Math Problem Statement

What is the area of this figure?

Solution

To find the area of this irregular polygon, we can break the shape into smaller, simpler shapes (rectangles, triangles) and calculate the area of each one individually, then sum them.

Looking at the figure, it can be divided into:

  1. A large rectangle on the left:

    • Dimensions: 9 ft by 14 ft
    • Area = 9ft×14ft=126square feet9 \, \text{ft} \times 14 \, \text{ft} = 126 \, \text{square feet}
  2. A smaller rectangle on the right top:

    • Dimensions: 3 ft by 6 ft
    • Area = 3ft×6ft=18square feet3 \, \text{ft} \times 6 \, \text{ft} = 18 \, \text{square feet}
  3. A right triangle on the bottom right:

    • Base: 9 ft (horizontal length)
    • Height: 6 ft (vertical length)
    • Area = 12×9ft×6ft=27square feet\frac{1}{2} \times 9 \, \text{ft} \times 6 \, \text{ft} = 27 \, \text{square feet}

Total Area:

Now, sum the areas of the three parts:

Total Area=126sq ft+18sq ft+27sq ft=171square feet\text{Total Area} = 126 \, \text{sq ft} + 18 \, \text{sq ft} + 27 \, \text{sq ft} = 171 \, \text{square feet}

Thus, the total area of the figure is 171 square feet.

Would you like a more detailed breakdown, or do you have any questions?

Here are some related questions to expand on this:

  1. How do you calculate the area of a trapezoid?
  2. What are the different ways to decompose irregular shapes into simpler shapes?
  3. How does the Pythagorean theorem relate to calculating the area of triangles?
  4. How would you calculate the perimeter of this figure?
  5. What are the formulas for the area of polygons, like parallelograms or trapezoids?

Tip: When working with irregular shapes, dividing them into rectangles and triangles makes calculating areas simpler.

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Math Problem Analysis

Mathematical Concepts

Geometry
Area Calculation
Decomposition of Shapes

Formulas

Area of a rectangle = length × width
Area of a triangle = 1/2 × base × height

Theorems

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Suitable Grade Level

Grades 6-8