Key takeaway: The area of a sector is the fraction of the full circle’s area that the sector covers, found by multiplying the central angle over 360 by pi times the radius squared.
Grade 10 – GeometryG-C.B.5Geometry and Measurement
Definition of Area of a Sector
The area of a sector is the flat space inside a wedge cut from a circle by two radii, like a slice of pizza or a piece of a pie chart.
A wedge covering \(\frac{\theta}{360}\) of the turn covers the same fraction of the circle’s area, \(\pi r^2\). So the area is that fraction of the whole disk. The same idea gives arc length from the circumference. Square the radius, not the whole product, and report the answer in square units.
Formula and notation
The area of a sector with radius \(r\) and central angle \(\theta\) in degrees is:
\[ A = \frac{\theta}{360}\cdot \pi r^2 \]
In radians the formula becomes \(A = \frac{1}{2} r^2 \theta\). Units are squared: ft², cm², and so on.
Key facts
- A 180 degree sector is half the disk, and 90 degrees is a quarter.
- Doubling the radius makes the area four times as large.
- The result uses square units, unlike arc length.
- The same fraction of the turn applies to both the area and the rim length.
Area of a Sector example
A lawn sprinkler waters a sector of radius 24 ft while it sweeps through \(110^{\circ}\). What area of grass does it cover on each sweep?
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Area of the full circle: \(\pi(24)^2 = 576\pi\) ft².
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Fraction of the turn: \(\frac{110}{360}\).
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Multiply: \(\frac{110}{360}\cdot 576\pi = 176\pi \approx 552.9\) ft².
The sprinkler covers \(176\pi \approx 552.9\) ft² of lawn.
Common mistake
Watch out
Vera squares the whole product and writes \((\frac{110}{360}\cdot 24)^2\pi\), getting about 169 ft².
Why it is wrong
Only the radius is squared when finding the area of a sector. The fraction \(\frac{110}{360}\) multiplies \(\pi r^2\) once. Her answer is far below the right value of 552.9 ft², and a quick estimate shows why: a bit under a third of a 1,810 ft² circle is about 550 ft².
Related terms and lessons
Browse the glossary: A–Z list · Geometry vocabulary · Geometry and Measurement