A sector of a circle is essentially a "slice of pie." It is the portion of a circle enclosed by two radii and an arc. Calculating the area of a sector is a fundamental skill in geometry, trigonometry, and various engineering fields.
The Formula
The formula for the area depends on whether your central angle is measured in degrees or radians:
Degrees: Area = (θ / 360) × π × r² Radians: Area = 0.5 × r² × θ
Where:
r is the radius of the circle.
θ (Theta) is the central angle.
π (Pi) is approximately 3.14159.
Step-by-Step Calculation Example
Suppose you have a circle with a radius of 5 cm and a central angle of 60 degrees.
Identify the radius (r = 5) and the angle (θ = 60).
Use the degree formula: Area = (60 / 360) × π × 5².
Simplify the fraction: 60/360 = 1/6.
Calculate the square of the radius: 5² = 25.
Multiply: (1/6) × 3.14159 × 25 ≈ 13.09 cm².
Practical Applications
Determining the area of a sector is useful in many real-world scenarios, such as:
Architecture: Designing curved rooms or circular windows.
Mechanical Engineering: Calculating the surface area of gears or specialized mechanical plates.
Agriculture: Measuring the coverage area of a central pivot irrigation system that only completes a partial rotation.
Graphics Design: Creating accurate pie charts and circular infographics.
function calculateSectorArea() {
var radius = parseFloat(document.getElementById("radiusInput").value);
var angle = parseFloat(document.getElementById("angleInput").value);
var unit = document.getElementById("unitType").value;
var resultBox = document.getElementById("resultBox");
var areaOutput = document.getElementById("areaOutput");
var formulaUsed = document.getElementById("formulaUsed");
if (isNaN(radius) || isNaN(angle) || radius <= 0 || angle 360) {
alert("Angle in degrees typically should not exceed 360 for a single sector.");
}
area = (angle / 360) * pi * Math.pow(radius, 2);
formulaUsed.innerHTML = "Used Degree Formula: ( " + angle + " / 360 ) × π × " + radius + "²";
} else {
if (angle > (2 * pi)) {
alert("Angle in radians typically should not exceed 2π (approx 6.28) for a single sector.");
}
area = 0.5 * Math.pow(radius, 2) * angle;
formulaUsed.innerHTML = "Used Radian Formula: 0.5 × " + radius + "² × " + angle;
}
areaOutput.innerHTML = area.toLocaleString(undefined, {minimumFractionDigits: 2, maximumFractionDigits: 4}) + " square units";
resultBox.style.display = "block";
}