Segment of a Circle

Segment of a Circle

A part of the interior of a circle bounded by a chord and an arc.

Shaded circular segment with radius r and central angle θ. Area = ½r²(θ - sinθ) radians or ½r²(π/180·θ - sinθ) degrees.

Key Formula

A = \frac{r^2}{2}\left(\theta - \sin\theta\right)

Where:

$A$ = Area of the segment
$r$ = Radius of the circle
$\theta$ = Central angle subtended by the arc, measured in radians

Worked Example

Problem: A circle has radius 10 cm. A chord subtends a central angle of 90° (π/2 radians). Find the area of the minor segment.

Step 1: Convert the central angle to radians if needed. Here 90° = π/2 radians.

\theta = \frac{\pi}{2} \approx 1.5708

Step 2: Write down the segment area formula and substitute the known values.

A = \frac{r^2}{2}\left(\theta - \sin\theta\right) = \frac{10^2}{2}\left(\frac{\pi}{2} - \sin\frac{\pi}{2}\right)

Step 3: Evaluate sin(π/2) = 1.

A = \frac{100}{2}\left(\frac{\pi}{2} - 1\right) = 50\left(\frac{\pi}{2} - 1\right)

Step 4: Compute the numerical value.

A = 50(1.5708 - 1) = 50 \times 0.5708 \approx 28.54 \text{ cm}^2

Answer: The area of the minor segment is approximately 28.54 cm².

Another Example

This example uses a different central angle (120° instead of 90°) and a different radius, reinforcing the same formula while showing that sin θ is not always a convenient value like 1.

Problem: A circle has radius 6 cm. A chord subtends a central angle of 120° at the centre. Find the area of the minor segment.

Step 1: Convert 120° to radians.

\theta = 120° \times \frac{\pi}{180°} = \frac{2\pi}{3} \approx 2.0944

Step 2: Substitute r = 6 and θ = 2π/3 into the formula.

A = \frac{6^2}{2}\left(\frac{2\pi}{3} - \sin\frac{2\pi}{3}\right) = 18\left(\frac{2\pi}{3} - \sin 120°\right)

Step 3: Evaluate sin 120° = √3/2 ≈ 0.8660.

A = 18\left(2.0944 - 0.8660\right) = 18 \times 1.2284

Step 4: Compute the final value.

A \approx 22.11 \text{ cm}^2

Answer: The area of the minor segment is approximately 22.11 cm².

Frequently Asked Questions

What is the difference between a segment and a sector of a circle?

A sector is the 'pie-slice' region bounded by two radii and an arc. A segment is the region between a chord and its arc — essentially a sector with the triangle removed. You can find the segment area by subtracting the triangle area from the sector area, which is exactly what the formula A = (r²/2)(θ − sin θ) does.

How do you find the area of a major segment?

First find the area of the minor segment using the formula with the minor central angle θ. Then subtract it from the total area of the circle: Major segment area = πr² − minor segment area. Alternatively, use the formula directly with the reflex central angle (2π − θ) in place of θ.

Does the segment formula work when the angle is in degrees?

The formula A = (r²/2)(θ − sin θ) requires θ in radians. If your angle is in degrees, either convert to radians first (multiply by π/180) or use the degree-based version: A = (r²/2)(πα/180 − sin α), where α is the angle in degrees.

Segment of a Circle vs. Sector of a Circle

	Segment of a Circle	Sector of a Circle
Definition	Region between a chord and its arc	Region between two radii and the arc they enclose
Boundary	One chord + one arc	Two radii + one arc
Area formula (radians)	A = (r²/2)(θ − sin θ)	A = (r²/2)θ
Relationship	Segment = Sector − Triangle	Sector = Segment + Triangle
Shape analogy	Like a "cap" cut by a straight line	Like a "pie slice"

Why It Matters

Segments of circles appear in many geometry and trigonometry problems, especially when calculating areas of overlapping circles or cross-sections of cylinders. You will encounter them in standardized tests (SAT, ACT, GCSE, and IB) as well as in real-world contexts like finding the cross-sectional area of water in a partially filled pipe. Mastering the segment area formula also strengthens your understanding of how radians, the sine function, and area formulas work together.

Common Mistakes

Mistake: Using degrees directly in the formula A = (r²/2)(θ − sin θ)

Correction: The θ in (θ − sin θ) must be in radians. If you plug in degrees, the subtraction θ − sin θ is dimensionally inconsistent and gives a wildly incorrect answer. Always convert degrees to radians first.

Mistake: Confusing a segment with a sector

Correction: A sector includes the triangular region formed by the two radii; a segment does not. If a problem asks for the segment, you need to subtract the triangle area from the sector area. The formula A = (r²/2)(θ − sin θ) already accounts for this subtraction.

Related Terms

Circle — The shape that contains the segment
Chord — Straight-line boundary of the segment
Arc of a Circle — Curved boundary of the segment
Area of a Segment of a Circle — Formula and derivation for segment area
Radian — Angle unit required in the segment formula
Sine — Trig function used in the area formula
Interior — A segment is part of a circle's interior
Degree — Alternative angle unit (must convert to radians)