How to Find the Area of a Shaded Region

Finding the area of a shaded region is a common geometry problem. These questions usually show a large shape with a smaller shape, circle, triangle, or another section shaded or unshaded.

The main idea is simple:

Shaded Area = Total Area − Unshaded Area

In some diagrams, you may need to add several shaded sections instead:

Shaded Area = Area 1 + Area 2 + Area 3 + ...

The challenge is usually not the arithmetic. It is identifying the correct shapes, choosing the right formulas, and deciding whether to add or subtract.

This guide explains how to find the area of a shaded region step by step, with examples involving rectangles, circles, triangles, and composite figures.

What Is the Area of a Shaded Region?

The area of a shaded region is the amount of space covered by the shaded part of a two-dimensional figure.

For example, imagine a large rectangle containing a smaller unshaded rectangle. The shaded region is everything in the large rectangle except the smaller rectangle.

You can calculate it by:

Shaded Area = Area of Large Shape − Area of Unshaded Shape

Area is measured in square units such as square inches, square feet, square centimeters, or square meters. The SI unit for area is the square meter (m²). NIST

How to Find the Area of a Shaded Region

Use these steps for most shaded-area problems.

Step 1: Identify the entire figure

Look at the outside boundary of the diagram.

Ask:

What is the total shape?

It could be a:

• Rectangle

• Square

• Circle

• Triangle

• Semicircle

• Trapezoid

• Composite shape

Step 2: Identify the unshaded section

Next, determine which part is not shaded.

This is important because the unshaded portion may need to be subtracted from the total area.

Step 3: Find the required dimensions

Look for measurements such as:

• Length

• Width

• Base

• Height

• Radius

• Diameter

If a required measurement is missing, calculate it before finding the area.

Step 4: Calculate each area

Use the appropriate area formula for each shape.

Step 5: Add or subtract

If the shaded region is what's left after removing an unshaded section:

Shaded Area = Total Area − Unshaded Area

If the shaded parts are separate sections:

Shaded Area = Area 1 + Area 2 + Area 3

Step 6: Write the answer in square units

For example:

48 cm²

not simply:

48 cm

Common Area Formulas for Shaded Regions

You may need several basic formulas when solving shaded-area problems.

If you need to calculate the area of different geometric shapes quickly, the Area Calculator can calculate areas for rectangles, circles, triangles, trapezoids, annuli, and other shapes.

Example 1: Shaded Rectangle

Suppose a large rectangle measures 12 cm by 8 cm. Inside it is an unshaded rectangle measuring 6 cm by 3 cm.

Find the shaded area.

Step 1: Find the area of the large rectangle

Area = Length × Width

Area = 12 × 8

Area = 96 cm²

Step 2: Find the area of the unshaded rectangle

Area = 6 × 3

Area = 18 cm²

Step 3: Subtract

Shaded Area = 96 − 18

Shaded Area = 78 cm²

Answer

The shaded region has an area of 78 cm².

Example 2: Shaded Circle Inside a Square

Suppose a circle is drawn inside a square.

The square has a side length of 10 cm, and the circle touches all four sides of the square.

Find the area of the shaded part if the region inside the square but outside the circle is shaded.

Step 1: Find the square's area

Area = Side × Side

Area = 10 × 10

Area = 100 cm²

Step 2: Find the circle's radius

Because the circle fits exactly inside the square:

Diameter = 10 cm

Therefore:

Radius = 10 ÷ 2 = 5 cm

Step 3: Find the circle's area

Area = π × Radius × Radius

Area = π × 5 × 5

Area ≈ 78.54 cm²

Step 4: Subtract

Shaded Area = 100 − 78.54

Shaded Area ≈ 21.46 cm²

Answer

The shaded area is approximately 21.46 cm².

Example 3: Shaded Region Between Two Circles

Suppose a large circle has a radius of 8 cm, while a smaller circle at its center has a radius of 5 cm.

The region between the two circles is shaded.

This shape is called an annulus.

Step 1: Find the large circle's area

Area = π × 8 × 8

Area ≈ 201.06 cm²

Step 2: Find the small circle's area

Area = π × 5 × 5

Area ≈ 78.54 cm²

Step 3: Subtract

Shaded Area = 201.06 − 78.54

Shaded Area ≈ 122.52 cm²

Answer

The shaded region is approximately 122.52 cm².

The important idea is that the smaller circle is not part of the shaded region, so its area must be removed from the larger circle's area.

Example 4: Shaded Triangle Inside a Rectangle

Suppose a rectangle measures 14 m by 10 m. An unshaded triangle inside the rectangle has a base of 14 m and a height of 6 m.

Find the remaining shaded area.

Step 1: Find the rectangle's area

Area = 14 × 10

Area = 140 m²

Step 2: Find the triangle's area

Area = 1/2 × Base × Height

Area = 1/2 × 14 × 6

Area = 42 m²

Step 3: Subtract

Shaded Area = 140 − 42

Shaded Area = 98 m²

Answer

The shaded area is 98 m².

Example 5: Two Separate Shaded Regions

Sometimes the shaded portion is made up of several separate shapes.

Suppose a diagram contains:

• Shaded rectangle: 20 cm²

• Shaded triangle: 12 cm²

• Shaded circle section: 8 cm²

The total shaded area is found by adding them.

Shaded Area = 20 + 12 + 8

Shaded Area = 40 cm²

Answer

The total shaded area is 40 cm².

