Area of Shaded Region Calculator

Area of Shaded Region Calculator

Calculate the area of a shaded region, specifically a rectangle with a circular cutout from its center.

Calculation Details:

2

Shaded Area: " + shadedArea.toFixed(2) + " square units

Understanding the Area of a Shaded Region

The "area of a shaded region" refers to the area of a specific portion of a geometric figure that is highlighted or distinguished from the rest. This concept is fundamental in geometry, design, engineering, and even art, where understanding the space occupied by a particular part of a shape is crucial.

What is a Shaded Region?

In many geometric problems, you'll encounter diagrams where a part of a larger shape is colored, hatched, or otherwise "shaded." The goal is often to find the area of just this shaded portion. This usually involves combining or subtracting the areas of simpler, known geometric shapes.

Common Scenarios for Shaded Regions

While there are countless variations, some common scenarios include:

• Shape within a Shape: A smaller shape (e.g., a circle) inscribed within a larger shape (e.g., a square), where the shaded region is the area of the larger shape minus the area of the smaller shape.
• Overlapping Shapes: Two or more shapes that intersect, and the shaded region is the area of their intersection or the area of one shape that is not covered by another.
• Composite Shapes: A complex shape made up of several simpler shapes, where the shaded region might be the sum of the areas of these simpler components.

How This Calculator Works: Rectangle with a Circular Cutout

Our calculator focuses on a very common and practical scenario: determining the area of a rectangle from which a circular section has been removed from its center. Imagine a rectangular piece of material with a perfectly round hole drilled through it. The shaded region, in this case, is the remaining material.

To calculate this, we follow a straightforward two-step process:

1. Calculate the Area of the Larger Shape (Rectangle): The area of a rectangle is found by multiplying its length by its width.
2. Calculate the Area of the Smaller Shape (Circle): The area of a circle is calculated using the formula πr², where 'r' is the radius of the circle.
3. Subtract to Find the Shaded Area: The area of the shaded region is then the area of the rectangle minus the area of the circle.

Formulas Used:

• Area of Rectangle (A_rect): Length × Width
• Area of Circle (A_circ): π × Radius²
• Area of Shaded Region (A_shaded): A_rect – A_circ

Example Calculation:

Let's say you have a rectangular sheet of metal with the following dimensions:

• Rectangle Length = 10 units
• Rectangle Width = 8 units
• A circular hole with a Radius = 3 units is cut out from the center.

Here's how the calculation proceeds:

1. Area of the Rectangle:
   A_rect = Length × Width = 10 units × 8 units = 80 square units
2. Area of the Circle:
   A_circ = π × Radius² = π × (3 units)² = π × 9 ≈ 28.27 square units
3. Area of the Shaded Region:
   A_shaded = A_rect – A_circ = 80 – 28.27 = 51.73 square units

The shaded area, representing the remaining material, is approximately 51.73 square units.

Using the Calculator:

Simply input the length and width of your rectangle, and the radius of the circular cutout. The calculator will instantly provide the area of the shaded region, ensuring the circle can fit within the rectangle's dimensions.