Arc Length and Sector Area

Arc length

An angle of $\theta$θ radians at the center of a circle of radius r cuts off an arc of length $s=r\theta$s = rθ, because each radian cuts off one radius of arc. On a circle of radius 6, an angle of 2 radians cuts off an arc of 6 × 2 = 12.

Check it in degrees. 2 radians is $2\times \frac{180^{\circ }}{\pi }$2 × 180°/π, which is 114.59° to 2 decimal places. That is $\frac{114.59}{360}$114.59/360 of the full turn, and the circumference is $2\pi \times 6$2π × 6, about 37.70, so the arc is 0.3183 × 37.70 = 12.0.

An angle of 2 radians: the arc is two radius lengths, s = 2r, one colored length for each radian. For a radius of 6 that is 12. Drag the end round the circle: the number of radius lengths in the arc is the angle in radians.

The sector as a fraction of the circle

A sector is the slice of a circle between two radii and the arc that joins their ends. A full turn is $2\pi$2π radians, so a sector with an angle of $\theta$θ radians is $\frac{\theta }{2\pi }$θ/2π of the whole circle.

The arc length fits this: $\frac{\theta }{2\pi }$θ/2π of the circumference $2\pi r$2πr is $r\theta$rθ, as before. The sector with an angle of 2 is $\frac{2}{2\pi }=\frac{1}{\pi }$2/2π = 1/π of the circle, which is about 0.318, a little less than a third.

A sector of a circle of radius 6 with an angle of 2 radians. Its arc is 12, and the sector is $\frac{1}{\pi }$1/π of the circle, about 0.318.

Sector area: $\frac{1}{2}{r}^2\theta$½r²θ

The area of the whole circle is $\pi r^2$πr². The sector is $\frac{\theta }{2\pi }$θ/2π of the circle, so its area is $\frac{\theta }{2\pi }\times \pi r^2$θ/2π × πr². The $\pi$π on top cancels with the $\pi$π below, leaving $\frac{\theta r^2}{2}$θr²/2. So the area of a sector is $\frac{1}{2}{r}^2\theta$½r²θ, with $\theta$θ in radians.

With r = 6 and $\theta =2$θ = 2, the area is $\frac{1}{2}\times 6^2\times 2=\frac{1}{2}\times 36\times 2=36$½ × 6² × 2 = ½ × 36 × 2 = 36. Square the radius first, then multiply. No $\pi$π appears in the answer, because the angle was a plain number of radians.

Check it in degrees: $\frac{114.59}{360}$114.59/360 of the full area $\pi \times 36$π × 36, about 113.10, is 0.3183 × 113.10 = 36.0.

The formula can also be read as $\frac{1}{2}\times r\times r\theta$½ × r × rθ, which is ½ × radius × arc: here ½ × 6 × 12 = 36. A very thin sector is close to a triangle whose base is the arc and whose height is the radius, and a triangle's area is half its base times its height.

A sector of radius 3 with an angle of 40°, which is $\frac{2\pi }{9}$2π/9 radians. Its area is $\frac{1}{2}\times 9\times \frac{2\pi }{9}=\pi$½ × 9 × 2π/9 = π, about 3.142, and the triangle inside it has area ½ × 9 × sin 40°, about 2.893. Shrink the angle: the triangle closes in on the sector. Then switch to the rectangle $r\times r\theta$r × rθ: it is exactly twice the sector at every angle, which is what leaving out the ½ measures.

Exact answers, and working back

When the angle is a fraction of $\pi$π, the answers are exact multiples of $\pi$π. For a sector of radius 10 cm with an angle of $\frac{\pi }{5}$π/5, the arc is $10\times \frac{\pi }{5}=2\pi$10 × π/5 = 2π cm, about 6.28 cm, and the area is $\frac{1}{2}\times 100\times \frac{\pi }{5}=10\pi$½ × 100 × π/5 = 10π cm², about 31.42 cm². In degrees, $\frac{\pi }{5}$π/5 is 36°, and $\frac{36}{360}$36/360 of $\pi \times 100$π × 100 is $10\pi$10π again.

The formulas also run backwards. A sector of a circle of radius 4 has an area of 20. Then $\frac{1}{2}\times 16\times \theta =20$½ × 16 × θ = 20, so $8\theta =20$8θ = 20 and $\theta =2.5$θ = 2.5 radians, about 143.2°. Its arc is 4 × 2.5 = 10, and its perimeter, the arc and two radii, is 10 + 4 + 4 = 18.

The usual mistakes

Leaving out the ½. With r = 6 and $\theta =2$θ = 2, $r^2\theta =72$r²θ = 72 is twice the area. The fraction $\frac{\theta }{2\pi }$θ/2π keeps a 2 in the denominator after the $\pi$π cancels.

Mixing up the arc and the area. $r\theta =12$rθ = 12 is the length of the curved edge; $\frac{1}{2}{r}^2\theta =36$½r²θ = 36 is the area inside.

Putting degrees into $\frac{1}{2}{r}^2\theta$½r²θ. With 114.59 in place of 2, the formula gives about 2063, more than eighteen times the area of the whole circle. The formula needs radians.

Putting radians into the degree formula. $\frac{\theta }{360}\times \pi r^2$θ/360 × πr² with $\theta =2$θ = 2 gives about 0.63, far too small. That formula needs the angle in degrees, 114.59°.

A slice of pizza and a pendulum

In the first application below, the crust and the radius of a slice of pizza give its angle in radians, $\theta =s/r$θ = s / r, and then its area, $\frac{1}{2}{r}^2\theta$½r²θ. In the second, a pendulum swings through an angle given in radians, and the arc of the swing is $r\theta$rθ.