A sector of a circle is a portion of a disk enclosed by two radii and an arc. Think of it like a slice of pizza or pie. The area of this sector depends on two key measurements: the radius of the circle from which it's cut, and the central angle that defines the slice.
The formula for the area of a sector is derived from the area of the entire circle (πr²). Since a sector represents a fraction of the whole circle, we multiply the total area by the ratio of the sector's angle to the total angle in a circle (360 degrees or 2π radians).
The Formula
If the angle (θ) is measured in degrees, the formula for the area of a sector is:
Area = (θ / 360) * π * r²
Where:
θ is the central angle of the sector in degrees.
r is the radius of the circle.
π (Pi) is a mathematical constant, approximately 3.14159.
If the angle is measured in radians, the formula is simpler:
Area = (1/2) * r² * θ (where θ is in radians)
Our calculator uses the formula for angles in degrees.
Use Cases
Calculating the area of a sector has practical applications in various fields:
Engineering and Design: Determining the surface area of curved components or the volume of materials needed for specific shapes.
Architecture: Planning circular or curved structures, like rotundas or pathways.
Geometry and Mathematics: Fundamental calculations in geometry problems, understanding proportions within circles.
Resource Management: Estimating the area of land or water bodies that are part of a circular region.
How to Use the Calculator
1. Enter the Radius (r) of the circle.
2. Enter the Angle (θ) of the sector in degrees.
3. Click the "Calculate Area" button.
The calculator will then display the calculated area of the sector, using the standard formula.
function calculateSectorArea() {
var radius = parseFloat(document.getElementById("radius").value);
var angle = parseFloat(document.getElementById("angle").value);
var resultDiv = document.getElementById("result");
if (isNaN(radius) || isNaN(angle)) {
resultDiv.innerHTML = "Please enter valid numbers for radius and angle.";
return;
}
if (radius < 0 || angle < 0) {
resultDiv.innerHTML = "Radius and angle cannot be negative.";
return;
}
var pi = Math.PI;
var area = (angle / 360) * pi * Math.pow(radius, 2);
resultDiv.innerHTML = "Area: " + area.toFixed(4) + " units²";
}