In this type of problem, there is nothing to subtract because all three sections are shaded.

Example 6: Shaded Region of an L-Shaped Figure

Consider an L-shaped figure made from a large rectangle with a smaller rectangular section removed.

Suppose the large rectangle measures:

• Length = 15 m

• Width = 10 m

The missing rectangle measures:

• Length = 6 m

• Width = 4 m

Step 1: Find the large rectangle's area

15 × 10 = 150 m²

Step 2: Find the missing rectangle's area

6 × 4 = 24 m²

Step 3: Subtract

Shaded Area = 150 − 24

Shaded Area = 126 m²

Answer

The shaded area is 126 m².

For more practice with breaking complex figures into simpler shapes, see this guide on calculating the area of irregular shapes. It covers methods such as dividing figures into rectangles and triangles and subtracting missing sections.

How to Find a Missing Measurement First

Some shaded-area problems do not give every dimension directly.

You may need to find a missing length before calculating the area.

For example, suppose a large rectangle has a total width of 12 cm. The unshaded section has a width of 5 cm.

The remaining width is:

12 − 5 = 7 cm

You can then use 7 cm in the area calculation.

This is especially common with L-shaped figures and composite diagrams.

Shaded Region With a Semicircle

Semicircles often appear in shaded-area geometry problems.

Suppose a rectangle has a semicircle attached to it, and the semicircular section is unshaded.

First calculate the rectangle's area.

Then calculate the semicircle's area:

Semicircle Area = 1/2 × π × Radius × Radius

Finally:

Shaded Area = Rectangle Area − Semicircle Area

Always check whether the measurement given is the diameter or radius. If the diameter is given, divide it by 2 to find the radius.

Shaded Region With a Quarter Circle

A quarter circle represents one-fourth of a complete circle.

Its area is:

Quarter Circle Area = 1/4 × π × Radius × Radius

If the quarter circle is unshaded inside another shape, subtract it from the total area.

For example:

Shaded Area = Rectangle Area − Quarter Circle Area

The same add-or-subtract principle works regardless of the specific shape.

How to Decide Whether to Add or Subtract

A quick way to solve shaded-region problems is to ask:

Is the shaded region made of separate pieces?

If yes, add the areas.

Shaded Area = Area 1 + Area 2 + Area 3

Is the shaded region the part left after removing another shape?

If yes, subtract.

Shaded Area = Total Area − Unshaded Area

Is the diagram complicated?

Break it into smaller familiar shapes first.

Then decide which areas should be added and which should be subtracted.

Common Mistakes to Avoid

1. Using the Wrong Shape Formula

Make sure you identify the shape correctly before calculating.

A triangle and rectangle do not use the same formula.

2. Using Diameter Instead of Radius

For circle area, use the radius:

Area = π × r × r

If the diameter is given:

Radius = Diameter ÷ 2

3. Adding an Unshaded Section

If a section is not shaded, you usually need to subtract its area rather than add it.

4. Forgetting Square Units

Area is always expressed in square units.

Examples include:

• cm²

• m²

• ft²

• in²

5. Mixing Units

Do not calculate using measurements such as centimeters and meters without converting them first.

For example, convert:

2 m = 200 cm

before combining it with a measurement in centimeters.

6. Double-Counting a Section

When dividing a complicated figure, make sure each part is counted only once.

7. Assuming the Drawing Is to Scale

A diagram may not be drawn perfectly to scale. Use the measurements provided in the question rather than estimating distances visually.

A Simple Strategy to Remember

Use this four-step approach:

Identify → Calculate → Add/Subtract → Check

Identify

Find the complete shape and the shaded or unshaded sections.

Calculate

Use the correct area formula for each section.

Add or Subtract

Add shaded pieces or subtract unshaded pieces from the total.

Check

Make sure your answer has the correct units and is reasonable compared with the size of the original figure.

Area of a Shaded Region: Quick Reference

Frequently Asked Questions

What is the formula for the area of a shaded region?

There is no single formula for every shaded region. The most common method is:

Shaded Area = Total Area − Unshaded Area

If the shaded region contains multiple separate pieces, add their areas instead.

How do you find the shaded area between two shapes?

Calculate the area of the larger or outside shape and subtract the area of the smaller or inside shape.

Shaded Area = Larger Area − Smaller Area

How do you find the shaded area of a circle inside a square?

Find the square's area, find the circle's area, and subtract:

Shaded Area = Square Area − Circle Area

If the circle fits exactly inside the square, its diameter equals the square's side length.

How do you find the shaded area of an irregular shape?

Break the figure into familiar shapes such as rectangles, triangles, circles, and trapezoids. Calculate each section and then add or subtract the areas as required.

What units are used for shaded area?

Area is expressed in square units, such as cm², m², ft², or in².

What is the most important step in shaded-area problems?

Correctly identifying which parts are shaded and which parts are not is usually the most important step. Once the regions are identified, choose the appropriate area formula and combine the results.

Final Takeaway

Learning how to find the area of a shaded region becomes much easier when you stop treating the diagram as one complicated figure.

Instead:

Break the figure into familiar shapes.

Then:

Calculate each area.

Finally:

Add the shaded sections or subtract the unshaded sections.

The most useful formula to remember is:

Shaded Area = Total Area − Unshaded Area

For more complicated diagrams, you may need several formulas, but the basic strategy remains the same. Keep your measurements consistent, use square units, and check your final answer against the original figure.