Arcs and Sectors: Step-by-Step Guide
An arc is a piece of the edge of a circle. A sector is a slice of the circle, like a slice of pizza, with two radii and an arc as its edges. Both are a fraction of the whole circle. The fraction depends on the central angle (the angle at the center): arc length = (angle / 360) x 2 x pi x r, and sector area = (angle / 360) x pi x r^2.
Math Racer skill guide · Geometry · Grade 10
How to solve it, step by step
- Find the radius and the central angle. Write down the radius r and the central angle in degrees.\[r,\;\theta \;(\text{in degrees})\]
- Write the fraction of the circle. The slice is the angle divided by 360 of the whole circle.\[\text{fraction} = \frac{\theta}{360^\circ}\]
- Find the arc length. Multiply that fraction by the circumference of the full circle, 2 x pi x r.\[L = \frac{\theta}{360^\circ} \cdot 2\pi r\]
- Find the sector area. Multiply that same fraction by the area of the full circle, pi x r^2.\[A = \frac{\theta}{360^\circ} \cdot \pi r^2\]
- Write the units. Arc length uses regular units (like cm). Sector area uses square units (like square cm).\[L:\;\text{units},\quad A:\;\text{units}^2\]
Worked example
A circle has radius 12 cm. A sector has a central angle of 60 degrees. Find the arc length and sector area.
Step 1: The radius is 12 cm and the central angle is 60 degrees.
Step 2: The fraction of the circle is 60 / 360 = 1/6.
Step 3: Arc length = (1/6) x 2 x pi x 12 = (1/6) x 24 x pi = 4 x pi cm, which is about 12.57 cm.
Step 4: Sector area = (1/6) x pi x 12^2 = (1/6) x 144 x pi = 24 x pi square cm, which is about 75.40 square cm.
Check: 60 degrees is one sixth of 360 degrees. One sixth of the full circumference (24 x pi) is 4 x pi, and one sixth of the full area (144 x pi) is 24 x pi. It matches.
Tips and common mistakes
- If the angle is in radians, the formulas are simpler: arc length = r x angle, and sector area = (1/2) x r^2 x angle.
- To change degrees to radians, multiply by pi / 180.
- Arc length is a length and sector area is an area. Do not mix up their units.
- Simplify the fraction first. 60/360 = 1/6 is easier to work with.
- A half circle has an angle of 180 degrees, so its fraction is 1/2.
- A quarter circle has an angle of 90 degrees, so its fraction is 1/4.
- Leave your answer in terms of pi unless the problem asks for a decimal.
- An arc is curved. Do not mix it up with the straight line (a chord) connecting its two endpoints.