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The Triangle Angle Calculator computes all three interior angles of a triangle when you know all three side lengths. Using the Law of Cosines, this tool converts side measurements into precise angle values in degrees. Knowing the angles of a triangle is essential for navigation, structural analysis, surveying, trigonometry coursework, and countless engineering applications.
The Law of Cosines generalizes the Pythagorean theorem to all triangles. For any triangle with sides $$a$$, $$b$$, $$c$$ and opposite angles $$A$$, $$B$$, $$C$$, the relationship is $$c^2 = a^2 + b^2 - 2ab\cos(C)$$. By rearranging this formula, we can solve for each angle: $$C = \arccos\left(\frac{a^2 + b^2 - c^2}{2ab}\right)$$. Applying this three times with appropriate variable substitutions yields all three angles.
One powerful property of triangles is that the interior angles always sum to exactly 180 degrees. The calculator provides an angle sum check as a verification measure. Due to floating-point arithmetic, the displayed sum may differ from 180 by a tiny fraction, but any significant deviation indicates the input sides do not form a valid triangle.
Understanding angles reveals the triangle's classification. If all angles are less than 90 degrees, it is an acute triangle. If one angle equals 90 degrees, it is a right triangle. If one angle exceeds 90 degrees, it is an obtuse triangle. Furthermore, equal angles indicate an isosceles triangle (two equal angles) or an equilateral triangle (all three angles are 60 degrees).
In practical applications, surveyors use angle calculations to map terrain, engineers use them to analyze force distributions in truss structures, and navigators apply them in triangulation to determine positions. Computer graphics rely heavily on triangle angle computations for lighting calculations, normal vector determination, and mesh quality assessment. A well-shaped triangle mesh (with angles close to 60 degrees) produces better numerical results in finite element analysis.
The calculator handles all valid triangle configurations, from nearly flat (angles approaching 0 or 180 degrees) to perfectly equilateral. For degenerate inputs where the sides cannot form a triangle, the arccos function returns NaN, clearly indicating invalid input.
Each angle is found by rearranging the Law of Cosines:
$$A = \arccos\left(\frac{b^2 + c^2 - a^2}{2bc}\right)$$
$$B = \arccos\left(\frac{a^2 + c^2 - b^2}{2ac}\right)$$
$$C = \arccos\left(\frac{a^2 + b^2 - c^2}{2ab}\right)$$
The arccos function returns radians, which are converted to degrees by multiplying by $$\frac{180}{\pi}$$. The results are verified by checking that $$A + B + C = 180°$$.
Each angle is measured in degrees and corresponds to the vertex opposite its named side. Angle A is opposite side a, Angle B is opposite side b, and Angle C is opposite side c. The largest angle is always opposite the longest side. The angle sum check should equal 180° (minor floating-point deviations are normal). If any output shows NaN, the input sides do not satisfy the triangle inequality.
Inputs
Results
An obtuse triangle with the largest angle (95.74°) opposite the longest side (c = 9).
Inputs
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All three angles are exactly 60° in an equilateral triangle.
Yes. The Law of Cosines allows you to compute all three angles from three known side lengths. Each angle is found using the arccos of an expression involving the three sides.
This is a fundamental theorem of Euclidean geometry. It can be proven by drawing a line parallel to one side through the opposite vertex, creating alternate interior angles that together with the vertex angle span a straight line (180°).
Then the triangle is a right triangle, and the side opposite the 90° angle is the hypotenuse. In this special case, the Law of Cosines reduces to the Pythagorean theorem since cos(90°) = 0.
If the largest angle is less than 90°, the triangle is acute. If the largest angle equals 90°, it is a right triangle. If the largest angle exceeds 90°, it is obtuse. The largest angle is always opposite the longest side.
NaN (Not a Number) means the arccos function received a value outside the valid range [-1, 1]. This happens when the three input sides violate the triangle inequality and cannot form a valid triangle.
No. This calculator uses Euclidean geometry where the angle sum is always 180°. In spherical geometry the sum exceeds 180°, and in hyperbolic geometry it is less than 180°. Different formulas are required for those geometries.
Roboculator Team
The Roboculator Team explains calculations, planning tools, and practical formulas in clear language for real-life situations.
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