Exterior Angles Of A Triangle Calculator - 360 Degree Sum Check

An exterior angles of a triangle calculator solves a triangle the moment you know any two of its three interior angles. Enter two interior angles in degrees or radians, and the calculator returns the missing third interior angle plus the three principal exterior angles, one at each vertex, and confirms the 360-degree exterior sum.

• • Geometry homework and exam checks: Confirm the missing interior angle on a worksheet that hands you two angles, then read out the three principal exterior angles as the 360-degree sum check.
• • Woodworking miter and saw setups: Convert a measured inside corner and one rafter pitch into the third inside corner plus the exterior cut angle at each joint for a miter saw.
• • Roof truss and rafter pitch problems: Turn two inside-corner angles of a roof truss into the apex interior angle and the three exterior angles that drive the birdsmouth and ridge cuts.
• • Navigation and bearing worksheets: Convert two measured bearings of a triangle into the third interior angle plus the exterior (supplement) of each bearing so the bearings line up around 360 degrees.

A triangle has three interior angles and six exterior angles because extending each side produces two exterior angles at each vertex. The three principal exterior angles, one at each vertex, are what most worksheets, miter setups, and bearing problems actually need.

If your problem hands you only sides or sides plus an angle instead of two interior angles, the Interior and Exterior Triangle Angles Calculator accepts those inputs and returns the full interior and exterior angle set using the law of cosines and the law of sines.

The calculator reads your two given interior angles and the unit selector, applies the exterior angle theorem and the supplementary angle rule, 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, in degrees by default or radians if the unit selector is set to radians.
• c: The missing third interior angle, recovered from the triangle interior angle sum as 180 deg - a - b.
• d, e, f: The three principal exterior angles, one at each vertex. d sits at the vertex opposite c and equals a + b by the exterior angle theorem; e and f sit at the two given vertices and equal 180 minus the interior angle at that vertex.
• sum check: d + e + f should always equal 360 deg (or 2 pi rad).

The exterior angle theorem converts two interior angles into the exterior angle at the third vertex without any side measurement. The other two principal exterior angles are the supplements of the two interior angles you typed.

According to Wolfram MathWorld, the exterior angle of a triangle at any vertex equals the sum of the two remote interior angles, and the three principal exterior angles of any triangle always add up to 360 degrees (or 2 pi radians).

When you know one side plus two angles or three sides and need the full angle set, the Triangle Angle Calculator handles those side-driven inputs while this tool stays focused on the two-interior-angle workflow.

These four ideas are the rules the calculator relies on.

Two given interior angles fix the third through the angle sum identity. The exterior angle theorem converts that pair into the principal exterior angle at the third vertex, and the linear pair plus the 360-degree sum identity produces the other two principal exteriors.

The linear pair rule that gives the supplementary angle at each vertex is the same identity behind Supplementary Angles, so that tool is the natural place to confirm why interior plus exterior at the same vertex always equals 180 degrees.

Pick a single unit for both inputs, type the two interior angles you already know, and read the missing 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, trigonometry, or physics problems.
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 every angle in the same unit you selected.
5. 5 Read the angles and check the 360-degree sum: The third interior angle sits in its own row. The principal exterior angles appear in three rows labeled at C, at A, and at B, and the exterior sum should read 360 deg (or 2 pi rad).

A roof truss has measured inside corners of 55 deg and 65 deg at the base. Enter 55 and 65 in degrees. The calculator returns 60 deg for the missing apex interior angle, 120 deg for the exterior at the apex, 125 deg for the exterior at the 55 deg base, and 115 deg for the exterior at the 65 deg base. The exterior sum reads 360 deg, confirming the inputs.

When one of the two given interior angles works out to exactly 90 degrees and you also know a side, the Right Triangle Calculator takes over to return the missing sides and the remaining angles using the Pythagorean theorem.

Working the exterior angle theorem out by hand takes two or three steps that are easy to slip on, especially when the inputs come in radians.

• • Catches impossible angle combinations: If the two given interior angles already sum to 180 deg (or pi rad), the calculator flags the input before any exterior angle is computed.
• • Skips the unit conversion step: You can enter either degrees or radians through the unit selector. The calculator does the radian-to-degree work internally and returns every result row in the same unit.
• • Returns every principal exterior angle at once: All three exterior angles appear together, so the user gets the exterior angle theorem and the supplementary angle rule in the same result panel.
• • Includes a built-in 360-degree sum check: The principal exterior angle sum row confirms the inputs are consistent. Any drift from 360 deg (or 2 pi rad) flags the offending input immediately.
• • Works for every triangle type: Acute, right, and obtuse triangles all use the same two identities, so the same form returns every principal exterior angle correctly.

These benefits matter most when the two interior angles come from a real measurement (a protractor, a theodolite, a miter gauge) rather than a textbook.

If your source data is in gradians or turns instead of degrees or radians, the Angle Converter lets you convert each interior angle before entering it here so the principal exterior angles come back in the unit you need.

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

• • The calculator only returns angles. It cannot recover side lengths from interior angles alone, because infinitely many triangles share the same three interior angles.
• • Each principal exterior angle is the supplement of the interior angle at the same vertex. Extending a side the other way produces a vertical pair with the same numeric value.

According to Wikipedia, the exterior angle theorem states that the measure of an exterior angle of a triangle equals the sum of the measures of the two remote interior angles, which follows directly from the triangle interior angle sum of 180 degrees.

For the special case where the two base angles of an isosceles triangle are equal, the Isosceles Triangle Angles walks through that symmetry and shows how the principal exterior angles simplify to two copies of the same value.