The Easiest Way To Calculate Triangle Angles!

The Easiest Way To Calculate Triangle Angles!

The Angles of a triangle are the spaces formed between the sides of a triangle. A triangle has three interior angles and six exterior angles where the interior angles add up to 180 degrees, and the exterior angles equal the sum of the two interior angles that are not adjacent. There are different types of triangles that have different properties of their angles. 

Triangles can be classified based on their angles. An equilateral triangle is a polygon and has three equal angles. An isosceles triangle has two equal angles and sides, and a scalene triangle has no equal angles or sides. In the Triangle calculator all the angles of a triangle sum to 180°. This angle fact is useful for solving problems that require students to find missing angles in triangles. Angle problems don't have difficult fractions, but answers need to be rounded to a degree of accuracy.

The angles in a triangle are:

  • Angles of 10, 20, and 150 degrees – obtuse triangle
  • Angles of 30, 60, and 90 degrees – right triangle
  • The angle of 40 degrees and two 70-degree angles – isosceles triangle
  • The angle of 90 degrees and two 45-degree angles – right isosceles triangle
  • Three 60 degree angles – equilateral triangle

An obtuse triangle is an oblique triangle that does not have a 90-degree angle. 

A right angle is an angle with a measure of exactly 90 degrees which is the two shortest sides of the Right triangle that are perpendicular to one another. The other two angles are complementary.

An acute angle is an angle with a measure of fewer than 90 degrees, a type of oblique triangle.

In an Isosceles triangle, two of its angles are equal, which means the two sides across from those angles have the same length. 

In an Equilateral triangle, all three of its angles are equal. It means that three sides across from those angles have the same length. 

How to find the angle of a triangle

There are some formulas used to calculate the angles of a Triangle that depend on what you give:

Triangle with three sides

The formula transformed from the law of cosines

Cos (α) = (b2 + c2 – a2 / 2bc)


α = arccos (b2 + c2 – a2 / 2bc)

For the second angle

Cos (β) = (a2 + c2 – b2 / 2ac)


β = arccos (a2 + c2 – b2 / 2ac)

For the third angle

Cos (γ) = (a2 + b2 – c2 / 2ab)


γ = arccos (a2 + b2 – c2 / 2ab)

Two triangles and one angle

The formula to find the missing angles

C = √a2 + b2 – 2ab x cos (γ)

Substitute c, 

α = arccos ((b2 + c2 – a2)/ (2bc))

β = 180o – α – γ

use the law of sines; if the angle is not between the given sides,

a / sin(α) = b / sin(β)


β = arcsin (b x sin (α) / a)

Two angles are given

Use the triangle angle sum theorem to find the missing angle

α = 180o – β - γ

β = 180o – α - γ

γ = 180o – α - β

The sum of angles in a triangle

For this, use the triangle angle sum theorem that states the interior angles of a triangle add to 180o

α + β + γ = 180o

Exterior angles of a triangle

For this, use the exterior triangle angle theorem equal to the sum of the opposite interior angles.

  • Triangle has six interior angles.
  • The exterior angles are taken one at each vertex, always summing up to 360o
  • The exterior angle is supplementary to its adjacent triangle interior angle 

How will you find the missing angles in a triangle

  • Choose your formula based on the sides and angles given. E.g., if you have two sides and one angle, then choose the angle and two side options
  • Type in the given values in the required field. α = 9 in, β = 14 in, and α = 30o, use the law of sines:

a / sin(α) = b / sin(β)
Β = arcsin (b x sin (α) / a)
Arcsin (14 in x sin (30o) / 9 in)
= arcsin (7/9) = 51.06o

The theorem about the sum of angles in a triangle, γ = 180o – α – β = 180o – 30o – 51.06o = 98.94o

  • The triangle angle calculator finds the missing angles in a triangle. They are equal to one we calculate manually; β = 51.06o, γ = 98.94o, and also you can determine the side length, c = 17.78 in.

Triangles have interior angles that add up to 180o. Angles in a triangle are the sum of the angles at each vertex in a triangle. The above formula and steps help you to find the triangle angles most easily and simply. Make use of this and get an accurate result.

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