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Area of A Sector: Definition and Examples | EDU.COMEDU.COMResourcesBlogGuidePodcastPlanBackHomesvg]:size-3.5">Math Glossarysvg]:size-3.5">Area of A SectorArea of A Sector: Definition and ExamplesTable of ContentsArea of a Sector of a Circle Definition of Sector Area

A sector of a circle is the region enclosed by an arc and two radii of a circle. It represents a portion of the circle's area and is formed by two radii and an arc. The area of a sector is measured in square units, depending on the unit of the radius.

The area of a sector can be calculated using two different formulas depending on how the central angle is expressed. When the angle θ is measured in degrees, the formula is θ360∘×πr2\frac{θ}{360^{\circ}} \times \pi r^{2}360∘θ​×πr2. When the angle θ is measured in radians, the formula becomes θ2×r2\frac{θ}{2} \times r^{2}2θ​×r2, where r is the radius of the circle in both formulas.

Examples of Area of a Sector Example 1: Finding the Area of a Sector with Angle in Degrees Problem:

Find the area of a sector with a central angle of 45°45°45° and a radius of 555 units.

Finding the Area of a Sector with Angle in Degrees

Step-by-step solution:

Step 1, Write down the given values. We have radius r=5r = 5r=5 units and angle θ=45∘θ = 45^{\circ}θ=45∘.

Step 2, Choose the correct formula. Since our angle is in degrees, we'll use the formula: Area =(θ360∘)×πr2= (\frac{θ}{360^{\circ}}) \times \pi r^{2}=(360∘θ​)×πr2.

Step 3, Substitute the values into the formula.

Area =(45∘360∘)×π(5)2= (\frac{45^{\circ}}{360^{\circ}}) \times \pi (5)^{2}=(360∘45∘​)×π(5)2

Step 4, Simplify the fraction.

Area =18×3.14×25= \frac{1}{8} \times 3.14 \times 25=81​×3.14×25

Step 5, Calculate the final answer.

Area =9.8125= 9.8125=9.8125 square units

Example 2: Finding the Area of a Sector with Angle in Radians Problem:

Find the area of a sector with a central angle of π4\frac{\pi}{4}4π​ radians and a radius of 777 units. (Use π=3.14\pi = 3.14π=3.14).

Finding the Area of a Sector with Angle in Radians

Step-by-step solution:

Step 1, Identify the given values. We have radius r=7r = 7r=7 units and angle θ=π4θ = \frac{\pi}{4}θ=4π​ radians.

Step 2, Since our angle is in radians, we'll use the formula: Area =(θ2)×r2= (\frac{θ}{2}) \times r^{2}=(2θ​)×r2.

Step 3, Substitute the values into the formula.

Area =(π4×2)×(7)2= (\frac{\pi}{4 \times 2}) \times (7)^{2}=(4×2π​)×(7)2

Step 4, Simplify the expression.

Area =(3.144×2)×(7)2= (\frac{3.14}{4 \times 2}) \times (7)^{2}=(4×23.14​)×(7)2

Area =(3.148)×49= (\frac{3.14}{8}) \times 49=(83.14​)×49

Step 5, Calculate the final result.

Area = 19.232519.232519.2325 square units

Example 3: Finding the Central Angle from Area and Radius Problem:

A sector has an area of 16π16π16π square units and a radius of 444 units. Find the central angle of the sector in radians.

Step-by-step solution:

Step 1, List what we know. We have radius r=4r = 4r=4 units and area of sector =16π= 16π=16π square units.

Step 2, We need to find the angle in radians, so we'll use the formula: Area =(θ2)×r2= (\frac{θ}{2}) \times r^{2}=(2θ​)×r2 and solve for θθθ.

Step 3, Substitute the known values into the formula.

16π=(θ2)×(4)216π = (\frac{θ}{2}) \times (4)^{2}16π=(2θ​)×(4)2

Step 4, Simplify the equation.

16π=(θ2)×1616π = (\frac{θ}{2}) \times 1616π=(2θ​)×16

Step 5, Solve for θθθ.

θ2=π\frac{θ}{2} = π2θ​=π

θ=2πθ = 2πθ=2π

Step 6, Interpret the result. A central angle of 2π2π2π radians means the given area is actually the area of the whole circle with radius 444 units.

Comments(4)FFrenchTutorHopeNovember 5, 2025I've been struggling to explain sector area to my students. This page's def and examples made it so much clearer! Thanks!

MGMs. GarciaNovember 4, 2025I've been struggling to explain sector area to my students. This page's def and examples made it super easy! Thanks!

PProducerFinnNovember 4, 2025I've been struggling to teach area of a sector. This glossary page was a lifesaver! Clear defs and examples helped my students finally get it.

GGardenerUmaNovember 4, 2025I've been struggling to explain sector area to my students. This page's clear def and examples made it so much easier! Thanks!

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