Arc Length and Sector Area

Arc Length

  • Definition: Arc length (s) is the distance between two points (A and B) along the circumference of a circle.

  • Variables:

    • ss = arc length

    • rr = radius of the circle

    • θ\theta = angle in radians

Formula

The arc length ss is calculated using the formula:

s=θ∗rs = \theta * r

where θ\theta (theta) is the angle in radians.

Example 1

Given an angle of 5 radians and a radius of 12 feet:

r=12 feetr = 12 \text{ feet}
θ=5 radians\theta = 5 \text{ radians}

s=5∗12=60 feets = 5 * 12 = 60 \text{ feet}

The arc length is 60 feet.

Conceptual Understanding

  • One full circle contains 2π2\pi radians (approximately 6.28 radians).

  • Half a circle contains π\pi radians (approximately 3.14 radians).

Example 2

Given a radius of 8 cm and an angle of 150 degrees, calculate the arc length.

  • Convert degrees to radians:

θ (radians)=θ (degrees)∗π180\theta \text{ (radians)} = \theta \text{ (degrees)} * \frac{\pi}{180}

θ=150∗π180=5π6 radians\theta = 150 * \frac{\pi}{180} = \frac{5\pi}{6} \text{ radians}

  • Calculate arc length:

s=5π6∗8=40π6=20π3 cms = \frac{5\pi}{6} * 8 = \frac{40\pi}{6} = \frac{20\pi}{3} \text{ cm}

  • Decimal approximation:

s≈20.944 cms \approx 20.944 \text{ cm}

Alternative Formula (Degrees)

If the angle θ\theta is given in degrees, use the following formula:

s=θ360∗2πrs = \frac{\theta}{360} * 2\pi r

where 2πr2\pi r is the circumference of the circle.

  • Full Circle: If θ=360∘\theta = 360^\circ, then s=2πrs = 2\pi r (the entire circumference).

  • Half Circle: If θ=180∘\theta = 180^\circ, then s=πrs = \pi r (half the circumference).

  • Quarter Circle: If θ=90∘\theta = 90^\circ, then s=14∗2πr=12πrs = \frac{1}{4} * 2\pi r = \frac{1}{2} \pi r (one-quarter of the circumference).

Example using Degrees Formula

Using the previous example (radius = 8 cm, angle = 150°):

s=150360∗2π∗8=512∗16π=20π3 cms = \frac{150}{360} * 2\pi * 8 = \frac{5}{12} * 16\pi = \frac{20\pi}{3} \text{ cm}

which is approximately 20.944 cm.

Summary of Arc Length Formulas

  1. If θ\theta is in radians: s=θ∗rs = \theta * r

  2. If θ\theta is in degrees: s=θ360∗2πrs = \frac{\theta}{360} * 2\pi r

Area of a Sector

Formulas

Radians

If the angle is in radians:

A=12θr2A = \frac{1}{2} \theta r^2

Degrees

If the angle is in degrees:

A=θ360∗πr2A = \frac{\theta}{360} * \pi r^2

  • πr2\pi r^2 is the area of the entire circle.

  • θ360\frac{\theta}{360} is the fraction of the circle represented by the sector.

Degrees to Radians

To convert from radians to degrees, multiply by 2π360\frac{2\pi}{360}. So rewriting the above:

A=θ360∗πr2A = \frac{\theta}{360} * \pi r^2

Given an angle of 90° and a radius of 10 cm, find the area of the sector.

  • The area of the entire circle: πr2=π(102)=100π\pi r^2 = \pi (10^2) = 100\pi

  • The sector is 90/360 = 1/4 of the circle.

So, the area of the sector is 14∗100π=25π\frac{1}{4} * 100\pi = 25\pi

Example 1

Given an angle of 60° and a radius of 5 feet, find the area of the sector.

A=60360∗π∗(52)=16∗25π=25π6 square feetA = \frac{60}{360} * \pi * (5^2) = \frac{1}{6} * 25\pi = \frac{25\pi}{6} \text{ square feet}

Decimal approximation: approximately 13.1 square feet.

Example 2

Given an angle of 2 radians and a radius of 8 cm, calculate the area of the sector.

A=12∗2∗(82)=1∗64=64 cm2A = \frac{1}{2} * 2 * (8^2) = 1 * 64 = 64 \text{ cm}^2

The area is 64 square cm.