Area Of Shaded Region

Area of Shaded Regions

Sometimes, you may be required to calculate the area of shaded regions. Usually, we would subtract the area of a smaller inner shape from the area of a larger outer shape in order to find the area of the shaded region. If any of the shapes is a composite shape then we would need to subdivide it into shapes that we have area formulas, like the examples below.

How To Find The Area Of a Shaded Region?

The following diagram gives an example of how to find the area of a shaded region. Scroll down the page for more examples and solutions.

 

Geometry Worksheets
Practice your skills with the following worksheets:
Printable & Online Geometry Worksheets

Steps to Find The Area Of a Shaded Region

1. Break down complex figures into simpler shapes (circles, rectangles, triangles, etc.).
2. Find the area of the larger shape (often the background or outer shape).
3. Find the area of the smaller shape(s) (the unshaded parts).
4. Subtract the unshaded area from the total area:
   Area of shaded region = area of larger shape – area of smaller shape

Some area formulas that you may need:
Rectangle: A = length×width
Square: A = side²
Triangle: A = 1/2×base×height
Circle: A = πr²
Semi-circle: A = 1/2 πr²
Trapezoid: A = 1/2 (base1 + base2)×height

Example:
Find the area of the shaded region:

Solution:
Step 1: Find area of inner square = 2 cm × 2 cm = 4 cm²
Step 2: Find area of outer shape = (2 cm × 3 cm) + (10 cm × 3 cm)
                                                  = 6 cm² + 30 cm²
                                                  = 36 cm²

Step 3: Area of shaded region = area of outer shape – area of inner square
                                = 36 cm² – 4 cm²
                                = 32 cm²

• Worksheets On Finding Areas Of Shaded Regions

  Shaded Area Worksheets (printable)
  Involving Polygons
  Circles And Polygons 1
  Circles And Polygons 2
  Circles And Polygons 3
  

How To Find The Area Of Shaded Region Of A Rectangle Within Another Rectangle?

Step 1: Find the area of the larger shape and the smaller shape.

Step 2: Subtract the areas.

Step 3: Write the units.

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How To Find The Area Of A Shaded Region Of A Circle With An Inscribed Triangle?

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Area And Circumference Examples

Some examples involving the area of triangles and circles. Also, some examples to find the area of a shaded region.

Examples:

1. Find the area and perimeter of the following triangle.
   
2. Find the area and circumference of a circle with radius 8.
   
3. Find the area and circumference of a circle with diameter 10.
   
4. Find the area of the shaded region if the circle has diameter 6.
   
5. Find the area of the shaded region if the square has side 8.

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Area For Rectangular Shaded Region When The Dimensions Are Given As Variable Expressions

Example:
Determine a formula for the area of the orange shaded region.

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Area For A Shaded Region Between An Inscribed Circle And A Square

Example:
In the figure, a circle is inscribed in a square.
a. Find the area of the circle.
b. Find the area of the shaded region.

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Area For A Shaded Region Between A Square Inscribed In A Circle

Example:
In the diagram, the square ABCD is inscribed in circle O with diagonal AC = 8. Find the number of square units in the area of the shaded region in terms of π.

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Area Of A Shaded Region In A Circle

Use area of sector and area of segment.

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GMAT - Find The Area Of The Shaded Region

Example:
Viewed from the outside inward, the figure below depicts a square-circle-square-circle, each enclosed within the other. If the area of square ABCD is 2 square units, then which of the following expresses the area of the darkened corners of square EFGH?

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