Degrees to Radians Converter

degrees to radians — quick reference

The formula, worked line by line

Begin with the defining equivalence π rad = 180°. Dividing both sides by 180 gives π/180 rad = 1°. That quotient is the degrees-to-radians factor, so it is derived from the definition rather than memorised as an unrelated constant.

Multiply the degree number by π/180 and cancel the degree unit. If the input is rational, reduce the numerical fraction before attaching π. The exact answer is then a rational multiple of π, while any decimal obtained by substituting digits for π must be labelled approximate.

The converter evaluates π numerically and displays four decimal places. That display rule does not change the underlying equality: 90° = π/2 rad is exact, whereas 90° ≈ 1.5708 rad uses an approximation sign because the decimal has been rounded.

π rad = 180°
1° = π/180 rad
radians = degrees × π ÷ 180
90° → 90 × π ÷ 180 = π/2 rad ≈ 1.5708 rad

One equivalence, both conversion directions

Dividing π rad = 180° by 180 produced π/180 radians per degree. Dividing the same equivalence by π instead produces 180/π degrees per radian. The two factors are reciprocals, so there is no need to memorise them as separate facts.

This derivation resolves three common mix-ups at once. One radian is not 180°; it is approximately 57.2958°. A full 360° turn is not one radian; it is 2π rad. And radians to degrees requires multiplication by 180/π, never by 180π.

A useful magnitude check is that 180° maps to about 3.1416 rad and 360° maps to about 6.2832 rad. If a positive angle smaller than a full turn produces hundreds of radians, the factor was probably inverted or π was moved to the wrong side.

How to use the degrees to radians converter

Enter an angle in degrees and the radian value appears to four decimal places. The page opens at 90°, giving approximately 1.5708 rad. For handwritten mathematics, keep the stronger answer beside it: 90° is exactly π/2 radians, while 1.5708 is only a rounded decimal representation.

There is only one relationship to remember: π rad = 180°. Divide both sides by 180 and the conversion factor follows as 1° = π/180 rad. Multiplying any degree value by π/180 therefore replaces the degree unit with radians without introducing measurement uncertainty.

The symbolic and decimal answers do different jobs. A reduced multiple of π preserves the exact relationship, making cancellation and trigonometric identities easier to see. A decimal is convenient for software, measurements and numerical formulas, but every finite decimal involving π is an approximation because π is irrational.

This direction has an especially useful pattern. A rational degree value becomes a rational multiple of π radians, so familiar angles often look cleaner: 30° is π/6 rad exactly and approximately 0.5236 rad; 45° is π/4 and approximately 0.7854; 60° is π/3 and approximately 1.0472.

Calculator mode matters after the conversion. If a formula expects radians, enter the radian result and use RAD mode. Feeding 90 into a sine key while the calculator is in RAD mode means 90 radians, not 90°, so the familiar right-angle result will disappear.

For geometry beyond a unit change, the Circle Calculator connects angles to circumference, arcs and sectors, while the Triangle Calculator handles side-and-angle relationships. Use those pages when the angle is part of a shape problem rather than an isolated conversion.

Why degree answers keep a clean π form

Suppose the degree input is rational. Multiplying by π/180 leaves a rational coefficient attached to π. Reduction produces familiar exact forms and their decimal approximations: π/6 is about 0.5236, π/4 about 0.7854, π/3 about 1.0472 and π/2 about 1.5708 rad. Nothing has to be guessed.

The reduction is ordinary fraction arithmetic. For 270°, write 270 × π/180, cancel the common factor of 90, and obtain 3π/2 rad exactly, approximately 4.7124 rad. Stopping at 270π/180 is equivalent, but it hides the three half-turns contained in the angle and misses the standard form.

A negative angle follows the same rule. For example, −90° becomes −π/2 rad exactly and approximately −1.5708 rad. The minus sign records direction around the circle; it does not alter the conversion factor or the need to distinguish the exact π expression from its decimal.

Angles beyond one turn also convert without a new rule. A 540° angle is exactly 3π rad and approximately 9.4248 rad because 540/180 reduces to 3. Whether it is later treated as coterminal with 180° depends on the problem, but conversion should preserve the turns.

Why technical formulas expect radians

A radian describes an angle through the ratio of arc length to radius. Because both lengths use the same unit, their units cancel, making the radian dimensionless in SI. That ratio is why radians fit naturally into formulas involving circles, rotation, waves and angular velocity.

Calculus also singles out radians. The small-angle relationship sin θ ≈ θ works directly only when θ is measured in radians. In degrees, an extra factor of π/180 remains, so silently sending degrees into a formula derived for radians changes the mathematics rather than merely changing the label.

In code, many standard trigonometric functions take radian arguments even when the interface displaying an angle uses degrees. Convert at the boundary, retain enough numerical precision internally, and round only the final displayed value. That sequence avoids magnifying a premature four-place rounding through later calculations.

Every degrees value, worked out

119 common degrees figures each get their own page, with the answer at full precision, the arithmetic, what rounding costs, and the nearest real-world reference point on the scale.

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