Arc Length and Sectors
An arc and a sector are fractions of the whole circle.
About this topic
An arc is a portion of a circle's circumference and a sector is a slice bounded by two radii and an arc. If the central angle is θ degrees, the arc length is (θ/360) × 2πr and the sector area is (θ/360) × πr².
In radians the formulas simplify to s = rθ and A = ½r²θ. They are proportional parts of the whole circle, so you can think of them as fractions, such as a quarter circle being 90/360 of the whole.
Worked example: a sector has radius 6 and a central angle of 60°. The arc length is (60/360) × 2π(6) = 2π ≈ 6.28, and the sector area is (60/360) × π(36) = 6π ≈ 18.85. In radians, 60° is π/3, and s = 6 × π/3 = 2π, the same result. A slice of pie cut at 45° from a 12 inch pie takes one eighth of the area.
Windshield wipers, pie charts, speedometer dials and radar screens use arcs and sectors. A typical mistake is mixing arc length (a length) with sector area (square units), or using the diameter instead of the radius. Make sure the calculator is in the right mode: degrees for the first formulas and radians for s = rθ. Always set up the fraction of the circle first. Check by asking whether the answer is a sensible part of the whole circumference or area.
Key ideas
- Arc = (θ/360) × 2πr
- Sector area = (θ/360) × πr²
- Radians make it simpler
Quick check
Use cases: where Arc Length and Sectors helps
- Homework: review the key idea behind Arc Length and Sectors before you start a problem set.
- Exam revision: use the quick check to test whether you can recall Arc Length and Sectors without notes.
- Visual learning: play with the interactive lab to see how Arc Length and Sectors changes when you adjust its inputs.
- Teaching a friend: explain Arc Length and Sectors in your own words, then compare your version with the key ideas.
Number and algebra skills are used in science, engineering, budgeting and computing.
Frequently asked questions
What is Arc Length and Sectors?
An arc and a sector are fractions of the whole circle.
What will I learn in Arc Length and Sectors?
Arc = (θ/360) × 2πr. Sector area = (θ/360) × πr². Radians make it simpler.
Is Arc Length and Sectors part of Mathematics?
Yes. It is one of 200 Mathematics topics on LearnViz HighSchoolEdu, with 3 key ideas, a live animation and a quick check.
Does the Arc Length and Sectors lab work on a phone or tablet?
Yes. The animation resizes to fit your screen and supports touch, so you can drag, tap and slide with your finger.
How should I study Arc Length and Sectors?
Read the summary and key ideas first, play with the interactive lab to see the idea in motion, then answer the quick check. If you get a question wrong, go back to the key idea it tests.
Do I need an account, and is my progress saved?
No account is needed. Marking a topic as learned is saved in your own browser, so it stays on that device and is not sent to a server.