Find The Area Of The Shaded Figure Calculator

Calculate Shaded Area

This calculator finds the area of the shaded region formed by a rectangle with a concentric circle cutout.

Understanding the Area of a Shaded Figure

What is the Area of the Shaded Figure?

The area of the shaded figure refers to the measure of the surface covered by the colored or shaded portion of a geometric figure. It's often calculated by finding the area of a larger, enclosing shape and subtracting the area(s) of one or more smaller, unshaded shapes within it or overlapping with it. This concept is fundamental in geometry and has practical applications in various fields.

Anyone dealing with geometric shapes and their dimensions might need to find the area of the shaded figure. This includes students learning geometry, engineers, architects, designers, and landscapers. For example, an architect might calculate the usable floor space (shaded area) excluding internal structures, or a landscaper might find the area of a lawn (shaded area) excluding a circular pond.

A common misconception is that the area of the shaded figure is always found by subtracting the area of a smaller shape from a larger one. While this is often the case (like a cutout), shaded areas can also be formed by the intersection or union of shapes, requiring different approaches.

Area of the Shaded Figure Formula and Mathematical Explanation (Rectangle with Circle Cutout)

For the specific case of a rectangle with a concentric circle cutout, the formula to find the area of the shaded figure (A_shaded) is:

A_shaded = A_rectangle – A_circle

Where:

• A_rectangle is the area of the outer rectangle.
• A_circle is the area of the inner circle cutout.

The area of the rectangle is calculated as:

A_rectangle = Width × Height = W × H

The area of the circle is calculated as:

A_circle = π × radius² = π × r²

So, the combined formula becomes:

A_shaded = (W × H) – (π × r²)

Variables Table

Variable	Meaning	Unit	Typical Range
W	Width of the outer rectangle	Length units (e.g., cm, m, inches)	Positive number
H	Height of the outer rectangle	Length units (e.g., cm, m, inches)	Positive number
r	Radius of the inner circle	Length units (e.g., cm, m, inches)	Positive number, 2r ≤ min(W, H)
π	Pi (mathematical constant)	Dimensionless	~3.14159
Ashaded	Area of the shaded figure	Area units (e.g., cm^2, m^2, inches^2)	Positive number

Practical Examples (Real-World Use Cases)

Let's look at how to find the area of the shaded figure in practical scenarios.

Example 1: Garden Lawn

Imagine a rectangular garden measuring 20 meters wide and 15 meters high. In the center, there is a circular pond with a radius of 4 meters. We want to find the area of the lawn (the shaded area).

• Rectangle Width (W) = 20 m
• Rectangle Height (H) = 15 m
• Circle Radius (r) = 4 m

Area of rectangle = 20 * 15 = 300 m²

Area of circle = π * 4² ≈ 3.14159 * 16 ≈ 50.265 m²

Area of the shaded figure (lawn) = 300 – 50.265 = 249.735 m²

Example 2: Metal Plate with Hole

A rectangular metal plate is 10 cm by 8 cm. A circular hole with a radius of 2 cm is drilled through its center. What is the remaining surface area of the plate (the shaded area)?

• Rectangle Width (W) = 10 cm
• Rectangle Height (H) = 8 cm
• Circle Radius (r) = 2 cm

Area of rectangle = 10 * 8 = 80 cm²

Area of circle = π * 2² ≈ 3.14159 * 4 ≈ 12.566 cm²

Area of the shaded figure (plate) = 80 – 12.566 = 67.434 cm²

Understanding how to calculate the area of the shaded figure is vital for material estimation and design. For more on basic shapes, see our Area of Rectangle Calculator and Area of Circle Calculator.

How to Use This Area of the Shaded Figure Calculator

1. Enter Rectangle Dimensions: Input the width (W) and height (H) of the outer rectangle into the respective fields.
2. Enter Circle Radius: Input the radius (r) of the inner circle cutout. Ensure the circle can fit within the rectangle (diameter 2r should be less than or equal to both W and H).
3. Calculate: Click the "Calculate" button or simply change the input values. The calculator will automatically update the results.
4. Review Results: The calculator will display:
   • The primary result: The area of the shaded figure.
   • Intermediate values: The area of the outer rectangle and the area of the inner circle.
   • The formula used for the calculation.
5. Analyze Table and Chart: The table and chart provide a visual breakdown of the areas.
6. Reset: Click "Reset" to clear the inputs and results to their default values.
7. Copy Results: Click "Copy Results" to copy the main result, intermediate values, and input values to your clipboard.

This area of the shaded figure calculator helps you quickly find the area remaining after a circular cutout from a rectangle.

Key Factors That Affect Area of the Shaded Figure Results

• Dimensions of the Outer Shape (Rectangle): The width and height directly determine the rectangle's area. Larger dimensions lead to a larger initial area before the cutout.
• Dimensions of the Inner Shape (Circle): The radius of the circle determines its area. A larger radius means a larger cutout area, thus a smaller area of the shaded figure.
• Relative Sizes: The ratio of the circle's diameter to the rectangle's dimensions is crucial. If the circle is too large to fit, the calculation for this specific scenario isn't valid. Our calculator checks this.
• Units Used: Consistency in units is vital. If you measure the rectangle in meters and the circle in centimeters, you must convert them to the same unit before calculating the area of the shaded figure. The result's unit will be the square of the input unit (e.g., m², cm²).
• Precision of π (Pi): The value of π used affects the accuracy of the circle's area and thus the shaded area. More decimal places in π give a more precise result.
• Shape of the Cutout: This calculator assumes a circular cutout. If the cutout were a different shape (e.g., another rectangle, a triangle), the formula for its area, and consequently the area of the shaded figure, would change. Our geometry calculators page has tools for various shapes.

Frequently Asked Questions (FAQ)

What if the inner shape is not a circle?

If the inner shape is different (e.g., a square, triangle), you would calculate its area using the appropriate formula and subtract it from the outer rectangle's area to find the area of the shaded figure. This calculator is specifically for a circular cutout.

What if the circle is not centered?

As long as the circle is fully contained within the rectangle, its position doesn't change its area or the area of the rectangle. So, the area of the shaded figure would be the same. However, if the circle overlapped the edges or was outside, the problem would be different.

Can I use this for units other than meters or centimeters?

Yes, you can use any unit of length (inches, feet, yards, etc.) as long as you are consistent for all measurements (W, H, and r). The resulting area of the shaded figure will be in the square of that unit (e.g., square inches, square feet).

What if the "shaded" area is the circle itself?

If the circle were shaded and the area between the rectangle and circle were unshaded, you would simply be calculating the area of the circle.

How do I find the area of the shaded figure between two concentric circles?

That's an annulus. You find the area of the larger circle and subtract the area of the smaller circle. The formula is A = π(R² – r²), where R is the outer radius and r is the inner radius.

What is the area of a shaded figure between a square and an inscribed circle?

If a circle is inscribed in a square of side 's', the circle's radius is s/2. The area of the square is s², and the area of the circle is π(s/2)². The shaded area between them is s² – π(s/2)². You can find more with our shape area guide.

Does the thickness of the lines matter?

In theoretical geometry problems, lines are considered to have no thickness, so they don't affect the area. In real-world applications, if the lines or boundaries have significant thickness, that might need to be accounted for, but it's usually negligible for calculating the area of the shaded figure.

Where can I find other math calculators?

We have a collection of math calculators for various purposes, including geometry and algebra.