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Area of a Sector of a Circle Formula

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Unlocking Circle Sector Perimeter and Area: Formulas, Calculations, and Applications

Comprehensive Definition, Description, Examples & Rules 

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Introduction to Circle Sectors and Their Significance

A sector of a circle is that part of a circle that is made of the arc of the circle and joining it with the two radii. A sector is the portion of the circle that you can form by the circumference and the radius at the end of the arc. You can compare the size of the sector of a circle to a slice of pizza. 

Importance of Circle Sectors in Geometry:

  • The circle sectors are very important to calculate and solve the circle problems and help to determine the exact central angle measurement of the circle. 
  • The sector also helps to find the exact calculation of the circle’s radius, which is very important to calculate any geometrical formula. 

Importance of Circle Sectors in Real-Life Applications:

  • The primary real-life sector where you can see circle sectors is a pizza or pie slice. 
  • You can also see that the circle sectors are visible on the tire wheels.
  • When you cut a cake into a triangle, it is divided into a sector.

Calculating the perimeter and the sector area must be relevant to each other as both help to find a particular measurement of a space and provide advanced Mathematics calculations of algebra and calculus. There are special formulas where you can determine the angle of the sector of a circle. 

The formula for Calculating the Perimeter of a Sector

You need to use a particular formula for calculating the perimeter of the sector of a circle. 

The Perimeter of a Sector formula to use is:

  • 2 Radius + ((θ/360) × 2πr )

If you break the formula, then

The perimeter of the sector of the circle is radius plus radius plus arc length.

So Radius plus Radius becomes 2 Radius

Arc Length is ((θ/360) × 2πr ).

It is how you can break down the formula and use it for calculating the perimeter of the sector of the circle. 

Step-by-step Guide to Finding the Perimeter of a Sector

You need to use the formula effectively as your guide to find the perimeter of a sector of a circle. The step that you have to follow includes:

  • Determine the length of the radius of the diameter of the circle.
  • Find the exact size of the angle that is creating the arc of the sector. 
  • Determine the arc length of the sector together with the arc length and the two radii.
  • You can find your answer by following the steps.

Example:

A circular arc whose radius is 15 cm makes an angle of 30° at the center. Find the perimeter of the sector formed. Using π = 3.14.

Solution:

Given that r = 15 cm,

θ = 30° = 30° × (π/180°) = π/6

The perimeter of the sector formula is:

P = 2 r + r θ

P = 2 (15) + 15 ( π/6)

P = 30 + 7.85 = 37.85

Hence, the Perimeter of the sector is 37.85 cm

The formula for Calculating the Area of a Sector

The area of a sector of a circle formula is

With respect to angle:

  • A = (θ/360) × π × r2

To define this formula, you use the arc length and then multiply it with the basic formula of the area of a circle. 

(θ/360) is the arc length of the circle

πr² is the area of the circle. 

With respect to the length of the arc:

  • A = (lr) / 2

Here,

l is the length of the arc, r is the radius of the circle

Calculating the Area of a Sector: A Practical Approach

The steps to calculate the area of a sector are:

  • Determine the angle of the arc that is the primary part of calculating the area of a sector. 
  • Find the exact radius of the circle if it is not given, and use the circumference to calculate its radius. 
  • Use the mentioned formula to calculate the area of the sector of the circle that you can use efficiently.

Examples:

With respect to the angle of the arc:

Find the area of the sector of the circle with the angle of the arc as 60° and a radius of 10cm. Take π = 3.14

The formula for the area of the sector is: A = (θ/360) × π × r2

So, (60/360) × 3.14 × 10²

1 /6 × 314 

52.33 

So, the area of the sector is 52.33 cm²

With respect to the length of the arc:

Find the area of a sector where the arc length is 4 cm and the radius is 8 cm.

The formula for the area of the Sector is (lr) / 2

So, (4 × 8)/ 2 = 16

So, the area of the sector is 16 cm².

Real-Life Applications of Circle Sector Perimeter and Area

There are many real-life applications of the perimeter and the area of the circle, and the primary parts are:

  • When you slice a cake or a pizza, it is the situation when you are using the sector to slice the item. 
  • One of the primary examples of a sector of a circle is a wall clock.

The other areas where you can use the sector of the circle are:

  • Physics: The part of the sector of the circle is very effective in physics as calculating the area and perimeter of the sector of the circle makes many physics calculations very easy.
  • Engineering: The engineering sector is also very important as civil engineers use the area and perimeter of the circle sector to make a lot of calculations and make buildings and roads according to these calculations. 
  • Design: The designing area also uses the sector of the circle to make a lot of calculations.

