Last updated: July 31, 2026
Interior & Exterior Triangle Angles Calculator
Creators
Dharmendra SinghReviewers
Creators
Dharmendra SinghReviewers
Creators
Dharmendra SinghReviewers
Creators
Dharmendra SinghReviewers
Formula
The interior angles of a triangle sum to 180°, so C = 180° - A - B. Each exterior angle is supplementary to its interior angle: exteriorX = 180° - X.
Where:
- A=First interior angle(degrees)
- B=Second interior angle(degrees)
- C=Third interior angle(degrees)
- E_A=Exterior angle at vertex A(degrees)
- E_B=Exterior angle at vertex B(degrees)
- E_C=Exterior angle at vertex C(degrees)
Worked Examples
Equilateral triangle
If two angles are 60°, the third is also 60° and all exterior angles are 120°.
- 1Add the known interior angles: 60 + 60 = 120.
- 2Subtract from 180° to get C = 60°.
- 3Compute each exterior angle as 180° minus the matching interior angle.
Right triangle with a 45° angle
A 90° angle and a 45° angle leave a 45° third angle.
- 1Compute C = 180 - 90 - 45 = 45.
- 2Compute exteriorA = 180 - 90 = 90.
- 3Compute exteriorB = 180 - 45 = 135 and exteriorC = 180 - 45 = 135.
Scalene triangle example
Different interior angles produce different exterior angles.
- 1Compute C = 180 - 30 - 60 = 90.
- 2Exterior angles are 150°, 120°, and 90° respectively.
Decimal angle inputs
The calculator also works with decimal degree measurements.
- 1Compute C = 180 - 72.5 - 48.25 = 59.25.
- 2Subtract each interior angle from 180° to get the three exterior angles.
Introduction
Triangles are governed by two simple facts: the three interior angles always add to 180°, and each exterior angle forms a straight line with its matching interior angle. This calculator uses those relationships to fill in the missing third angle and report all three exterior angles instantly.
Interior angle sum theorem
Every triangle, no matter its shape, has interior angles that add to 180°. When you already know two of them, the third is determined exactly.
Add A and B.
Subtract that sum from 180°.
The result is angle C.
How exterior angles are found
An exterior angle is supplementary to the interior angle at the same vertex. That means the two angles lie on a straight line and add up to 180°.
Exterior A = 180° - A.
Exterior B = 180° - B.
Exterior C = 180° - C.
If an exterior angle looks larger than expected, remember that obtuse exterior angles are completely normal.
When two angles form a valid triangle
Each interior angle must be greater than 0° and less than 180°. Also, the two known angles together must be less than 180° so that the third angle stays positive.
A must satisfy 0 < A < 180.
B must satisfy 0 < B < 180.
A + B must be less than 180.
If A + B = 180°, the third angle would be 0°, which is not a triangle.
Using the results to classify a triangle
Once angle C is known, you can often identify the triangle type immediately. Equal angles imply equal opposite sides, while a 90° angle identifies a right triangle.
60°, 60°, 60° indicates an equilateral triangle.
A single 90° angle indicates a right triangle.
Three different angles indicate a scalene triangle.
Why exterior angles matter
Exterior angles appear in geometry proofs, polygon reasoning, and trigonometry diagrams. Knowing them quickly helps with supplementary-angle checks and theorem applications.
Exterior angle theorems rely on them.
They help verify drawings and constructions.
They can reveal mistakes in angle-labelling problems.
Common mistakes to avoid
Most errors come from mixing interior and exterior angles or forgetting the triangle sum theorem.
Do not add an exterior angle into the 180° interior-angle sum.
Check that the two given angles are both interior angles.
Make sure your calculator is not using radians; this tool expects degrees.
Working with decimal angles
Measured triangles often produce decimal angles, especially in surveying or computer graphics. The calculator keeps six-decimal precision in its numeric outputs.
Whole numbers and decimals are both accepted.
Outputs are rounded consistently.
You can use the results in later geometry calculations.
FAQs
Why do triangle interior angles add to 180°?
In Euclidean geometry, the three interior angles of a triangle always sum to a straight angle, which is 180°.
Can angle A or B be 0°?
No. An interior angle of 0° would not form a valid triangle.
What if angle A plus angle B equals 180°?
Then angle C would be 0°, so the shape would collapse into a straight line instead of a triangle.
How is an exterior angle different from an interior angle?
An interior angle is inside the triangle, while an exterior angle is formed by extending one side beyond a vertex.
Do the exterior angles have to be acute?
No. Exterior angles are often obtuse because they are supplementary to the interior angles.
Can I use decimal values for the angles?
Yes. The calculator accepts decimal degree measurements and rounds outputs to six decimal places.
Does this calculator work for non-Euclidean geometry?
No. It is based on the standard Euclidean triangle angle sum of 180°.
How can I check if my triangle is right, acute, or obtuse?
Look at the largest interior angle after computing C. If it is 90°, the triangle is right; less than 90° means acute; greater than 90° means obtuse.
Why is the exterior angle sometimes useful in proofs?
Because an exterior angle equals the sum of the two remote interior angles, making it a common theorem in geometry problems.