Radians: why 180 degrees equals π

TL;DR:

• One radian is the angle at the centre of a circle whose arc is as long as the radius.

• A full turn is 2π radians, so 180° = π radians and 1 rad ≈ 57.3°.

• Arc length is s = rθ only when θ is in radians, and calculus formulas for sine are cleanest in radians.

Degrees come from dividing a circle into 360 parts, a choice that goes back to old counting systems. It is a fine human choice, but the circle itself has a more natural unit. Radians measure an angle by the arc it cuts off, and that makes many formulas shorter.

The definition

Draw a circle of radius r. Mark an arc along the edge that is exactly r long, as if you bent a piece of string the length of the radius around the rim. The angle at the centre that this arc subtends is one radian. Because it is defined by a length ratio, arc divided by radius, a radian has no unit of its own and does not depend on how big the circle is.

Worked example: an arc equal to the radius

1. Take a circle of radius 5 cm. An arc of 5 cm gives an angle of 5 / 5 = 1 radian.

2. An arc of 10 cm gives 10 / 5 = 2 radians.

3. The whole circumference is 2π × 5 = 10π ≈ 31.4 cm, which gives 10π / 5 = 2π radians.

So a full turn is 2π ≈ 6.28 radians, and half a turn is π ≈ 3.14 radians. That is why 180° = π radians. The relationship s = rθ links arc length s, radius r and angle θ in radians: check with r = 5 and θ = 2, giving s = 10, matching the 10 cm arc.

Converting between degrees and radians

Use the fact that 180° = π radians. To go from degrees to radians, multiply by π / 180. To go the other way, multiply by 180 / π.

• 60° × π/180 = π/3 radians.

• 135° × π/180 = 3π/4 radians, since 135/180 = 3/4.

• 5π/6 radians × 180/π = 150°.

• One radian is 180/π ≈ 57.3°. Check: 57.3 × π / 180 ≈ 1.0.

You do not need to memorize this table if you remember π radians is half a turn. Then 90° is half of that, so π/2. And 45° is half of 90°, so π/4. And 60° is a third of 180°, so π/3.

Arc length and sector area become simple

With degrees, the length of an arc is (θ / 360) × 2πr. With radians it is just s = rθ. A pendulum of length 2 m swings through 0.3 rad, so the tip travels about 2 × 0.3 = 0.6 m. The sector area is ½r²θ, which for r = 5 and θ = 2 gives ½ × 25 × 2 = 25 square cm. Check against the full circle: θ = 2π gives ½ × 25 × 2π = 25π, which is π r². Correct.

Try two conversions yourself

Convert 7π/12 radians to degrees: 7π/12 × 180/π = 7 × 15 = 105°. Now convert 250° to radians: 250 × π/180 = 25π/18 ≈ 4.36 radians. Check the second against a full turn: 250° is a little more than half a turn, so the answer should be a little more than π ≈ 3.14, and 4.36 is. Compare with 3π/2 ≈ 4.71 (270°), which is a bit further round.

A wheel of radius 0.3 m turns through 5 radians. A point on its rim travels s = rθ = 0.3 × 5 = 1.5 m. This is how rolling distance and rotation connect: one full turn of 2π radians moves a wheel forward 0.3 × 2π ≈ 1.88 m.

Why calculus prefers radians

For small angles in radians, sin θ ≈ θ. Try θ = 0.1: sin 0.1 ≈ 0.0998, very close to 0.1. In degrees, sin 5.73° ≈ 0.0998, and the approximation would need an extra factor of π/180. Derivatives work the same way: the derivative of sin x is cos x only if x is in radians. In degrees, a factor π/180 appears everywhere. Radians remove that clutter.

How to check your answer

• Reverse the conversion. Convert your radian answer back to degrees.

• Anchor to π. Is your angle a sensible fraction of π or 2π?

• Compare sizes. One radian is a bit less than 60°, so 2 radians is a bit less than 120°.

• Calculator mode. Use radian mode for radian inputs. sin(π/2) should give 1.

Common mistakes

• Multiplying by 180/π when you needed π/180. A degrees-to-radians result should be a small number, under 6.3 for one turn.

• Treating π as 180. It is 3.14. The 180° equals π radians.

• Leaving the calculator in degree mode while entering radians.

• Using s = rθ with θ in degrees.

• Forgetting that radians are unitless, so the units of rθ are those of r.

Where you will use it next

Radians turn the angles of the unit circle into the tidy values the graphs use in sine and cosine graphs: amplitude, period and shift. In physics, angular speed is measured in radians per second for the same reason: a point on a rim moves r times that number in metres per second.

Frequently asked questions

Why is a radian about 57.3 degrees?

Because 180/π ≈ 57.3. If the arc is as long as the radius, the angle is a little less than a sixth of a full turn, since the circumference is about 6.28 radii.

Can radians be written without π?

Yes. 1 radian, 2 radians and 0.5 radians are all valid. π is only common because full turns and half turns involve it.