Comparing Sectors with Different Angles

The different angles of the sector help to determine its area and perimeter, and the angle of the sector impacts your answer, as all angles have different sectors. While using the calculation, the size of the angle is different, so the area and perimeter of the sector will also be completely different. 

Examples:

  • Pie chart: 180°
  • Slice of Pizza: 60°

Perimeter vs. Area: Key Differences

The perimeter and the area of a shape have funda­mental differ­ences. The area is deter­mined by measuring two dimen­sions, while the perimeter is measured along a single dimen­sion. When calcu­lating the area, you find the extent of space within the bound­aries of the shape. On the other hand, the perimeter represents the outer boundary or edge of the shape. The area is expressed in square units, while linear units measure the perim­eter. 

When you want to choose one of the two, if you want to determine the space inside a shape, you will use the area, and when you want to determine the length of the shape, you will use the perimeter. 

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Key Takeaways

  1. The area of the sector is written in square units, while the perimeter is written in linear units. 
  2. The formula to calculate the sector’s area and perimeter is different, and you need to keep that in mind. 
  3. In regular real-life applications, you can use the area and perimeter of the sector formula.

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Frequently Asked Questions

The central angle is the measure of the arc it inter­cepts and helps determine the area and perimeter of a sector. Knowing the central angle, you can calculate the precise measurements for the perimeter and area. If the central angle changes, the area and perimeter will also change accordingly.

There is a funda­mental disti­nction between a sector and a circle segment. A sector refers to a significant portion of the circle, including an arc and two radii. On the other hand, a segment refers to the region enclosed by an arc and the chord conne­cting its endpo­ints.

For calculating and converting the central angle from a degree to radiance, you need to multiply the number of degrees with (π/180)

For example, If you want to convert 60-degree angles into radians, your answer will be π/3.

You can use the sector perimeter and area calculation in various areas in your practical life. These are:

  • Engineering: Civil engineers use the geometry formula the most as it allows them to calculate the floor area of the house or a particular area of fencing at the park, which allows them to build things.
  • Trigonometry: Trigonometry and geometry are co-related and have direct relations with the circle sector. 
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2D Shapes2cosacosb Formula30-60-90 Formulas3D ShapesAbsolute Value FormulaAcute AngleAcute Angle triangleAdditionAlgebra FormulasAlgebra of MatricesAlgebraic EquationsAlgebraic ExpressionsAngle FormulaAnnulusAnova FormulaAnti-derivative FormulaAntiderivative FormulaApplication of DerivativesApplications of IntegrationArc Length FormulaArccot FormulaArctan FormulaArea Formula for QuadrilateralsArea FormulasArea Of A Sector Of A Circle FormulaArea Of An Octagon FormulaArea Of Isosceles TriangleArea Of ShapesArea Under the Curve FormulaArea of RectangleArea of Regular Polygon FormulaArea of TriangleArea of a Circle FormulaArea of a Pentagon FormulaArea of a Square FormulaArea of a Trapezoid FormulaArithmetic Mean FormulaArithmetic ProgressionsArithmetic and Geometric ProgressionAssociative Property FormulaAsymptote FormulaAverage Deviation FormulaAverage Rate of Change FormulaAveragesAxioms Of ProbabilityAxis of Symmetry FormulaBasic Math FormulasBasics Of AlgebraBinary FormulaBinomial Probability FormulaBinomial distributionBodmas RuleBoolean AlgebraBusiness MathematicsCalculusCelsius FormulaCentral Angle of a Circle FormulaCentral Limit Theorem FormulaCentroid of a Trapezoid FormulaChain RuleChain Rule FormulaChange of Base FormulaChi Square FormulaCirclesCircumference FormulaCoefficient of Determination FormulaCoefficient of Variation FormulaCofactor FormulaCompound Interest FormulaConditional Probability FormulaConeConfidence Interval FormulaCongruence of TrianglesCorrelation Coefficient FormulaCos Double Angle FormulaCos Square theta FormulaCos Theta FormulaCosec Cot FormulaCosecant FormulaCosine FormulaCovariance FormulaCubeCylinderDiscover the world of MathsEllipseEquilateral triangleEuler’s formulaHeptagonIntegrationIntegration by partsLinesLocusNatural numbersNumber lineParallelogramPerimeterPolygonPrismProbabilityPyramidSetsSphereSquareStandard deviation formulaSubtractionSymmetryTimeVector
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