Arc length and sector area
Arc Length and Sector Area
Arc Length and Sector Area is a topic in Circles in the Common Core State Standards. It is aligned to Standard G-C.5, which requires students to derive the fact that the length of the arc intercepted by an angle is proportional to the radius, and to find arc lengths and areas of sectors.
Arc length is \(\dfrac{\theta}{360^\circ}\cdot 2\pi r\) and sector area is \(\dfrac{\theta}{360^\circ}\cdot \pi r^2\), each a fraction of the whole circle.
Theory
A sector is a wedge of a circle between two radii, and an arc is the curved part of its boundary. Both are the same fraction \(\dfrac{\theta}{360}\) of the whole circle, where \(\theta\) is the central angle in degrees:
The two formulas:
How to find an arc or sector
- Write the fraction \(\dfrac{\theta}{360}\).
- Arc: multiply by the circumference \(2\pi r\).
- Sector: multiply by the area \(\pi r^2\).
- Solve for \(\theta\) or \(r\) if that is the unknown.
Arc length is the fraction \(\dfrac{\theta}{360}\) of the circumference.
| \(\dfrac{90}{360}\cdot 2\pi(6)\) | \(=\) | \(\dfrac{1}{4}\cdot 12\pi=3\pi\) |
Sector area is \(\dfrac{\theta}{360}\) of the circle's area.
| \(\dfrac{90}{360}\cdot\pi(6)^2\) | \(=\) | \(\dfrac{1}{4}\cdot 36\pi=9\pi\) |
Solve \(\dfrac{\theta}{360}\cdot\pi(10)^2=15\pi\).
| \(\dfrac{\theta}{360}\cdot 100\pi\) | \(=\) | \(15\pi\) |
| \(\dfrac{\theta}{360}\) | \(=\) | \(\dfrac{15}{100}\) |
| \(\theta\) | \(=\) | \(54^\circ\) |
Use arc \(=\dfrac{\theta}{360}\cdot 2\pi r\).
| \(\dfrac{60}{360}\cdot 2\pi r\) | \(=\) | \(4\pi\) |
| \(\dfrac{\pi r}{3}\) | \(=\) | \(4\pi\) |
| \(r\) | \(=\) | \(12\) |
Common pitfalls
Frequently asked questions
How do you find arc length?
Multiply the fraction \(\dfrac{\theta}{360}\) by the circumference \(2\pi r\).
How do you find the area of a sector?
Multiply the fraction \(\dfrac{\theta}{360}\) by the circle's area \(\pi r^2\).
How do you find the central angle from a sector?
Set the sector formula equal to the given value and solve for \(\theta\).
What is a sector?
A wedge-shaped region of a circle bounded by two radii and an arc.