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Sector of a Circle

Sector of a Circle

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

 

Circle with shaded sector showing radius r and angle θ. Area = (1/2)r²θ (radians) = (θ/360)πr² (degrees)

 

 

See also

Radians, degrees, area of a sector of a circle

Key Formula

A=θ360°×πr2(degrees)orA=12 r2 θ(radians)A = \frac{\theta}{360°} \times \pi r^2 \qquad \text{(degrees)} \qquad \text{or} \qquad A = \frac{1}{2}\,r^2\,\theta \qquad \text{(radians)}
Where:
  • AA = Area of the sector
  • rr = Radius of the circle
  • θ\theta = Central angle of the sector (in degrees for the first formula, in radians for the second)

Worked Example

Problem: Find the area of a sector with a radius of 10 cm and a central angle of 90°.
Step 1: Write down the sector area formula using degrees.
A=θ360°×πr2A = \frac{\theta}{360°} \times \pi r^2
Step 2: Substitute the given values: r = 10 cm and θ = 90°.
A=90°360°×π(10)2A = \frac{90°}{360°} \times \pi (10)^2
Step 3: Simplify the angle fraction. 90 divided by 360 equals 1/4, so the sector is one-quarter of the full circle.
A=14×100πA = \frac{1}{4} \times 100\pi
Step 4: Multiply to get the final area.
A=25π≈78.54 cm2A = 25\pi \approx 78.54 \text{ cm}^2
Answer: The area of the sector is 25π≈78.5425\pi \approx 78.54 cm².

Another Example

This example uses radians instead of degrees and also computes the arc length, showing how the radian formula is often simpler when the angle is already in radians.

Problem: A sector has a radius of 6 m and a central angle of 2 radians. Find its area and the length of its arc.
Step 1: Use the radian form of the sector area formula.
A=12 r2 θA = \frac{1}{2}\,r^2\,\theta
Step 2: Substitute r = 6 m and θ = 2 radians.
A=12(6)2(2)=12(36)(2)A = \frac{1}{2}(6)^2(2) = \frac{1}{2}(36)(2)
Step 3: Compute the area.
A=36 m2A = 36 \text{ m}^2
Step 4: Now find the arc length using the arc length formula for radians.
s=r θ=6×2=12 ms = r\,\theta = 6 \times 2 = 12 \text{ m}
Answer: The sector has an area of 36 m² and an arc length of 12 m.

Frequently Asked Questions

What is the difference between a sector and a segment of a circle?
A sector is bounded by two radii and an arc — it looks like a pizza slice. A segment is bounded by a chord and the arc it cuts off — it looks like the piece you get when you slice straight across a circle. To find a segment's area, you subtract the triangle's area from the sector's area.
How do you find the arc length of a sector?
In degrees, the arc length is s=θ360°×2πrs = \frac{\theta}{360°} \times 2\pi r. In radians, it simplifies to s=rθs = r\theta. The arc length is the curved distance along the boundary of the sector, not the area.
When do you use the degree formula vs. the radian formula for a sector?
Use whichever matches the units of the angle you are given. If the problem states the angle in degrees (e.g., 60°), use θ360°×πr2\frac{\theta}{360°} \times \pi r^2. If the angle is in radians (e.g., π3\frac{\pi}{3}), use 12r2θ\frac{1}{2}r^2\theta. Both formulas give the same result when the angle is correctly converted.

Sector vs. Segment

SectorSegment
DefinitionRegion between two radii and an arcRegion between a chord and the arc it subtends
Shape"Pizza slice" — has a vertex at the center"Cut-off" piece — no vertex at the center
Area formula (degrees)θ360°πr2\frac{\theta}{360°} \pi r^2θ360°πr2−12r2sin⁡θ\frac{\theta}{360°} \pi r^2 - \frac{1}{2}r^2 \sin\theta
Boundary componentsTwo radii + one arcOne chord + one arc

Why It Matters

Sectors appear throughout geometry and real-world applications — calculating the area of a slice of pizza, the sweep of a windshield wiper, or the region covered by a radar beam all rely on the sector formula. In trigonometry and calculus, the radian-based sector area formula 12r2θ\frac{1}{2}r^2\theta is fundamental, serving as the basis for deriving the areas of polar curves. You will encounter sectors on standardized tests (SAT, ACT, GCSE) regularly, so fluency with both the degree and radian forms is essential.

Common Mistakes

Mistake: Using the degree formula when the angle is in radians (or vice versa).
Correction: Always check the units of the angle first. If in degrees, use θ360°πr2\frac{\theta}{360°} \pi r^2. If in radians, use 12r2θ\frac{1}{2}r^2\theta. Mixing units will give a wildly incorrect answer.
Mistake: Confusing sector area with arc length.
Correction: Area measures the region inside the sector (in square units), while arc length measures the curved edge (in linear units). The formulas are different: area uses πr2\pi r^2 (or r2r^2), while arc length uses 2πr2\pi r (or rr).

Related Terms