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Area of a Segment of a Circle — Formula & Examples

Area of a Segment of a Circle

The formula is given below.

 

Circle with shaded segment, radius r, angle θ. Area = ½r²(θ − sinθ) radians or ½r²(π/180·θ − sinθ) degrees.

 

 

See also

Segment of a circle, radians, degrees, sine

Key Formula

A=r22(θ−sin⁡θ)A = \frac{r^2}{2}\left(\theta - \sin\theta\right)
Where:
  • AA = Area of the segment
  • rr = Radius of the circle
  • θ\theta = Central angle of the sector, measured in radians

Worked Example

Problem: Find the area of a segment of a circle with radius 10 cm and a central angle of 60°.
Step 1: Convert the central angle from degrees to radians.
θ=60°×π180°=π3 radians\theta = 60° \times \frac{\pi}{180°} = \frac{\pi}{3} \text{ radians}
Step 2: Write the segment area formula and substitute the known values.
A=r22(θ−sin⁡θ)=1022(π3−sin⁡π3)A = \frac{r^2}{2}(\theta - \sin\theta) = \frac{10^2}{2}\left(\frac{\pi}{3} - \sin\frac{\pi}{3}\right)
Step 3: Evaluate the sine. Since sin(π/3) = √3/2 ≈ 0.8660, compute the expression inside the parentheses.
π3−32≈1.0472−0.8660=0.1812\frac{\pi}{3} - \frac{\sqrt{3}}{2} \approx 1.0472 - 0.8660 = 0.1812
Step 4: Multiply to find the area.
A=1002×0.1812=50×0.1812=9.06 cm2A = \frac{100}{2} \times 0.1812 = 50 \times 0.1812 = 9.06 \text{ cm}^2
Answer: The area of the segment is approximately 9.06 cm².

Another Example

This example uses the 'sector minus triangle' approach explicitly, showing the two-part logic behind the combined formula. The 90° angle also makes the triangle calculation straightforward.

Problem: Find the area of a segment of a circle with radius 8 cm and a central angle of 90°.
Step 1: Convert the central angle to radians.
θ=90°×π180°=π2 radians\theta = 90° \times \frac{\pi}{180°} = \frac{\pi}{2} \text{ radians}
Step 2: Find the area of the sector (the 'pie slice' containing the segment).
Asector=r2θ2=82⋅π22=64π4=16π≈50.27 cm2A_{\text{sector}} = \frac{r^2 \theta}{2} = \frac{8^2 \cdot \frac{\pi}{2}}{2} = \frac{64\pi}{4} = 16\pi \approx 50.27 \text{ cm}^2
Step 3: Find the area of the triangle formed by the two radii and the chord. With a 90° angle, this is a right triangle with legs equal to the radius.
Atriangle=12r2sin⁡θ=12(64)sin⁡π2=12(64)(1)=32 cm2A_{\text{triangle}} = \frac{1}{2} r^2 \sin\theta = \frac{1}{2}(64)\sin\frac{\pi}{2} = \frac{1}{2}(64)(1) = 32 \text{ cm}^2
Step 4: Subtract the triangle area from the sector area to get the segment area.
Asegment=16π−32≈50.27−32=18.27 cm2A_{\text{segment}} = 16\pi - 32 \approx 50.27 - 32 = 18.27 \text{ cm}^2
Answer: The area of the segment is approximately 18.27 cm².

Frequently Asked Questions

What is the difference between a segment and a sector of a circle?
A sector is the 'pie slice' region enclosed by two radii and an arc. A segment is the region between a chord and the arc it subtends. You can think of a segment as a sector with the central triangle removed. Consequently, the area of a segment equals the area of the sector minus the area of that triangle.
How do you find the area of a segment when the angle is given in degrees?
You can either convert the angle to radians first and then use A = (r²/2)(θ − sin θ), or use the degree-based formula: A = (πr²θ/360) − (r²/2)sin θ, where θ is in degrees. Both give the same result. Most textbooks prefer the radian version because it is more compact.
What is a major segment vs a minor segment?
A chord divides a circle into two segments. The smaller one (with the shorter arc) is the minor segment, and the larger one is the major segment. The formula A = (r²/2)(θ − sin θ) directly gives the minor segment area when θ < π radians (180°). To find the major segment, subtract the minor segment area from the total area of the circle: A_major = πr² − A_minor.

Segment of a Circle vs. Sector of a Circle

Segment of a CircleSector of a Circle
DefinitionRegion between a chord and its arcRegion between two radii and an arc ('pie slice')
Formula (radians)A = (r²/2)(θ − sin θ)A = r²θ/2
BoundaryBounded by one chord and one arcBounded by two radii and one arc
RelationshipSegment = Sector − TriangleSector = Segment + Triangle

Why It Matters

You encounter segments of circles in geometry courses when studying chord and arc relationships. The concept also appears in real-world problems — for example, calculating the cross-sectional area of water in a partially filled pipe or the area of a circular window above a horizontal frame. Mastering this formula reinforces your ability to combine sector and triangle area formulas, a skill tested frequently on standardized exams.

Common Mistakes

Mistake: Using degrees directly in the formula A = (r²/2)(θ − sin θ) without converting to radians.
Correction: The formula requires θ in radians. If your angle is in degrees, either convert it to radians first (multiply by π/180) or use the degree version: A = (πr²θ/360) − (r²/2) sin θ.
Mistake: Forgetting to subtract the triangle and just calculating the sector area.
Correction: The segment is the region between the chord and the arc, not the full 'pie slice.' You must subtract the triangular area formed by the two radii and the chord from the sector area.

Related Terms

  • Segment of a Circle — Defines the geometric region this formula measures
  • Sector — The 'pie slice' whose area is part of this calculation
  • Radian — Angle unit required by the standard formula
  • Degree — Alternative angle unit needing conversion
  • Sine — Used to compute the triangle area within the sector
  • Formula — General concept of mathematical formulas
  • Area of a Circle — Total circle area from which segments are derived
  • Chord — Line segment forming one boundary of the segment