Calculating Areas: A Step-by-Step Guide

Hey guys! Let's dive into some geometry and figure out how to calculate the areas of different shapes. We'll break down the problems step-by-step, so it's super easy to follow along. This guide is all about finding areas, so grab your pencils and let's get started. We'll explore two different figures, each with its own set of shapes. This will help you understand how to calculate areas of combined shapes, so you can solve problems like a pro. Get ready to flex those math muscles and learn some cool stuff! By the end of this guide, you will be able to solve the area of complex shapes.

Figure 1: Decomposing into Rectangles

Okay, let's start with Figure 1. This one is a bit of a jigsaw puzzle, but don't worry, we'll put it together piece by piece. Figure 1 is composed of three rectangles. Each rectangle has its own height and base, and we need to find the area of each one and add them up. It's like building with blocks – each block (rectangle) has its own size, and when you put them all together, you get the total area. Remember, the area of a rectangle is calculated by multiplying its height by its base. This is the foundation of our calculation, so let's get into it.

Rectangle 1: Dimensions and Area

Let's start with the first rectangle. It has a height of $5x$ and a base of $6y$. To find the area, we simply multiply these two dimensions together. So, the area of the first rectangle is $5x * 6y$. When we multiply these terms, we get $30xy$. Therefore, the area of the first rectangle is $30xy$ square units. Not too bad, right? We've successfully calculated the area of the first part of our figure. Keep in mind that we're using $x$ and $y$ as variables, meaning we don't have exact numerical values, but we can still express the area in terms of these variables. This approach is really common in algebra and geometry, so it's a super useful skill to have. Knowing the area helps you find how much space the rectangle is taking up, and it's essential when we want to get the total area of the entire figure.

Rectangle 2: Dimensions and Area

Alright, moving on to the second rectangle. It has a height of $4x$ and a base of $2y$. Following the same formula, the area of this rectangle is $4x * 2y$. Multiplying these terms gives us $8xy$. So, the area of the second rectangle is $8xy$ square units. We're making great progress! By understanding these individual areas, you are able to take on any shape. Keep in mind that as the base and height change so does the area. Keep going, and we are going to calculate the area of the third and final rectangle.

Rectangle 3: Dimensions and Area

Last one for Figure 1! The third rectangle has a height of $3x$ and a base of $3y$. Calculating the area, we get $3x * 3y$, which equals $9xy$. Therefore, the area of the third rectangle is $9xy$ square units. We've now found the areas of all three rectangles that make up Figure 1. Now we'll combine these individual areas to find the total area. Good job on finding the final piece to this figure!

Total Area of Figure 1

To find the total area of Figure 1, we need to add up the areas of all three rectangles we just calculated. So, we add $30xy$ (from Rectangle 1) + $8xy$ (from Rectangle 2) + $9xy$ (from Rectangle 3). Adding these together, we get $30xy + 8xy + 9xy = 47xy$. Therefore, the total area of Figure 1 is $47xy$ square units. That wasn't so tough, right? We broke down the complex shape into simpler parts, calculated their individual areas, and then added them up to find the total area. This strategy is super helpful for solving a lot of geometry problems, not just this one. Understanding how to break down complex shapes helps us when finding the area of multiple shapes! Awesome!

Figure 2: Combining a Square and a Rectangle

Now, let's move on to Figure 2. This figure is a combo of a square and a rectangle. The strategy here is similar: calculate the area of each shape separately and then add them together. This time, we're dealing with a square and a rectangle, so we need to remember the specific formulas for each. It's like mixing ingredients in a recipe – each ingredient (shape) contributes to the final result (total area). Let's go through this together, and you will understand how easy this is!

The Square: Dimensions and Area

First, let's calculate the area of the square. A square is a special type of rectangle where all sides are equal. Let's assume the side length of the square is $s$. The area of a square is calculated by multiplying the side length by itself (side * side, or $s^2$). So, if the side length of our square is, for instance, $5$, then the area would be $5 * 5 = 25$ square units. We'll use the variable $s$ to represent the side length, so the area of our square is $s^2$ square units. Understanding how squares and rectangles are calculated can help when working with different shapes.

The Rectangle: Dimensions and Area

Next, let's look at the rectangle that is attached to the square. Remember, the area of a rectangle is found by multiplying its length by its width. Let's say the length of the rectangle is $l$ and the width is $w$. The area of the rectangle would then be $l * w$ square units. For example, if the length is $10$ and the width is $4$, the area would be $10 * 4 = 40$ square units. Now, you should easily calculate the area of the second figure.

Total Area of Figure 2

To find the total area of Figure 2, we add the area of the square ($s^2$) to the area of the rectangle ($l*w$). So, the total area of Figure 2 is $s^2 + l*w$ square units. Just remember to add all the shapes together to calculate the area. You've now learned how to find the area of two different complex shapes. You are awesome!

Conclusion: Mastering Area Calculations

Wow, you've made it to the end, guys! We've successfully calculated the areas of two different figures by breaking them down into simpler shapes and using the formulas for rectangles and squares. Remember, the key is to take it step by step, calculate the area of each individual shape, and then add them up to find the total area. Keep practicing, and you'll become a pro at these problems! We hope this guide was super helpful. Keep up the great work! Always remember the different shapes and their formulas. You've got this!