Area of a Sector
In short: The area of a sector is the amount of flat space inside a pie-shaped wedge of a circle, equal to the central angle’s fraction of the whole circle’s area.
GeometryGeometryG-C.B.5
What is Area of a Sector?
To find the area of a sector, the wedge between two radii and the rim, decide what fraction of the circle the wedge is, then take that fraction of the full circle area \(\pi r^2\).
With the central angle \(\theta\) in degrees, \(A = \frac{\theta}{360} \times \pi r^2\). In radians the formula becomes \(A = \frac{1}{2}r^2\theta\). The answer is measured in square units, which separates it from the curved length of the rim, measured in plain units.
How to spot it
- A problem shows a wedge, slice or fan shape with a radius and an opening angle at the center.
- A sprinkler, windshield wiper sweep, lighthouse beam or pizza slice covers part of a circle.
- The question asks for covered ground, sauce or paint, so the area of a sector in square units is needed.
- A percentage or fraction of a circle is given together with the radius.
The method
- For the area of a sector, write the fraction of the circle from the central angle: \(\frac{\theta}{360}\).
- Compute the area of the whole circle, \(\pi r^2\).
- Multiply the two and simplify, leaving \(\pi\) exact unless a decimal is requested.
- Attach square units, such as square inches, to the final number.
Area of a Sector: solved example
Problem
A pizza has radius 10 inches, and a slice cut with a \(135^{\circ}\) angle at the center is served to a customer. What is the area of the slice?
Solution
The fraction is \(\frac{135}{360} = \frac{3}{8}\). The whole pizza has area \(\pi(10)^2 = 100\pi\) square inches.
Slice area: \(\frac{3}{8} \times 100\pi = 37.5\pi \approx 117.8\). The slice covers about 117.8 square inches.
Try it
Question 1
A sprinkler on a lawn waters a wedge of radius 6 meters and central angle \(60^{\circ}\). Find the watered area in terms of \(\pi\).
Show the answer
\(\frac{60}{360} \times \pi(6)^2 = \frac{1}{6} \times 36\pi = 6\pi\) square meters, about 18.85.
Question 2
A wedge of a circle with radius 6 has area \(12\pi\). Find its central angle.
Show the answer
\(12\pi = \frac{\theta}{360} \times 36\pi\), so \(\theta = 120^{\circ}\).
Area of a Sector vs. Arc Length
Both use the same fraction \(\frac{\theta}{360}\), but they share different things. The curved distance along the rim takes that fraction of the circumference \(2\pi r\) and is measured in plain units. The wedge’s flat space takes it from the area \(\pi r^2\) and is measured in square units.
Keep going
Lessons
Worksheets and quizzes
Related words
Whole dictionary: A–Z · Geometry · Geometry