Degrees to radians formula (and why it works)

The core conversion is beautifully simple:

• Radians = Degrees × π / 180

The reason is the circle itself. A full circle is 360°. The same full turn in radians is 2π rad. That’s not a random fact — it comes from how radians are defined. In the unit circle (a circle of radius 1), the circumference is 2π. One full revolution sweeps an arc length of 2π, so the angle measure in radians for a full revolution is 2π.

Now set the two “full circles” equal to each other as a ratio:

• 360° ↔ 2π rad

Divide both sides by 2 to get the most useful anchor:

• 180° ↔ π rad

From here, a conversion is just a scaling problem. If 180 degrees equals π radians, then 1 degree equals π/180 radians. Multiply any degree value by π/180 and you’re done. That’s exactly what the calculator does.

The opposite direction (radians to degrees) simply inverts the factor:

• Degrees = Radians × 180 / π

In trig and calculus, you often want answers written in terms of π because it keeps the relationships exact. For example, sin(π/2) = 1 is exact, while sin(1.5707963268) is the same idea but looks like a rounding approximation. This calculator can display both: a clean π fraction (when it matches a common rational multiple) and a decimal version for quick numeric work.

Step-by-step conversion examples

Use the formula radians = degrees × π / 180:

• 90° × π/180 = (90/180)π = (1/2)π = π/2

Decimal check: π/2 ≈ 1.5707963268.

45° × π/180 = (45/180)π = (1/4)π = π/4 ≈ 0.7853981634.

225° × π/180 = (225/180)π. Reduce the fraction 225/180 by dividing by 45:

• 225/180 = 5/4 → radians = 5π/4

Decimal check: 5π/4 ≈ 3.9269908170.

Switch to “Radians → Degrees” and apply the reverse formula:

• degrees = 2.5 × 180/π ≈ 2.5 × 57.295779513 = 143.2394488°

Negative angles convert the same way. For example, −30° is just −(30°): −30° × π/180 = −π/6. In physics and trig, negative angles are normal (clockwise rotations, phase shifts, angular velocity direction).

What the calculator actually does (simple + transparent)

Under the hood, this tool follows a short, reliable process. There’s no hidden trick — it’s pure unit conversion based on the definition of radians.

The calculator reads your number from the input field. You can enter integers like 90, decimals like 12.5, or even values written with π when converting the other direction (for example pi/3 or 3pi/2). For convenience, the radians input accepts pi or the symbol π.

If you’re converting degrees to radians, it multiplies by π/180. If you’re converting radians to degrees, it multiplies by 180/π. That’s it.

When “π form” is enabled (it’s on by default), the calculator tries to express the radian value as a simplified rational multiple of π, like 7π/12. This is especially useful for common angles that appear in trig tables and unit-circle problems. If your value doesn’t match a clean fraction within a sensible tolerance, the calculator still shows the accurate decimal value and tells you it’s “decimal best-effort”. You get the best of both worlds: an exact-looking expression when possible and a numeric result when needed.

Different classes and tasks want different rounding. Homework might want π form. Engineering might want 4–10 decimals. This is why there’s a precision selector. Internally, the calculator keeps full floating‑point precision and only rounds for display.

A quick conversion is often something you want to send to a friend, paste into a note, or keep for later. The Save button stores your latest conversions in your browser’s local storage (on this device only), and the share buttons generate a ready-to-post message — perfect for group chats, study sessions, and “help me check this” moments.

Frequently Asked Questions

• What is a radian, in plain English?

  A radian measures an angle by comparing it to the radius of a circle. If you take a circle and mark off an arc whose length equals the radius, the angle that arc makes at the center is 1 radian. Because the circumference of a unit circle is 2π, one full turn is 2π radians.

• Why do math and physics prefer radians?

  Many formulas become cleaner in radians. Calculus is the big one: derivatives like d/dx(sin x) = cos x are “naturally” true when x is in radians. In physics, angular velocity, oscillations, and waves often use radians because they map directly to circle geometry and periodic motion.

• Is π form always exact?

  π form is exact when your angle is a rational fraction of 180°. For example, 30° gives π/6 exactly. But if you type a value like 17°, the “best” π fraction depends on whether you want an approximation (like 17π/180) or a decimal. This tool will show a simplified π expression when it’s clean and stable, and it always shows the decimal value as well.

• How many decimals should I use?

  For typical homework, 4–6 decimals is plenty if a decimal is required. For numerical simulation, you might keep 10+ decimals. If your teacher asks for exact answers, use π form.

• Can I enter π directly for radians?

  Yes. In “Radians → Degrees” mode, you can enter values like pi, π, pi/2, 3pi/4, or 2π. The calculator parses common patterns and converts them.

• What about grads (gons) or turns?

  Those are other angle units (400 grads per full circle, or 1 turn per full circle). This calculator focuses on the most common academic and practical conversion: degrees ↔ radians. If you want, you can add a “turns” converter later as a separate tool for SEO coverage.