https://file.aiquickdraw.com/mathbot/file/17268543528465umm9n2x.jpg

The given image shows an irregular polygon, and we are tasked with calculating its area. The polygon is composed of several rectangles and right-angled trapezoids. Let's break down the shape to compute the area step by step.

We are provided with the following dimensions:
- \( a = 8 \, \text{cm} \)
- \( b = 6 \, \text{cm} \)
- \( c = 4 \, \text{cm} \)

### Step-by-Step Area Calculation

1. **Divide the shape:**
   The polygon can be divided into three key areas:
   - Two rectangles (on the left and right sides)
   - One central trapezoid

2. **Calculate the areas of the rectangles:**
   - The left rectangle has dimensions \( a \times c \), so its area is:
     \[
     A_{\text{left rectangle}} = a \times c = 8 \, \text{cm} \times 4 \, \text{cm} = 32 \, \text{cm}^2
     \]
   - The right rectangle has the same dimensions as the left, so its area is also:
     \[
     A_{\text{right rectangle}} = 32 \, \text{cm}^2
     \]

3. **Calculate the area of the trapezoid:**
   The central trapezoid has two parallel sides of lengths \( a \) and \( b \), and its height is \( c \). The formula for the area of a trapezoid is:
   \[
   A_{\text{trapezoid}} = \frac{1}{2} \times (a + b) \times c
   \]
   Substituting the values:
   \[
   A_{\text{trapezoid}} = \frac{1}{2} \times (8 \, \text{cm} + 6 \, \text{cm}) \times 4 \, \text{cm} = \frac{1}{2} \times 14 \, \text{cm} \times 4 \, \text{cm} = 28 \, \text{cm}^2
   \]

4. **Total area:**
   Now, sum the areas of all the regions:
   \[
   A_{\text{total}} = A_{\text{left rectangle}} + A_{\text{right rectangle}} + A_{\text{trapezoid}} = 32 \, \text{cm}^2 + 32 \, \text{cm}^2 + 28 \, \text{cm}^2 = 92 \, \text{cm}^2
   \]

Thus, the total area of the polygon is:
\[
A = 92 \, \text{cm}^2
\]

Would you like further clarification on any of these steps, or do you have any related questions?

Here are five related questions to expand your understanding:
1. How do you calculate the area of a triangle given its base and height?
2. What is the formula for the area of a parallelogram, and how does it differ from a rectangle?
3. How can you find the area of a polygon if it cannot be easily divided into rectangles or trapezoids?
4. Can you apply the Pythagorean theorem to find missing side lengths in irregular polygons?
5. How would the calculation change if the polygon had different side lengths?

**Tip:** When working with irregular polygons, always try to break the shape into simpler geometric figures like triangles, rectangles, and trapezoids for easier area calculation.

Berechnen Sie den Flächeninhalt des Vielecks, wenn a = 8cm, b = 6cm und c = 4cm lang ist.

This tutorial walks through the step-by-step calculation of the area of a polygon with side lengths a = 8 cm, b = 6 cm, and c = 4 cm. The shape is decomposed into two rectangles and a trapezoid. Learn how to apply basic area formulas to solve similar problems.

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