What is a Sector Calculator?
This calculator computes the arc length and area of a circular sector from radius and central angle.
How it's calculated
Arc length = 2πr × (θ/360)
Area = πr² × (θ/360)
In radians: arc = rθ, area = ½r²θ
Step-by-step example
Area of a sector with radius 6cm and central angle 60°
- Step 1 Fraction of the full circle: 60÷360 = 1/6
- Step 2 Full circle area: π×6² = 36π
- Step 3 Sector area: 36π×(1/6) ≈ 18.84 cm²
Interpretation: A sector is just a fraction of the full circle, scaled by the central angle's share of 360°.
A sector is just a slice of a full circle
The math simply takes whatever fraction the central angle represents out of the full 360°, and applies that same fraction to the circle's arc and area. A 90° slice is 1/4 of the circle, a 60° slice is 1/6 — that's the entire idea. Using radians makes it even cleaner, collapsing directly to arc = rθ.
Good to know
- A slice of pizza is literally a sector. Cut into 8 equal pieces, each slice has a 45° central angle.
- Sector area can also be written as ½ × arc length × radius — the same shape as the triangle area formula, which makes intuitive sense if you imagine the sector as an infinitely thin triangle.
- Unrolling a cone produces a sector. This relationship is used directly to calculate a cone's lateral surface area, and it's the same geometry behind designing a paper party hat.
- The angle between a clock's hour and minute hands is a classic sector problem — at 3:30, the hands are 75° apart.
- Radar and sonar detection ranges, camera fields of view, and lighthouse beam coverage are all commonly represented as sectors.
Frequently asked questions
Q. What if the central angle is 180°?
It becomes a semicircle — exactly half the circle's area.
Q. How do I find the area if I only know the arc length?
Area = ½ × arc length × radius.
Q. What is a circular segment?
A segment is a sector minus the triangle formed by its two radii — the region bounded by a chord and an arc.