Rectangle
Circle
Triangle
Sector of a Circle
Annulus (Ring)
Result
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Understanding and Calculating the Area of Shaded Regions
In geometry, a shaded region often represents a specific part of a larger shape, or the area between two or more shapes. Calculating the area of these regions is a fundamental skill with applications in fields like design, engineering, architecture, and even art. The method for finding the shaded area depends entirely on the shapes involved and how they are combined.
Common Scenarios and Formulas:
This calculator handles several common scenarios:
Area of a single basic shape: For simple shapes like rectangles, circles, or triangles, the shaded area is just the area of that shape itself.
Rectangle: Area = Length × Width
Area = L × W
Circle: Area = π × Radius²
Area = πr²
Triangle: Area = ½ × Base × Height
Area = ½ × b × h
Area of a Sector of a Circle: A sector is a pie-shaped portion of a circle. Its area is a fraction of the entire circle's area, determined by its central angle.
Area = (Central Angle / 360°) × π × Radius²
Area = &frac;θ}{360°} × πr²
(Where θ is the angle in degrees)
Area of an Annulus (Ring): An annulus is the region between two concentric circles (circles with the same center). The shaded area is the area of the larger circle minus the area of the smaller circle.
Area = Area of Outer Circle – Area of Inner Circle
Area = (π × Outer Radius²) – (π × Inner Radius²)
Area = π × (Outer Radius² – Inner Radius²)
Area = π (R² – r²)
(Where R is the outer radius and r is the inner radius)
Shaded Area as a Difference: Often, a shaded region is formed by removing a smaller shape from a larger shape. In such cases, the area of the shaded region is found by subtracting the area of the unshaded (removed) part from the area of the larger shape. For example, if a circle is cut out from the center of a rectangle, the shaded area is the rectangle's area minus the circle's area. This calculator focuses on the individual shapes and the annulus, which is a specific case of this difference.
How to Use the Calculator:
1. Select the primary shape type that best represents the region you are interested in or the outer boundary of your shaded area.
2. Input the required dimensions for the selected shape (e.g., length and width for a rectangle, radius for a circle). Ensure you are using consistent units (e.g., all in centimeters, all in inches). The calculator does not automatically convert units.
3. For shapes like an Annulus or Sector, input the specific parameters required (outer/inner radii, angle).
4. Click the "Calculate Shaded Area" button.
5. The result will display the calculated area in the same units squared as your input dimensions.
Example Calculation (Rectangle):
Imagine a rectangular garden plot that is 15 meters long and 8 meters wide. If the entire garden is considered the "shaded region" we want to measure:
Length = 15 meters
Width = 8 meters
Area = 15 m × 8 m = 120 square meters
Using the calculator: Select "Rectangle", enter 15 for Length, and 8 for Width. The result would be 120.
Example Calculation (Annulus):
Consider a circular running track. The outer edge of the track has a radius of 50 meters, and the inner edge (the grassy field) has a radius of 40 meters. The area of the track itself (the shaded region) is calculated as follows:
Outer Radius (R) = 50 meters
Inner Radius (r) = 40 meters
Area = π × (50² – 40²)
Area = π × (2500 – 1600)
Area = π × 900 ≈ 2827.43 square meters
Using the calculator: Select "Annulus", enter 50 for Outer Radius, and 40 for Inner Radius. The result would be approximately 2827.43.
function getElement(id) {
return document.getElementById(id);
}
function updateInputs() {
var shapeType = getElement("shapeType").value;
// Hide all shape-specific input divs
var shapeDivs = document.querySelectorAll('.shape-inputs');
for (var i = 0; i < shapeDivs.length; i++) {
shapeDivs[i].style.display = 'none';
}
// Show the relevant input div
var selectedDiv = getElement(shapeType + "Inputs");
if (selectedDiv) {
selectedDiv.style.display = 'block';
}
}
function calculateShadedArea() {
var shapeType = getElement("shapeType").value;
var area = 0;
var pi = Math.PI;
var validInputs = true;
// Clear previous result
getElement("result").innerText = "–";
try {
if (shapeType === "rectangle") {
var length = parseFloat(getElement("rectLength").value);
var width = parseFloat(getElement("rectWidth").value);
if (isNaN(length) || isNaN(width) || length <= 0 || width <= 0) {
validInputs = false;
} else {
area = length * width;
}
} else if (shapeType === "circle") {
var radius = parseFloat(getElement("circleRadius").value);
if (isNaN(radius) || radius <= 0) {
validInputs = false;
} else {
area = pi * radius * radius;
}
} else if (shapeType === "triangle") {
var base = parseFloat(getElement("triangleBase").value);
var height = parseFloat(getElement("triangleHeight").value);
if (isNaN(base) || isNaN(height) || base <= 0 || height <= 0) {
validInputs = false;
} else {
area = 0.5 * base * height;
}
} else if (shapeType === "sector") {
var radius = parseFloat(getElement("sectorRadius").value);
var angle = parseFloat(getElement("sectorAngle").value);
if (isNaN(radius) || isNaN(angle) || radius <= 0 || angle 360) {
validInputs = false;
} else {
area = (angle / 360) * pi * radius * radius;
}
} else if (shapeType === "annulus") {
var outerRadius = parseFloat(getElement("annulusOuterRadius").value);
var innerRadius = parseFloat(getElement("annulusInnerRadius").value);
if (isNaN(outerRadius) || isNaN(innerRadius) || outerRadius <= 0 || innerRadius = outerRadius) {
validInputs = false;
} else {
area = pi * (outerRadius * outerRadius – innerRadius * innerRadius);
}
}
if (validInputs) {
// Format the result to 2 decimal places
getElement("result").innerText = area.toFixed(2);
} else {
getElement("result").innerText = "Invalid Input";
}
} catch (e) {
getElement("result").innerText = "Error";
console.error("Calculation error: ", e);
}
}
// Initialize the correct inputs on page load
document.addEventListener("DOMContentLoaded", updateInputs);