Triangle Sum Theorem Calculator - Missing Angle and Sum Solver

The triangle sum theorem calculator uses the Euclidean identity that the three interior angles of any triangle add up to 180 degrees (pi radians) to recover a missing interior angle or the combined sum of two missing interior angles.

• • Geometry homework checks: Confirm the missing angle on a worksheet that hands you two of the three interior angles, then read the family label to match the answer key.
• • Roof truss and rafter angles: Convert two measured inside corners of a roof truss into the third inside corner plus the acute or obtuse family label.
• • Navigation and bearing worksheets: Use mode B to convert one measured interior bearing into the combined sum of the other two interior angles for a triangle course correction.
• • Quick triangle classification drills: Run 60 / 60 or 90 / 45 inputs to confirm the equilateral and right triangle families without redoing the arithmetic by hand.

A Euclidean triangle has three interior angles and the theorem says those three angles always sum to 180 degrees or pi radians.

When you also need the three principal exterior angles at each vertex, Interior and Exterior Triangle Angles Calculator takes the same two given interior angles and returns the exterior angle set alongside the missing third interior angle.

The calculator reads the mode selector, the two entered angles, and the unit selector, applies the angle sum identity, and re-projects the result back into the unit you picked.

• A: The first known interior angle.
• B: The second known interior angle, used only in mode A.
• C: The recovered value, the missing third angle or the combined sum of the two missing angles.
• 180 deg / pi rad: The fixed interior angle sum identity used to recover C from A and B.
• Sum check: The total of the implied angle set, recomputed so the user can confirm the result by eye.

The unit selector handles degrees and radians internally. Radian inputs are converted to degrees for the identity, and the recovered value plus the sum check are converted back to radians before they are displayed.

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.

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

These four ideas are the rules the triangle sum theorem calculator uses and the only facts you need to apply the theorem by hand.

When mode B is used the implied angle set is not unique, so the family and angle labels both collapse to ambiguous. The recovered sum is still useful because it locks in how the remaining 180 - A degrees are split.

When the three implied interior angles are all 60 degrees, the family label becomes equilateral and Equilateral Triangle walks through why every interior angle of an equilateral triangle has to be 60 degrees.

Pick a mode, type the angle or angles you already know, choose a unit, and read the recovered value, sum check, and triangle classification from the result panel.

1. 1 Pick a mode that matches your problem: Use mode A when you know two interior angles, and mode B when you know only one.
2. 2 Enter interior angle A in the first field: Type the first known interior angle in degrees or radians. The value must be greater than 0 and at most 180 deg, or at most pi rad in radian mode.
3. 3 Enter interior angle B in mode A: Type the second known interior angle in the same unit as angle A. The solver ignores angle B in mode B.
4. 4 Confirm or change the unit selector: Set the unit selector to degrees for textbook problems and radians for trig or physics problems.
5. 5 Read the recovered value and classification: The primary card shows the recovered value. The classification rows report acute, right, obtuse, and equilateral, isosceles, scalene, or ambiguous.
6. 6 Use the 180 deg / pi rad sum check as a safety net: The sum check row recomputes the 180 degree (pi rad) identity from the implied angle set, so any drift away from 180 flags the offending input.

A roof truss has two measured interior angles of 55 and 65 degrees. Pick mode A, enter 55 and 65, leave the unit selector on degrees, and the solver returns the missing third angle as 60 degrees with a scalene acute classification.

When one of the recovered interior angles 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 triangle sum theorem calculator replaces a stack of separate triangle tools with one focused tool that returns a missing angle, a missing sum, and a classification label from a single short form.

• • Solves mode A and mode B in one form: The same form recovers the missing third angle in mode A and the combined sum of two missing angles in mode B.
• • Accepts degrees and radians through one selector: A single unit selector lets you use the solver with textbook problems in degrees and with trig or physics problems in radians.
• • Labels the triangle family and angle class: The result panel reports the family label (equilateral, isosceles, scalene, ambiguous) and the angle label (acute, right, obtuse, degenerate) so the answer matches a worksheet answer key.
• • Built-in 180 deg / pi rad sum check: The sum check row recomputes the 180 degree identity from the implied angle set, so any drift away from 180 flags the offending input.
• • Lightweight real-time updates: Every change to the mode selector, the angle inputs, or the unit selector updates the result panel immediately.

Mode B is the feature that distinguishes this solver from a plain third-angle calculator. It returns the combined sum of the two missing interior angles when the problem hands you only one interior angle, which is a common case in navigation and bearing worksheets.

When the two given interior angles are equal, the family label becomes isosceles and Isosceles Triangle Angles walks through that case in a dedicated solver that returns the missing vertex angle from a single base angle.

A few choices in the input form decide whether the recovered value and the classification label actually match the triangle you are trying to solve.

• • The solver only returns angles. It cannot recover side lengths from angles alone, because the same angle set fits infinitely many triangles of different sizes.
• • Mode B returns the combined sum of the two missing interior angles, not the two individual values. If you need the two individual missing angles, use mode A instead.

These limits come from the triangle sum theorem itself. The identity fixes angles but not sides.

According to Wolfram MathWorld, a triangle is acute when every interior angle is less than 90 degrees, right when one interior angle equals 90 degrees, and obtuse when one interior angle exceeds 90 degrees.

Once the missing third interior angle is in hand, Exterior Angles of a Triangle Calculator converts it into the three principal exterior angles and reports the 360 degree exterior sum as a consistency check.