Triangle Angle Calculator
Free Triangle angle Calculator for triangle. Enter values to get step-by-step solutions with formulas and graphs. See charts, tables, and visual results.
Reviewed for accuracy by Manoj Kumar, Mathematics Educator
Triangle Angle Calculator
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Formula: Angle Sum: A + B + C = 180 | Law of Cosines: cos(A) = (b^2 + c^2 - a^2) / (2bc)
Worked example — Angle C = 53 degrees | Acute Scalene Triangle
Formula
Angle Sum: A + B + C = 180 | Law of Cosines: cos(A) = (b^2 + c^2 - a^2) / (2bc)
The angle sum property states all triangle angles total 180 degrees. The Law of Cosines finds any angle from three known sides by relating the cosine of an angle to the lengths of all three sides.
Worked Examples
Example 1: Finding the Third Angle from Two Known Angles
Problem:A triangle has angles A = 55 degrees and B = 72 degrees. Find angle C and classify the triangle.
Solution:Using the angle sum property: Angle C = 180 - A - B Angle C = 180 - 55 - 72 = 53 degrees All angles are less than 90 degrees, so this is an Acute triangle. All angles are different, so this is a Scalene triangle. In radians: A = 0.9599, B = 1.2566, C = 0.9250 Verification: 0.9599 + 1.2566 + 0.9250 = 3.1416 = pi
Result:Angle C = 53 degrees | Acute Scalene Triangle
Example 2: Finding All Angles from Three Sides
Problem:A triangle has sides a = 6, b = 8, c = 10. Find all three angles.
Solution:Using the Law of Cosines: cos(A) = (8^2 + 10^2 - 6^2) / (2 x 8 x 10) = (64 + 100 - 36) / 160 = 128/160 = 0.8 A = arccos(0.8) = 36.87 degrees cos(B) = (6^2 + 10^2 - 8^2) / (2 x 6 x 10) = (36 + 100 - 64) / 120 = 72/120 = 0.6 B = arccos(0.6) = 53.13 degrees C = 180 - 36.87 - 53.13 = 90.00 degrees This is a Right triangle (3-4-5 Pythagorean triple scaled by 2)
Result:A = 36.87 deg | B = 53.13 deg | C = 90.00 deg | Right Scalene Triangle
Frequently Asked Questions
What is the triangle angle sum property?
The triangle angle sum property states that the three interior angles of any triangle always add up to exactly 180 degrees (or pi radians). This is one of the most fundamental theorems in Euclidean geometry and applies to all triangles regardless of their shape, size, or type. The proof follows from drawing a line through one vertex parallel to the opposite side and using alternate interior angles. This property means that if you know any two angles of a triangle, you can always find the third by subtracting their sum from 180. It also means no triangle can have more than one right angle (90 degrees) or more than one obtuse angle (greater than 90 degrees). This property does not hold in non-Euclidean geometries like spherical geometry.
How do you find the third angle when two angles are known?
Finding the third angle of a triangle is straightforward when two angles are known. Simply subtract the sum of the two known angles from 180 degrees. For example, if angle A = 45 degrees and angle B = 75 degrees, then angle C = 180 - 45 - 75 = 60 degrees. This works because the angle sum property guarantees all three angles total exactly 180 degrees. In radians, the process is identical but you subtract from pi instead of 180. If angle A = pi/4 and angle B = pi/3, then angle C = pi - pi/4 - pi/3 = pi - 3pi/12 - 4pi/12 = 5pi/12 radians. Always verify that each angle is positive and that none exceeds 180 degrees, as this would indicate an invalid triangle.
How do you calculate angles from three sides using the Law of Cosines?
The Law of Cosines allows you to find any angle when all three sides are known. The formula is cos(A) = (b^2 + c^2 - a^2) / (2bc), where A is the angle opposite side a, and b and c are the other two sides. Apply the inverse cosine (arccos) function to get the angle. For example, with sides a=5, b=7, c=8: cos(A) = (49 + 64 - 25) / (2 x 7 x 8) = 88/112 = 0.7857, so A = arccos(0.7857) = 38.21 degrees. Repeat for each angle or find the remaining angle by subtraction from 180. The Law of Cosines is a generalization of the Pythagorean theorem, and when the angle is 90 degrees, cos(90) = 0, which reduces the formula to a^2 = b^2 + c^2.
What are the different types of triangles based on angles?
