Calculating Area of a Sector

Area of a Sector Calculator

Radius (r):

Central Angle (θ in degrees):

function calculateSectorArea() { var radiusInput = document.getElementById("radius").value; var centralAngleInput = document.getElementById("centralAngle").value; var resultDiv = document.getElementById("result"); var radius = parseFloat(radiusInput); var centralAngle = parseFloat(centralAngleInput); if (isNaN(radius) || isNaN(centralAngle) || radius <= 0 || centralAngle <= 0) { resultDiv.innerHTML = "Please enter valid positive numbers for Radius and Central Angle."; return; } // Formula for Area of a Sector when angle is in degrees: // Area = (θ / 360) * π * r² var area = (centralAngle / 360) * Math.PI * Math.pow(radius, 2); resultDiv.innerHTML = "Area of the Sector: " + area.toFixed(4) + " square units"; }

Understanding the Area of a Sector

A sector of a circle is a region bounded by two radii and the intercepted arc. Think of it like a slice of pizza or a piece of a pie. Calculating the area of a sector is a fundamental concept in geometry and has applications in various fields, from engineering to design.

What is a Sector?

In simple terms, a sector is a portion of a circle enclosed by two radii and the arc connecting their endpoints. The size of the sector is determined by the radius of the circle and the angle between the two radii, known as the central angle.

The Formula for Area of a Sector

The area of a sector is a fraction of the total area of the circle. This fraction is determined by the ratio of the central angle of the sector to the total angle in a circle (360 degrees or 2π radians).

The most common formula for calculating the area of a sector, when the central angle (θ) is given in degrees, is:

Area = (θ / 360°) × π × r²

Where:

θ (theta) is the central angle of the sector in degrees.
π (pi) is a mathematical constant approximately equal to 3.14159.
r is the radius of the circle.

If the central angle (θ) is given in radians, the formula simplifies to:

Area = (1/2) × r² × θ

Our calculator uses the formula with the angle in degrees for convenience.

How to Use the Calculator

To find the area of a sector using the calculator above, simply follow these steps:

Enter the Radius (r): Input the length of the radius of the circle. This is the distance from the center to any point on the circle's edge.
Enter the Central Angle (θ in degrees): Input the angle formed by the two radii that define the sector, measured in degrees.
Click "Calculate Area": The calculator will instantly display the area of the sector in square units.

Examples

Let's look at a few examples to illustrate the calculation:

Example 1: A Quarter Circle

Radius (r): 10 units
Central Angle (θ): 90 degrees
Calculation: Area = (90 / 360) × π × 10² = (1/4) × π × 100 = 25π ≈ 78.5398 square units

Using the calculator with these values will yield approximately 78.5398.

Example 2: A Semicircle

Radius (r): 7 units
Central Angle (θ): 180 degrees
Calculation: Area = (180 / 360) × π × 7² = (1/2) × π × 49 = 24.5π ≈ 76.9690 square units

The calculator will show approximately 76.9690 for these inputs.

Example 3: A Smaller Sector

Radius (r): 5 units
Central Angle (θ): 60 degrees
Calculation: Area = (60 / 360) × π × 5² = (1/6) × π × 25 ≈ 13.0899 square units

Inputting these values into the calculator will give you approximately 13.0899.

Understanding how to calculate the area of a sector is crucial for various geometric problems and real-world applications, such as determining the area of a circular garden section or the volume of a conical shape.