Angle formulas this calculator uses (and why they work)

An angle measures rotation between two rays that share a common endpoint (the vertex). On paper it looks like ∠ABC; in real life it looks like “how much did we turn?”. Because angles measure rotation, we can represent them in different units and connect them to shapes with reliable “sum rules”.

Degrees are the unit most people learn first. A full turn is 360°. A half turn is 180°. A right angle is 90°. Radians are the “math-native” unit used in trigonometry, calculus, physics, and engineering. The key bridge between the two systems is:

• 180° = π radians

From that single fact we get the conversion formulas:

• Radians = Degrees × (π / 180)
• Degrees = Radians × (180 / π)

Why does π show up? Because radians come from circles. One radian is the angle that “cuts off” an arc length equal to the circle’s radius. Since a circle’s circumference is 2πr, a full rotation is 2π radians. Half of that is π radians, which matches 180°. That’s why the degree↔radian formulas are just a scale conversion.

For any triangle (acute, right, or obtuse), the sum of interior angles is always 180°. That gives a super-fast “solve the missing angle” formula:

• Angle C = 180° − Angle A − Angle B

This rule is true because of parallel lines: if you draw a line through one vertex parallel to the opposite side, the triangle’s three interior angles line up as a straight angle (180°). It’s one of those “once you see it, you can’t unsee it” geometry proofs, and it’s why triangle angle problems are often pure subtraction.

These are two of the most common angle relationships:

• Complementary: two angles that add to 90° (a right corner).
• Supplementary: two angles that add to 180° (a straight line).

The math is the same shape as the triangle rule: “total minus known gives unknown.” So the formulas become:

• Complement = 90° − given angle
• Supplement = 180° − given angle

You’ll see these constantly in worksheets about parallel lines, transversals, and linear pairs. In the real world, they show up when you’re checking if two edges meet cleanly at a right corner, or if you’re designing a straight “bend” that continues in a line.

Worked examples (so you can sanity-check your answers)

Convert 45° to radians:
Radians = 45 × (π / 180) = (45/180)π = (1/4)π ≈ 0.785398.

Convert 2 radians to degrees:
Degrees = 2 × (180 / π) ≈ 114.5916°.
This is why radians feel “smaller”: 2 radians is already a pretty big turn.

If a triangle has angles 50° and 60°, the third angle is:
180 − 50 − 60 = 70°.

If one angle is 35°, its complement is:
90 − 35 = 55°.

If one angle is 120°, its supplement is:
180 − 120 = 60°.

• If you’re converting degrees→radians: 180° should become π (≈3.14159).
• If you’re converting radians→degrees: π should become 180°.
• If you’re in triangle mode: the result must be greater than 0 and less than 180.
• If you’re in complement mode: the result must be between 0 and 90.
• If you’re in supplement mode: the result must be between 0 and 180.

What the calculator does behind the scenes

The calculator is intentionally “boring” under the hood—in a good way. It follows the same steps you would do by hand, but it does them instantly, formats the result cleanly, and stops you from accidentally entering impossible values.

• Step 1: You choose a mode (conversion or “sum rule” problem).
• Step 2: The calculator checks your inputs are numbers and within a reasonable range.
• Step 3: It applies one formula (multiply by π/180, 180/π, or subtract from 90/180).
• Step 4: It rounds the result to a clean number and shows a small explanation.
• Step 5: Optional: you can save results locally and share a screenshot.

Radians and π don’t “fit” neatly into base-10 decimals, so conversions often produce long decimal expansions. This calculator shows a rounded value for readability (by default up to 6 decimals), but the underlying math is using high precision. If you need more precision for engineering or physics, copy the result and keep extra digits.

In triangle mode, the only way a triangle can exist is if the third angle is between 0 and 180 and the two known angles add up to less than 180. So if you enter 100° and 90°, the calculator will stop you because the third angle would be -10° (impossible).

Frequently Asked Questions

• What is the difference between degrees and radians?

  Degrees are based on dividing a circle into 360 parts. Radians are based on arc length: an angle in radians is the arc length divided by the radius. In practice, radians are the “natural” unit for trig functions and most advanced math.

• Why is 180° equal to π radians?

  A full circle is 2π radians because the circumference is 2πr and radians measure arc length over radius. Half a circle is π radians, which corresponds to 180°.

• Can a triangle have a 0° or 180° angle?

  Not as a real triangle. A 0° or 180° “triangle” collapses into a line (degenerate case). Most geometry problems assume a non-degenerate triangle, where all angles are greater than 0° and less than 180°.

• Do complementary angles have to be adjacent?

  No. They just need to add to 90°. They might touch (adjacent) or be far apart in a diagram. The same idea applies to supplementary angles (sum to 180°).

• What angles should I memorize?

  The most useful are: 30°, 45°, 60°, 90°, 180°, and 360°, plus their radian versions: π/6, π/4, π/3, π/2, π, and 2π. These show up constantly in trigonometry.

• Is this calculator good for trig problems?

  Yes for unit conversion and triangle-angle relationships. If you need sides, trigonometric ratios, or full triangle solving, use the Triangle Calculator and Pythagorean Theorem tools too.

How to make this “shareable” (without being cringe)

Angles are one of those “tiny math things” that show up everywhere—so the most viral use is simple: share a quick before/after screenshot of a homework problem, a DIY cut, or a triangle diagram and say: “I always forget this rule… here’s the cheat.”

• Post “Triangle sum = 180°” with a real example (people save it).
• Post “180° = π” conversions (people share it to friends in class).
• Use the Save button to keep a mini “angle history” for a project.
• Link to Interior/Exterior Angles after this page (natural next click).

If you’re building a calculator empire: angle pages are great internal-link hubs because they connect geometry (shapes), trig (radians), and proofs (triangle sums) in one place.