Find the Area of the Shaded Region Worksheets With Answers - Free Printable

Let's go through each problem on the worksheet and verify the solutions provided, then explain how to find the area of the shaded region in each case.

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Problem 1:

- Shape: Rectangle with a right triangle cut out.
- Rectangle dimensions: 12 units (length) × 5 units (width)
- Triangle dimensions: Base = 4, Height = 3

Step-by-step:
- Area of rectangle = $ 12 \times 5 = 60 $ square units
- Area of triangle = $ \frac{1}{2} \times 4 \times 3 = 6 $ square units
- Shaded area = $ 60 - 6 = 54 $ square units ✔

✔ Correct.

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Problem 2:

- Shape: Large right triangle with a smaller right triangle removed from inside.
- Large triangle: Legs = 10 and 10 → Right triangle
- Small triangle: Legs = 7 and 8

Step-by-step:
- Area of large triangle = $ \frac{1}{2} \times 10 \times 10 = 50 $
- Area of small triangle = $ \frac{1}{2} \times 7 \times 8 = 28 $
- Shaded area = $ 50 - 28 = 22 $ square units ✔

✔ Correct.

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Problem 3:

- Shape: Square with an equilateral-like triangle cut out (but it’s actually a triangle with base = 6 and height = 6).
- Square side: 6 units
- Triangle base: 6, height: 6

Step-by-step:
- Area of square = $ 6 \times 6 = 36 $
- Area of triangle = $ \frac{1}{2} \times 6 \times 6 = 18 $
- Shaded area = $ 36 - 18 = 18 $ square units ✔

✔ Correct.

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Problem 4:

- Shape: Square with a rectangle removed from center.
- Square: 12 × 12
- Rectangle: 6 × 2

Step-by-step:
- Area of square = $ 12 \times 12 = 144 $
- Area of rectangle = $ 6 \times 2 = 12 $
- Shaded area = $ 144 - 12 = 132 $ square units ✔

✔ Correct.

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Problem 5:

- Shape: Large rectangle with two shapes removed:
- A rectangle (4 × 7)
- A right triangle (base 4, height 5)

Step-by-step:
- Area of large rectangle = $ 18 \times 12 = 216 $
- Area of inner rectangle = $ 4 \times 7 = 28 $
- Area of triangle = $ \frac{1}{2} \times 4 \times 5 = 10 $
- Shaded area = $ 216 - 28 - 10 = 178 $ square units ✔

✔ Correct.

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Problem 6:

- Shape: Trapezoid with a rectangle removed from the middle.
- Trapezoid: Bases = 6 and 10, height = 4
- Rectangle: 5 × 1

Step-by-step:
- Area of trapezoid = $ \frac{1}{2} \times (6 + 10) \times 4 = \frac{1}{2} \times 16 \times 4 = 32 $
- Area of rectangle = $ 5 \times 1 = 5 $
- Shaded area = $ 32 - 5 = 27 $ square units ✔

✔ Correct.

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✔ Final Summary:

All answers provided in red are correct! Here’s the breakdown:

| Problem | Total Area | Subtracted Area(s) | Shaded Area |
|--------|------------|--------------------|-------------|
| 1 | 60 | 6 | 54 |
| 2 | 50 | 28 | 22 |
| 3 | 36 | 18 | 18 |
| 4 | 144 | 12 | 132 |
| 5 | 216 | 28 + 10 = 38 | 178 |
| 6 | 32 | 5 | 27 |

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🔍 Key Strategy:

To find the area of the shaded region, use this general method:

> Shaded Area = Area of Entire Shape – Area of Unshaded (White) Parts

This works when the shaded region is the remaining part after removing unshaded shapes.

Common formulas used:
- Rectangle: $ \text{length} \times \text{width} $
- Triangle: $ \frac{1}{2} \times \text{base} \times \text{height} $
- Trapezoid: $ \frac{1}{2} \times (\text{base}_1 + \text{base}_2) \times \text{height} $

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✔ All problems are solved correctly. Great practice!