Radians

One radian

Take a circle with radius r. Starting from one point on the circle, measure a length r along the circumference, and join both ends of that arc to the center. The angle between the two radii is one radian.

So a radian is the angle at the center whose arc is exactly as long as the radius. That holds for every circle: double the radius and the arc doubles with it, so the angle stays the same.

An angle of one radian at the center. The arc it cuts off is the same length as the radius.

An arc one radius long, laid round the circle: an angle of 1 radian. Drag its end round to wrap more radius lengths, colored in turn, and stop halfway round the circle.

$\pi$π radians make half a turn

The circumference of a circle is $C=2\pi r$C = 2πr, so half the circumference is $\pi r$πr. Each radian uses up r of that arc, so the number of radians in half a turn is $\pi r/r=\pi$πr / r = π. Half a turn is 180°, so $180^{\circ }=\pi$180° = π radians, and a full turn is $2\pi$2π radians.

Since $\pi$π is about 3.14, a little more than three radians fit in half a turn. One radian is $180^{\circ }÷\pi =57.2958{\dots }^{\circ }$180° ÷ π = 57.2958…°, which is 57.3° to 1 decimal place.

A straight angle, half a turn: 180°, or $\pi$π radians.

Converting between degrees and radians

Since 180° is $\pi$π radians, 1° is $\frac{\pi }{180}$π/180 radians. To change degrees into radians, multiply by $\frac{\pi }{180}$π/180. For 135°, that gives $\frac{135\pi }{180}$135π/180, and $\frac{135}{180}$135/180 simplifies to $\frac{3}{4}$3/4, so 135° is $\frac{3\pi }{4}$3π/4 radians. In the same way 270° is $\frac{270\pi }{180}=\frac{3\pi }{2}$270π/180 = 3π/2 radians, and 20° is $\frac{20\pi }{180}=\frac{\pi }{9}$20π/180 = π/9 radians.

To change radians into degrees, multiply by $\frac{180}{\pi }$180/π. For $\frac{5\pi }{6}$5π/6 radians, the $\pi$π cancels: $\frac{5}{6}\times 180^{\circ }=150^{\circ }$5/6 × 180° = 150°. An angle in radians does not have to contain $\pi$π: 2 radians is $2\times \frac{180^{\circ }}{\pi }$2 × 180°/π, which is 114.6° to 1 decimal place.

Check a conversion against an angle you know. $\frac{\pi }{2}$π/2 should be a right angle, and 180 ÷ 2 = 90. An angle written without a degree sign is in radians.

Arc length: $s=r\theta$s = rθ

Write $\theta$θ for an angle in radians. Each radian of the angle cuts off an arc of length r, so $\theta$θ radians cut off an arc of length $r\times \theta$r × θ. The arc length is $s=r\theta$s = rθ.

Take a circle of radius 5 cm and an angle of 1.2 radians. The arc is s = 5 × 1.2 = 6 cm. Check it in degrees: 1.2 radians is $1.2\times \frac{180^{\circ }}{\pi }$1.2 × 180°/π, about 68.75°, and $\frac{68.75}{360}$68.75/360 of the circumference $2\pi \times 5$2π × 5 is 6 cm as well.

Turned around, $\theta =s/r$θ = s / r: an angle in radians is the arc length divided by the radius. Both are lengths, so the angle is a plain number. This is why calculus uses radians: with x in radians, the derivative of sin x is exactly cos x, and in degrees an extra factor of $\frac{\pi }{180}$π/180 appears.

A sector of radius 5 cm with an angle of 1.2 radians, about 68.75°. Its arc is 5 × 1.2 = 6 cm long.

The usual mistakes

Taking $\pi$π radians as a full turn. $\pi$π radians is half a turn, 180°, so 90° is $\frac{\pi }{2}$π/2 radians, not $\frac{\pi }{4}$π/4.

Dividing by 360 instead of 180. 60° is $\frac{60}{180}=\frac{1}{3}$60/180 = 1/3 of $\pi$π, which is $\frac{\pi }{3}$π/3. Dividing by 360 gives $\frac{1}{6}$1/6, the fraction of a full turn, and $\frac{\pi }{6}$π/6 is only 30°.

Putting degrees into $s=r\theta$s = rθ. For a 68.75° angle in a circle of radius 5 cm, 5 × 68.75 would give an arc of 343.75 cm, longer than the whole circumference. The formula needs the angle in radians, 1.2.

Leaving a calculator in degree mode. With x in radians, $\sin \left(\frac{\pi }{6}\right)$sin(π/6) is 0.5; a calculator set to degrees works out sin of 0.524°, which is about 0.009.

A slice of pizza

In the application below, a slice of pizza has a crust 10 cm long on a radius of 15 cm. The angle in radians is the arc divided by the radius, and the area of the slice uses the sector formula of the next lesson.