Triangle Sum Theorem Calculator
Know two interior angles of a triangle? Get the third instantly using α + β + γ = 180°, with step-by-step working, a labelled diagram and a degrees/radians toggle.
What is the triangle sum theorem?
The triangle sum theorem — also called the triangle angle sum theorem — states that the three interior angles of any triangle in the plane add up to exactly 180°, or π radians. It is one of the earliest results in Euclidean geometry and the reason a triangle's third angle is never a free choice: give the first two and the third is forced.
Written as a formula, if the interior angles are α, β and γ, then
Rearranging gives the working formula this calculator uses: γ = 180° − α − β. In radians the same identity reads γ = π − α − β. Because angles fix a triangle's shape but not its size, this result also explains why any two triangles with the same three angles are similar, not congruent.
Triangle sum theorem, case by case
Two angles in degrees
The most common case: two interior angles are known in degrees and you just need the third. Subtract both from 180° and you're done — no trigonometry required.
A right triangle (one angle is 90°)
If one angle is a right angle, the other two acute angles must add to exactly 90°. This is the fastest sanity check for any right-triangle problem — if your two acute angles don't complement, something is wrong.
Angles in radians
In radians the total switches from 180° to π. The rearranged formula reads γ = π − α − β. Handy when you're feeding results straight into calculus or a physics equation that already lives in radians.
Why it works — the parallel-line proof
Draw a line through vertex C parallel to side AB. Alternate interior angles pair the two outer angles at C with α at A and β at B, and the middle angle is γ. Together they fill a straight line, so α + γ + β = 180°.
The exterior angle shortcut
At any vertex the exterior angle equals the sum of the two non-adjacent interior angles. At vertex B, that says ext = α + γ. Since the exterior and interior angles at B form a straight line (ext + β = 180°), substituting gives α + β + γ = 180° in one step.
Formulas at a glance
| Situation | Formula | Notes |
|---|---|---|
| Third angle (degrees) | γ = 180° − α − β | Requires α + β < 180°. |
| Third angle (radians) | γ = π − α − β | Requires α + β < π. |
| Right triangle | β + γ = 90° | One angle is 90°; the other two are complementary. |
| Equilateral | α = β = γ = 60° | Only triangle with three equal angles. |
| Exterior angle | ext = α + γ | Exterior at B equals sum of the other two interior angles. |
| Degrees ↔ radians | rad = deg · π/180 | 180° = π radians. |
What this tool does for you
- Solves the third interior angle of any triangle from the other two, using γ = 180° − α − β (or γ = π − α − β in radians).
- Accepts input in degrees or radians with a one-click unit toggle — no manual conversion needed.
- Flags impossible inputs — zero, negative, or two angles that already sum to 180° or more — instead of returning a silently wrong answer.
- Renders a labelled SVG triangle with α, β and γ at their vertices so the geometry is obvious at a glance.
- Prints personalised step-by-step working with your actual numbers substituted in, ready to copy into homework or a report.
- Copy-to-clipboard, PNG snapshot, PDF export and print, all matching the shared result-actions used across the site.
- Runs entirely in the browser — nothing you type is uploaded, and the page works offline once loaded.
Frequently asked questions
+What is the triangle angle sum theorem?
The triangle angle sum theorem says that the three interior angles of any flat (Euclidean) triangle always add up to exactly 180°, or π radians. It doesn't matter whether the triangle is tiny or huge, acute, right or obtuse — the sum is fixed. That is why knowing any two interior angles instantly pins down the third: γ = 180° − α − β.
+Do three angles define a unique triangle?
No. Three angles tell you the shape of a triangle but not its size — any two triangles with the same three angles are similar, not congruent. Scaling every side by the same factor keeps all three angles unchanged. If you want to actually solve for side lengths, use the full Triangle Calculator, which handles SSS, SAS, ASA, AAS and SSA cases.
+How does this relate to the exterior angle theorem?
The exterior angle at any vertex equals the sum of the two non-adjacent interior angles. That's a direct consequence of the angle-sum theorem: if the interior angle at C is γ, its exterior angle is 180° − γ = α + β. In fact, that identity gives one of the shortest proofs that the interior angles must sum to 180°.
+What if my two given angles add up to 180° or more?
Then no triangle exists with those angles. Two interior angles must leave a positive amount for the third, so α + β has to be strictly less than 180° (or π radians). The calculator flags this instead of returning zero or a negative angle, because the situation is geometrically impossible, not just a rounding issue.
+Does this work for right triangles?
Yes. A right triangle has one 90° angle, so the other two acute angles must add to 90°. Type 90 and either acute angle and the calculator returns the remaining one. For deeper right-triangle work — legs, hypotenuse, altitude, inradius — use the dedicated Right Triangle Calculator.
+Does the theorem still hold on a curved surface?
No — only on a flat plane. On a sphere the angles of a triangle sum to more than 180°, and on a saddle-shaped (hyperbolic) surface they sum to less. This calculator, and the theorem it uses, assume ordinary Euclidean geometry, which is what almost every school and engineering problem needs.