What makes a triangle acute?

A triangle is acute when all three of its interior angles are strictly less than 90 degrees. Equivalently, for a triangle with sides a, b, and c (where c is the longest side), it is acute if and only if a squared plus b squared is greater than c squared. This is the reverse of the Pythagorean condition: in a right triangle a squared plus b squared equals c squared, and in an obtuse triangle a squared plus b squared is less than c squared. Because angles in any triangle add to 180 degrees, having all three angles under 90 degrees means no single angle dominates the shape.

How do you find the angles of an acute triangle from its sides?

Use the law of cosines: for angle A opposite side a, the formula is cos(A) = (b squared + c squared minus a squared) divided by (2bc). Rearranging gives A = arccos((b squared + c squared minus a squared) / (2bc)). Apply the same formula cyclically to find angles B and C. Once two angles are known, the third is 180 minus A minus B degrees. If all three resulting angles are less than 90 degrees and greater than zero, the triangle is acute. If any cosine value is negative, the corresponding angle exceeds 90 degrees and the triangle is obtuse.

Can an equilateral triangle be acute?

Yes. An equilateral triangle has all three angles equal to exactly 60 degrees, which are all less than 90 degrees, so it is always an acute triangle. It is in fact the most symmetric acute triangle. Any isosceles triangle with the apex angle less than 90 degrees is also acute, provided both base angles (which are equal) are also less than 90 degrees. For example, an isosceles triangle with angles 70, 70, and 40 degrees is acute.

Acute Triangle Calculator

Enter the three side lengths of a triangle to check whether it is acute (all angles less than 90 degrees) and instantly calculate every angle, the area, perimeter, altitudes, inradius, and circumradius.

Angles are found using the law of cosines. Area is calculated using Heron's formula. All angles and properties update as you type.

Calculate.co.nz is proud to be partnered with Health Based Building, a leader in sustainable and health-conscious building innovation. With over a century of experience, they develop high-performance systems like Foreverbreathe Specification, Magnum Board, and Foreverbreathe Paints to support energy-efficient, non-toxic living environments. Their commitment to healthier homes aligns with our belief that informed choices lead to better outcomes for Kiwi households.

Calculate.co.nz partner: Health Based Building

Standard geometry Law of cosines (ISO 80000-2) and Heron's formula. No rounding until display.

Side Lengths

Side a (any positive number)

units

Side b (any positive number)

units

Side c (any positive number)

units

Triangle Type Check

Enter three side lengths on the left to see the triangle type and a full breakdown of all properties.

Triangle inequality-

Triangle type-

Largest angle-

Triangle Properties

Angle A
-
Opposite side a

Angle B
-
Opposite side b

Angle C
-
Opposite side c

Area

Heron's formula

Measurements

Perimeter-

Semi-perimeter (s)-

Area-

Altitude from a (ha)-

Altitude from b (hb)-

Altitude from c (hc)-

Circles and Angles

Angle A (opposite a)-

Angle B (opposite b)-

Angle C (opposite c)-

Inradius (r)-

Circumradius (R)-

Sum of angles-

Result: Enter three side lengths above to check if the triangle is acute.

What Is an Acute Triangle?

An acute triangle is one where all three interior angles are strictly less than 90 degrees. Because the angles of any triangle must sum to exactly 180 degrees, having all angles less than 90 degrees means none of the three angles dominates the shape. Acute triangles are sometimes described as "sharp" triangles because all their corners are pointed rather than flat or wide.

The three main types of triangle by angle are:

Acute triangle: all angles are less than 90 degrees (for example, 60, 70, and 50 degrees)
Right triangle: exactly one angle is 90 degrees
Obtuse triangle: exactly one angle is greater than 90 degrees

How to Check if a Triangle Is Acute

Given three side lengths a, b, and c, first check whether they form a valid triangle using the triangle inequality: every pair of sides must sum to more than the third side. If that passes, identify the longest side and call it c. The triangle is:

Acute if a² + b² > c²
Right if a² + b² = c² (the Pythagorean theorem)
Obtuse if a² + b² < c²

This test only needs to be applied to the longest side because the shorter sides always produce a positive difference, and it is only the largest angle that can reach or exceed 90 degrees.

Formulas Used by This Calculator

Property	Formula
Angle A (law of cosines)	A = arccos((b² + c² − a²) / (2bc))
Angle B (law of cosines)	B = arccos((a² + c² − b²) / (2ac))
Angle C	C = 180 − A − B degrees
Semi-perimeter	s = (a + b + c) / 2
Area (Heron's formula)	Area = √(s(s−a)(s−b)(s−c))
Altitude ha	ha = 2 × Area / a
Inradius	r = Area / s
Circumradius	R = (a × b × c) / (4 × Area)

Worked Example

Using the default values of a = 5, b = 6, c = 7 (all units):

Triangle inequality: 5 + 6 = 11 > 7, 5 + 7 = 12 > 6, 6 + 7 = 13 > 5. Valid triangle.
Acuteness check (longest side c = 7): 5² + 6² = 25 + 36 = 61 > 49 = 7². The triangle is acute.
Angle A = arccos((36 + 49 − 25) / (2 × 6 × 7)) = arccos(60/84) = arccos(0.7143) ≈ 44.42°
Angle B = arccos((25 + 49 − 36) / (2 × 5 × 7)) = arccos(38/70) = arccos(0.5429) ≈ 57.12°
Angle C = 180 − 44.42 − 57.12 ≈ 78.46°
Semi-perimeter: s = (5 + 6 + 7) / 2 = 9
Area = √(9 × 4 × 3 × 2) = √216 ≈ 14.70 square units
Inradius: r = 14.70 / 9 ≈ 1.63 units
Circumradius: R = (5 × 6 × 7) / (4 × 14.70) ≈ 3.57 units

All three angles (44.42°, 57.12°, 78.46°) are less than 90 degrees, confirming the triangle is acute.

Related Calculators

Maths and Stats Calculators: full hub of maths tools.
Triangle Calculator (General): any triangle type by sides or angles.
Area of a Triangle (Heron's Formula): area from three side lengths.
Right Angle Triangle Calculator: solve for sides and angles with one 90-degree angle.
Equilateral Triangle Calculator: the special case where all sides and angles are equal.

Method: Angles calculated via the law of cosines (ISO 80000-2). Area by Heron's formula. Inradius r = Area / s. Circumradius R = abc / (4 × Area). All intermediate values are computed in full floating-point precision; results are rounded only for display.

This calculator is for educational and general-purpose use. Results depend on the side lengths you enter. For very large or very small values, normal floating-point rounding applies. Always verify critical measurements independently.