Geometry Challenge: Rectangles, Perimeter, And Area

Hey guys! Let's dive into a fun geometry problem that's all about rectangles, their perimeters, and their areas. This one's a classic, and it's a great way to brush up on some fundamental math skills. We're going to explore a figure made up of three identical rectangles, each with specific dimensions, and then we'll calculate its perimeter and area. Are you ready to get started?

Understanding the Problem: Three Rectangles in a Row

Okay, so the core of our problem involves a figure formed by three equal rectangles. These rectangles are lined up next to each other, creating a longer, combined shape. Each individual rectangle has a width of 4 cm and a length of p cm. Think of it like this: imagine three identical doors placed side by side. The length of each door is p, and the width is always 4.

Now, before we get into the calculations, let's make sure we're clear on the definitions. The perimeter of a shape is the total distance around its outside edge. It's like walking around the entire shape and measuring the total length of your walk. The area, on the other hand, is the amount of space the shape covers. It's usually measured in square units, like square centimeters (cm²).

The challenge has two main parts. In the first, we need to express the perimeter and area of the entire combined shape in terms of p. This means our answers will contain the variable p. In the second part, we will substitute p with a specific number (9 cm) to find the actual values of the perimeter and area. Let's break down the problem step by step, making sure we understand everything before moving forward.

So, to recap, we have:

• Three identical rectangles
• Each rectangle has dimensions of p cm and 4 cm
• We need to find the perimeter and area in terms of p
• We need to find the perimeter and area when p = 9 cm

Calculating the Perimeter of the Combined Figure

Alright, let's tackle the perimeter first. Remember, the perimeter is the total distance around the outside of the combined figure. Imagine walking around the entire shape, measuring each side. How do we go about this?

Step 1: Identify the sides.

Looking at our figure, we have:

• Three sides with a length of p cm (the longer sides of the rectangles)
• Two sides with a length of 4 cm (the shorter sides at the top and bottom)
• Two sides with a length of 4 cm (the shorter sides on the sides of combined shape)

Step 2: Add up all the sides.

To find the perimeter, we simply add the lengths of all the sides together. In our case:

Perimeter = p + p + p + 4 + 4 + 4 + 4

Step 3: Simplify the expression.

We can simplify the equation by combining like terms. We have three p's, so we can write that as 3p. We also have two 4's plus two 4's which equal 16. So the perimeter will be:

Perimeter = 3p + 16 cm

So there you have it. We've successfully expressed the perimeter of the combined figure in terms of p. The perimeter is (3p + 16) cm. Now that's pretty neat. Remember, this formula will work for any value of p.

Calculating the Area of the Combined Figure

Now, let's switch gears and calculate the area of the combined figure. Remember, the area tells us how much space the shape covers. The area of a rectangle is calculated by multiplying its length by its width (Area = length * width).

Step 1: Calculate the area of a single rectangle.

The area of one of the individual rectangles is p cm * 4 cm = 4p cm².

Step 2: Recognize the big picture.

Since we have three identical rectangles, the total area of the combined figure will be three times the area of a single rectangle.

Step 3: Calculate the total area.

To find the total area, we multiply the area of one rectangle (4p) by 3:

Total Area = 3 * (4p) = 12p cm²

So, we've found that the area of the combined figure, in terms of p, is 12p cm². Easy, right? The area depends directly on the value of p.

Finding the Perimeter and Area When p = 9 cm

We've expressed both the perimeter and the area in terms of p. Now, let's put it all together by finding the actual values of the perimeter and area when p = 9 cm. This is the final step in our problem. It’s all about plugging in the numbers!

Step 1: Substitute p = 9 into the perimeter formula.

Remember, the perimeter formula is 3p + 16 cm. We replace p with 9:

Perimeter = 3 * 9 + 16 = 27 + 16 = 43 cm

So, when p = 9 cm, the perimeter of the combined figure is 43 cm.

Step 2: Substitute p = 9 into the area formula.

The area formula is 12p cm². We replace p with 9:

Area = 12 * 9 = 108 cm²

So, when p = 9 cm, the area of the combined figure is 108 cm².

Solution Summary

Let’s summarize our findings:

1. Perimeter in terms of p: 3p + 16 cm
2. Area in terms of p: 12p cm²
3. Perimeter when p = 9 cm: 43 cm
4. Area when p = 9 cm: 108 cm²

Conclusion

Well done, guys! You've successfully worked through this geometry problem. We started with a figure made of three rectangles, calculated its perimeter and area in terms of p, and then found the specific values when p equaled 9 cm. This problem is an excellent exercise in understanding perimeter, area, and working with variables. Keep practicing, and you'll become a geometry pro in no time!

Keep in mind:

• Always remember the units (cm for perimeter, cm² for area).
• Carefully identify all the sides when calculating the perimeter.
• Understand that the area of the combined figure is the sum of the areas of the individual rectangles.
• Practice these types of problems to build your skills and confidence.

I hope you found this explanation helpful and easy to follow. Thanks for joining me on this geometry adventure! Feel free to ask questions or share your thoughts in the comments below. Have fun with math, and keep exploring! Keep learning and practicing, and you will master geometry!