find the perimeter and area of the polygon shown below

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31-03-2025

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Understanding how to calculate the perimeter and area of polygons is a fundamental skill in geometry. This guide will walk you through the process, providing clear explanations and examples. We'll focus on finding the perimeter and area of a specific polygon, but the principles apply to many different shapes.

Defining Perimeter and Area

Before we dive into calculations, let's clarify the terms:

• Perimeter: The total distance around the outside of a polygon. It's the sum of the lengths of all its sides.

• Area: The amount of space enclosed within a polygon. The calculation varies depending on the polygon's shape.

Example Polygon: A Composite Shape

Let's assume our polygon is a composite shape – a combination of simpler shapes. For example, imagine a shape that looks like a rectangle with a triangle attached to one side. (Insert image here of a rectangle with a triangle on top. The dimensions of the rectangle and triangle should be clearly labeled. Example: Rectangle – length 10cm, width 5cm; Triangle – base 10cm, height 4cm). Make sure the image is compressed for faster loading.

Calculating the Perimeter

1. Identify the Sides: List all the sides of the polygon and their lengths. In our example, we have two sides of the rectangle (10cm each), one side of the rectangle (5cm), another side of the rectangle (5cm) and the hypotenuse of the triangle.

2. Calculate the Hypotenuse (if necessary): If your polygon includes a triangle, you might need to use the Pythagorean theorem (a² + b² = c²) to find the length of any hypotenuse. In our example, the triangle's hypotenuse is √(4² + 5²) = √41 cm.

3. Sum the Sides: Add the lengths of all the sides together to find the perimeter. In our example: 10cm + 5cm + 10cm + 5cm + √41cm ≈ 30cm + 6.4cm ≈ 36.4cm. Therefore, the perimeter of the polygon is approximately 36.4 centimeters.

Calculating the Area

1. Divide into Simpler Shapes: Break down the complex polygon into simpler shapes like rectangles, triangles, squares, etc., whose area formulas you know. In our example, we have a rectangle and a triangle.

2. Calculate the Area of Each Shape:

   • Rectangle: Area = length × width = 10cm × 5cm = 50cm²
   • Triangle: Area = (1/2) × base × height = (1/2) × 10cm × 4cm = 20cm²

3. Sum the Areas: Add the areas of all the simpler shapes to find the total area of the polygon. In our example: 50cm² + 20cm² = 70cm². Therefore, the area of the polygon is 70 square centimeters.

Different Polygon Types: Formulas to Remember

For other common polygons, here are the area formulas:

• Square: Area = side²
• Rectangle: Area = length × width
• Triangle: Area = (1/2) × base × height
• Circle: Area = πr² (where r is the radius)

Remember to always include the correct units (e.g., cm, m, in) in your answers. For perimeter, the units are linear (cm, m, etc.). For area, the units are squared (cm², m², etc.).

How to Find the Perimeter and Area of Any Polygon

The steps outlined above can be adapted to find the perimeter and area of most polygons. The key is to:

1. Identify the shape: Determine the type of polygon (e.g., rectangle, triangle, irregular polygon).
2. Break it down (if necessary): Divide complex shapes into simpler, easily calculable shapes.
3. Apply the relevant formulas: Use the appropriate formula(s) for each component shape.
4. Combine the results: Add together the perimeters or areas of the individual shapes to get the final answer.

By mastering these techniques, you’ll be well-equipped to tackle any perimeter and area problem you encounter! Remember to always double-check your calculations and units.