Interior and Exterior Triangle Angles Calculator - 180 and 360 degree sums

The interior and exterior triangle angles calculator solves for every unknown angle of a triangle once you give it any two known interior angles. Enter angle A and angle B in degrees or radians, pick the unit selector, and the tool returns the missing third interior angle plus the three principal exterior angles at each vertex, all in the same unit you entered.

• • Geometry homework checks: Confirm the missing angle on a worksheet that hands you two of the three interior angles, then read out the three exterior angles as a 360 degree sum check.
• • Truss and roof pitch problems: Turn a measured inside corner and a rafter pitch into the third inside corner plus the exterior cut angle at each joint for a saw or miter setup.
• • Navigation and bearing worksheets: Convert two measured bearings of a triangle into the third interior angle, then read the exterior (supplement) of each bearing so the bearings line up around 360.
• • Special triangle drills: Run quick 60/60 or 90/45 inputs to confirm the equilateral and right triangle angle sets without redoing the arithmetic by hand.

A triangle has three interior angles and six exterior angles because extending each side produces two exterior angles per vertex. The calculator reports the three principal exterior angles, one at each vertex, which is what most worksheet problems need.

The two rules the calculator relies on are the interior angle sum theorem and the exterior angle theorem. The first fixes the third interior angle once two are known, and the second turns each pair of opposite interior angles into the exterior angle at the third vertex.

When the problem hands you sides instead of two angles, Triangle Angle Calculator accepts three sides, two sides plus an angle, or two angles and returns the full angle set using the law of cosines and the law of sines.

The calculator reads your two given interior angles and the unit selector, converts both inputs to degrees when needed, applies the two named angle theorems, and returns the missing third interior angle plus the three principal exterior angles in the unit you chose.

• a, b: The two interior angles you enter. The unit selector decides whether they are in degrees or radians. Each must be strictly between 0 and 180 deg (or 0 and pi rad).
• c: The missing third interior angle, recovered from the interior angle sum theorem as 180 deg - a - b.
• d, e, f: The three principal exterior angles, one at each vertex. Each is the sum of the two interior angles that do not sit at that vertex.
• sum check: The interior angle sum should always be 180 deg (or pi rad), and the exterior angle sum should always be 360 deg (or 2 pi rad).

The interior angle sum theorem is the rule that makes the third angle trivial: once two interior angles are known, the third is whatever number is needed to reach 180 degrees.

The exterior angle theorem turns the same two given angles into one of the exterior angles for free. Each exterior angle equals the sum of the two interior angles that are not at its vertex.

According to Wikipedia, the three interior angles of any Euclidean triangle always sum to 180 degrees, which follows directly from the parallel postulate and the angles-on-a-straight-line rule.

According to Wikipedia, pi radians equals 180 degrees exactly, so one radian equals 180 divided by pi degrees, which is approximately 57.29578.

These four ideas are the rules the calculator uses and the only facts you need to solve any triangle by angles alone.

Two given interior angles determine the third through the angle sum theorem, the exterior angle theorem converts that pair into one exterior angle, and the linear pair plus 360 degree sum rule produces the others.

For a triangle whose two base angles are equal, the interior angle sum theorem is what fixes the apex angle once you know one base angle, and Isosceles Triangle Angles walks through that case in a dedicated solver.

Pick a single unit for both inputs, type the two interior angles you already know, and read the third interior angle plus all three principal exterior angles from the result panel.

1. 1 Pick a single unit for both inputs: Use degrees for textbook or worksheet problems and radians for calculus or trig problems. Mixing units is the most common mistake.
2. 2 Enter interior angle A in the first field: Type the first known interior angle in degrees or radians. Anything strictly between 0 and 180 deg (or 0 and pi rad) is accepted.
3. 3 Enter interior angle B in the second field: Type the second known interior angle in the same unit. The two together must leave room for a positive third interior angle.
4. 4 Confirm or change the unit selector: Set the selector to degrees or radians. The result panel returns all six angles in the same unit you selected.
5. 5 Read the third interior angle and the three exterior angles: The third interior angle sits under Interior angle C. The three principal exterior angles appear under d, e, and f.
6. 6 Use the 180 and 360 sum checks as a safety net: The interior sum should be 180 deg (or pi rad) and the exterior sum should be 360 deg (or 2 pi rad).

A roof truss has interior angles 60, 70, and an unknown third corner. Enter 60 and 70 in degrees, and the calculator returns 50 degrees for the missing interior angle and 130, 110, and 120 degrees for the three exterior angles.

When one of the interior angles you enter is exactly 90 degrees, the triangle becomes a right triangle and Right Triangle Calculator takes over to return the missing sides and angles using the Pythagorean theorem.

The calculator replaces a stack of separate triangle tools with one fast angle solver that returns both interior and exterior sets from a single two input form.

• • Returns every angle in one pass: The third interior angle plus the three principal exterior angles appear together, so there is no need to switch tools.
• • Accepts degrees or radians: A single unit selector lets you use the calculator with textbook problems in degrees and with trig or physics problems in radians.
• • Built-in 180 and 360 sum checks: The interior and exterior sum fields confirm the inputs are consistent. Any drift away from 180 or 360 (or pi or 2 pi) flags the offending input immediately.
• • No side lengths needed: The angle rules work without any side measurement, so the calculator solves the full angle set for a triangle you have only sketched on paper.
• • Lightweight real-time updates: Every change to the inputs or unit selector updates the result panel immediately, so a 60/60 or 90/45 worksheet runs in a single browser session.

Entering 60 and 60 in degrees returns the classic 60-60-60 set plus three 120 degree exterior angles, and Equilateral Triangle confirms why every interior angle of an equilateral triangle has to be 60 degrees.

A few choices in the input form decide whether the recovered interior and exterior triangle angles actually match the triangle you are trying to solve.

• • The calculator only returns angles. It cannot recover side lengths from angles alone, because the same angle set fits infinitely many triangles of different sizes.
• • The exterior angle theorem returns the principal exterior angle at each vertex, which is the supplement of the interior angle. Extending a side the other way produces a vertical pair that carries the same numeric value, so the 360 degree sum still holds.

According to Wolfram MathWorld, the principal exterior angle at each vertex of a convex polygon equals the supplement of the interior angle at that vertex, so the three principal exterior angles of any triangle always add up to 2 pi radians (360 degrees).

Because each principal exterior angle is the supplement of the interior angle at the same vertex, the linear pair rule behind Supplementary Angles is what locks in the 360 degree exterior sum even when the triangle is scalene.