Math Problem Statement

What is the area of the shaded region in a figure where the larger rectangle has dimensions 14 ft by 12 ft, and the inner square has dimensions 8 ft by 8 ft?

Solution

To find the area of the shaded region, follow these steps:

  1. Calculate the area of the larger rectangle:

    • The dimensions are given as 14ft14 \, \text{ft} (width) and 12ft12 \, \text{ft} (height).
    • Area of the larger rectangle = 14×12=168square feet14 \times 12 = 168 \, \text{square feet}.
  2. Calculate the area of the inner (unshaded) square:

    • The dimensions are 8ft×8ft8 \, \text{ft} \times 8 \, \text{ft}.
    • Area of the inner square = 8×8=64square feet8 \times 8 = 64 \, \text{square feet}.
  3. Subtract the area of the inner square from the larger rectangle:

    • Area of the shaded region = 16864=104square feet168 - 64 = 104 \, \text{square feet}.

Thus, the area of the shaded region is 104 square feet.

Would you like further details or have any questions?


Here are 5 related questions to expand your understanding:

  1. How do you calculate the area of composite shapes that include circles?
  2. What would the shaded area be if the inner square had different dimensions?
  3. How would the solution change if the outer shape were a circle instead of a rectangle?
  4. Can you derive a formula for the shaded area if only the dimensions are given symbolically?
  5. How do perimeter calculations differ from area calculations in such problems?

Tip: Always ensure you subtract the area of the inner shape from the outer shape to get the shaded area correctly.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Geometry
Area Calculation
Subtraction of Areas

Formulas

Area of a rectangle = width × height
Area of a square = side × side
Shaded area = Area of larger shape - Area of inner shape

Theorems

Basic Area Theorem

Suitable Grade Level

Grades 6-8