions and area.

Last updated:

Creators Anna Szczepanek, PhD Mathematics
1,900 people find this calculator helpful

Understanding Acute Triangles

Triangles fall into three categories based on their largest interior angle. An acute triangle has all three angles strictly less than 90°. A right triangle contains exactly one 90° angle. An obtuse triangle has one angle greater than 90°.

Acute triangles can be further classified by side length. An acute equilateral triangle has three equal sides and three 60° angles. An acute isosceles triangle has two equal sides and a third distinct side, with all angles remaining acute. An acute scalene triangle has three unequal sides and three different acute angles.

The defining property is straightforward: if even one angle reaches or exceeds 90°, the triangle ceases to be acute. This makes the acute classification mutually exclusive with both right and obtuse categories.

Testing Acuteness From Side Lengths

When you know only the three side lengths, apply the Pythagorean inequality test. Square the two shorter sides and add them. Compare this sum to the square of the longest side:

If a² + b² > c², the triangle is acute

If a² + b² = c², the triangle is right

If a² + b² < c², the triangle is obtuse

This method derives from the law of cosines. When the sum of squares of the shorter sides exceeds the square of the longest side, all angles must be acute.

  • a, b — The lengths of the two shorter sides
  • c — The length of the longest side (opposite the largest angle)

Computing Angles From Three Sides

Use the law of cosines to find each angle when all three sides are known:

cos(α) = (b² + c² − a²) ÷ (2bc)

cos(β) = (a² + c² − b²) ÷ (2ac)

cos(γ) = (a² + b² − c²) ÷ (2ab)

Area = √[s(s−a)(s−b)(s−c)], where s = (a+b+c)÷2

  • a, b, c — The three side lengths
  • α, β, γ — The interior angles opposite sides a, b, and c respectively
  • s — The semi-perimeter, equal to half the triangle's perimeter

Input Modes and Calculator Features

This tool supports four common input scenarios:

  • Three angles (AAA): Provide all three angle measures. The calculator verifies they sum to 180° and checks whether all are below 90° for acute classification.
  • Three sides (SSS): Enter side lengths. The calculator reconstructs all angles using the law of cosines, computes area via Heron's formula, and applies the Pythagorean inequality test.
  • Two sides and included angle (SAS): Supply two side lengths and the angle between them. The law of cosines recovers the third side and remaining angles.
  • Two angles and a side (ASA): Enter two angles and the side connecting them. The third angle is found from the 180° rule, and the law of sines provides the missing sides.

After classification, the output includes the triangle's perimeter, area, side length ratios, and explicit angle measures—allowing you to verify acuteness through multiple approaches.

Common Pitfalls and Verification Tips

Avoid these mistakes when classifying or calculating acute triangles.

  1. Don't forget the Pythagorean inequality reversal — Many confuse the sign direction. Remember: if a² + b² <strong>exceeds</strong> c², the triangle is acute. If the sum is <strong>less than</strong> c², it's obtuse. Equality means a right triangle.
  2. Verify angle sum equals 180° — When entering three angles, always confirm they sum exactly to 180° before submitting. A small rounding error in manual calculation can shift an acute angle very close to 90° into obtuse territory, misleading your classification.
  3. Identify the longest side correctly — The Pythagorean test requires comparing the two shorter sides against the longest. If you reverse this comparison, your classification flips. Sort the sides first: a ≤ b < c, then apply the test.
  4. Precision matters near boundaries — A triangle with angles of 89°, 85°, and 6° is barely acute. Measurement or rounding errors can misclassify near-boundary cases. When stakes are high, use exact values or recalculate with higher precision.

Frequently Asked Questions

How do you determine if a triangle is acute using only its three side lengths?

Square each of the two shorter sides and add them together. Then square the longest side. If the sum of the shorter sides' squares exceeds the longest side's square, your triangle is acute. This rule stems from the law of cosines: when a² + b² > c², the angle opposite the longest side must be less than 90°, which forces all three angles below 90°.

What is the relationship between acute triangles and the law of cosines?

The law of cosines expresses each angle in terms of the three sides: cos(angle) = (adjacent² + adjacent² − opposite²) ÷ (2 × adjacent × adjacent). When all sides satisfy a² + b² > c², the cosine value of the largest angle remains positive, meaning that angle is less than 90°. Since the largest angle is acute, the other two must be acute as well, confirming the triangle is acute.

Can an acute triangle have two angles that sum to exactly 90°?

Yes, an acute triangle can have two angles summing to 90°. For example, a triangle with angles 60°, 60°, and 60° (equilateral) has pairs summing to 120°, but a triangle with 50°, 40°, and 90° is a right triangle, not acute. In an acute triangle like 70°, 60°, and 50°, the first two sum to 130°. The key constraint is that each individual angle must be below 90°, not their pairwise sums.

Why can't a right triangle be acute?

By definition, a right triangle contains one angle equal to exactly 90°. An acute triangle requires all three angles to be strictly less than 90°. Since 90° is not less than 90°, no triangle can simultaneously satisfy both conditions. The classifications—acute, right, and obtuse—are mutually exclusive categories based on the largest interior angle.

Is a triangle with sides 3, 4, 5 acute or right?

The 3-4-5 triangle is a right triangle. Applying the Pythagorean inequality: 3² + 4² = 9 + 16 = 25, and 5² = 25. Since the sum equals the square of the longest side (not exceeds it), the largest angle is exactly 90°. This is the famous Pythagorean triple, and it always produces a right angle.

How do you find all three angles if you only know two sides and the angle between them?

Use the law of cosines to find the third side first. If you know sides a and b with included angle γ, then c² = a² + b² − 2ab cos(γ). Once you have all three sides, apply the law of cosines again to recover the remaining two angles. Alternatively, use the law of sines: once c is known, sin(α)/a = sin(γ)/c, allowing you to solve for the remaining angles directly.

More math calculators (see all)

Circle Calculator LCM Calculator Hypotenuse Calculator Miracle Calculator Square Area Calculator Power Modulo Calculator Unit Circle Calculator Least Common Multiple Calculator