Calculate The Perimeter Of A Rectangle With A Length Of 8.4 Dm And A Width Of 25 Cm. Calculate The Area Of The Rectangle. Determine The Length Of The Side Of A Square That Has A Perimeter Equal To The Perimeter Of The Rectangle.

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This article will guide you through calculating the perimeter and area of a rectangle, as well as determining the side length of a square with the same perimeter. We will use the given dimensions of the rectangle – a length of 8.4 decimeters and a width of 25 centimeters – to perform these calculations step-by-step. This problem combines basic geometric concepts and unit conversions, making it a great exercise for reinforcing mathematical skills. Whether you're a student learning geometry or just looking to brush up on your math, this article will provide a clear and comprehensive explanation.

a) Calculate the Perimeter of the Rectangle

The perimeter of a rectangle is the total distance around its four sides. The formula for the perimeter (P) of a rectangle is given by:

P = 2 * (length + width)

However, before we can apply this formula, we need to ensure that the length and width are expressed in the same units. The length is given in decimeters (dm), and the width is given in centimeters (cm). Let's convert the length from decimeters to centimeters. We know that 1 decimeter is equal to 10 centimeters. Therefore:

Length = 8.4 dm * 10 cm/dm = 84 cm

Now that both the length and width are in centimeters, we can calculate the perimeter:

Length = 84 cm Width = 25 cm

P = 2 * (84 cm + 25 cm) P = 2 * (109 cm) P = 218 cm

Thus, the perimeter of the rectangle is 218 cm. To further understand the concept, let’s break down why the perimeter is calculated this way. A rectangle has two pairs of equal sides: two lengths and two widths. By adding the length and width together, we get the sum of two adjacent sides. Multiplying this sum by 2 accounts for all four sides of the rectangle. This approach ensures we include every side in our calculation, providing the total distance around the rectangle. Moreover, understanding the units is crucial. Converting decimeters to centimeters ensures that our final answer is in a consistent unit, making it easier to interpret and use in further calculations.

b) Calculate the Area of the Rectangle

The area of a rectangle is the amount of space it covers, calculated by multiplying its length by its width. The formula for the area (A) of a rectangle is:

A = length * width

We have already converted the length to centimeters in the previous section, so we can use the values directly:

Length = 84 cm Width = 25 cm

A = 84 cm * 25 cm A = 2100 cm²

Therefore, the area of the rectangle is 2100 square centimeters. Let’s delve deeper into why we multiply length and width to find the area. Imagine the rectangle as a grid of tiny squares, each with a side length of 1 cm. The length tells us how many of these squares fit along one side, and the width tells us how many rows of these squares there are. Multiplying these two numbers gives us the total number of squares, which is the area. In this case, we have 84 squares along one side and 25 rows of squares, resulting in a total of 2100 squares. It’s also important to note the unit of area. Since we are multiplying centimeters by centimeters, the resulting unit is square centimeters (cm²), indicating that we are measuring a two-dimensional space. This understanding of area as a measure of space helps to visualize and interpret the numerical result more effectively.

c) Determine the Length of the Side of a Square with the Same Perimeter as the Rectangle

We know that the perimeter of the rectangle is 218 cm. A square has four equal sides, so to find the length of one side, we divide the perimeter by 4. The formula for the perimeter (P) of a square is:

P = 4 * side

We can rearrange this formula to solve for the side length:

side = P / 4

Using the perimeter of the rectangle (218 cm):

side = 218 cm / 4 side = 54.5 cm

Thus, the length of the side of a square with the same perimeter as the rectangle is 54.5 cm. To further clarify, let’s consider why dividing the perimeter by 4 gives us the side length of the square. A square, by definition, has four equal sides. Therefore, the total distance around the square (the perimeter) is simply four times the length of one side. By dividing the perimeter by 4, we are essentially distributing the total length equally among the four sides. This approach ensures that each side is of the same length, which is a fundamental property of squares. Understanding this relationship between the perimeter and the side length of a square helps to easily calculate one from the other, making it a useful concept in various geometric problems. Moreover, knowing the side length allows us to calculate other properties of the square, such as its area.

In summary, we have successfully calculated the perimeter and area of the given rectangle and determined the side length of a square with the same perimeter. This exercise highlights the importance of understanding basic geometric formulas and unit conversions in solving mathematical problems. By breaking down each step and providing detailed explanations, we hope this article has clarified the concepts and enhanced your problem-solving skills. Remember, practice is key to mastering these concepts, so continue to explore similar problems and deepen your understanding of geometry.