Triangles are classified into three types based on their angles. An acute triangle has all three interior angles less than 90 degrees, such as a 60-60-60 equilateral triangle or a 50-60-70 triangle. A right triangle has exactly one 90-degree angle, with the side opposite the right angle called the hypotenuse. Common examples include the 3-4-5, 5-12-13, and 45-45-90 triangles. An obtuse triangle has one angle greater than 90 degrees, such as a 30-40-110 triangle. Every triangle must fall into exactly one of these categories because the angle sum is always 180 degrees. You can determine the type from sides alone: if a^2 + b^2 = c^2 it is right, if a^2 + b^2 > c^2 it is acute, and if a^2 + b^2 < c^2 it is obtuse.
What is the exterior angle theorem?
The exterior angle theorem states that an exterior angle of a triangle equals the sum of the two non-adjacent interior angles. An exterior angle is formed by extending one side of the triangle beyond a vertex. For example, if a triangle has angles 40, 60, and 80 degrees, the exterior angle at the 80-degree vertex equals 40 + 60 = 100 degrees. This makes sense because the exterior angle and its adjacent interior angle are supplementary (sum to 180), so the exterior angle = 180 - adjacent interior angle = sum of the other two angles. This theorem is extremely useful in geometry proofs and problem-solving because it establishes a relationship between interior and exterior angles without needing to know all three interior angles directly.
How do degrees and radians relate to each other?
Degrees and radians are two different units for measuring angles. A full circle contains 360 degrees or 2pi radians, so 1 radian equals approximately 57.2958 degrees, and 1 degree equals approximately 0.01745 radians. To convert degrees to radians, multiply by pi/180. To convert radians to degrees, multiply by 180/pi. Common conversions include: 30 degrees = pi/6, 45 degrees = pi/4, 60 degrees = pi/3, 90 degrees = pi/2, and 180 degrees = pi radians. Radians are preferred in calculus and higher mathematics because they simplify many formulas, such as the derivative of sin(x) equaling cos(x) only when x is measured in radians. Degrees are more intuitive for everyday use and practical applications like navigation and construction.
What is the Law of Sines and when is it used?
The Law of Sines states that in any triangle, the ratio of a side length to the sine of its opposite angle is constant: a/sin(A) = b/sin(B) = c/sin(C). This relationship is used to solve triangles when you know either two angles and one side (AAS or ASA) or two sides and an angle opposite one of them (SSA, the ambiguous case). For example, if A = 40 degrees, B = 60 degrees, and a = 10, then b = 10 x sin(60)/sin(40) = 10 x 0.8660/0.6428 = 13.47. The Law of Sines can also determine the circumradius R of the triangle since a/sin(A) = 2R. The SSA case requires caution because it can yield zero, one, or two valid triangles depending on the measurements.
Can a triangle have two right angles or two obtuse angles?
No, a triangle cannot have two right angles or two obtuse angles. Since the sum of all three interior angles must equal exactly 180 degrees, having two right angles (each 90 degrees) would total 180 degrees for just two angles, leaving 0 degrees for the third, which is impossible. Similarly, two obtuse angles would each exceed 90 degrees, making their sum greater than 180 degrees, which leaves a negative value for the third angle. Therefore, a triangle can have at most one right angle or at most one obtuse angle. If it has one right angle, the other two must be acute and sum to 90. If it has one obtuse angle, the other two must both be acute and sum to less than 90. These constraints are direct consequences of the angle sum property.
What are complementary and supplementary angles in triangles?
Complementary angles are two angles that sum to 90 degrees, while supplementary angles are two angles that sum to 180 degrees. In a right triangle, the two acute angles are always complementary since they must sum to 90 degrees (because the right angle accounts for the other 90 of the 180 total). For example, in a right triangle with a 35-degree angle, the other acute angle must be 55 degrees (35 + 55 = 90). Supplementary angles appear when considering an interior angle and its corresponding exterior angle at a triangle vertex. An interior angle of 120 degrees has a supplementary exterior angle of 60 degrees. Understanding these relationships helps solve geometry problems quickly without calculating all three angles.
How do you determine if three given angles can form a valid triangle?
Three angles can form a valid triangle if and only if they satisfy three conditions: all three angles must be positive (greater than 0 degrees), their sum must equal exactly 180 degrees, and no single angle can be 180 degrees or more. For example, angles 60, 70, and 50 form a valid triangle because they are all positive and sum to 180. Angles 90, 50, and 50 do not because they sum to 190. Angles 0, 90, and 90 fail because one angle is zero. When working with decimal or rounded values, allow for small floating-point errors (for example, accepting a sum between 179.99 and 180.01). In practical applications, also verify the triangle inequality theorem if side lengths are involved: any side must be less than the sum of the other two.
References
Reviewed for accuracy by Manoj Kumar, Mathematics Educator · Editorial policy
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