Hello my dear students! Welcome to today's mathematics lesson. I am so happy to see you all here, ready to learn something new and interesting. Today, we are going to study Chapter 6, which is all about Perimeter and Area. Now, before we begin, let me tell you that this is a very practical chapter. You will see how these concepts are used in our everyday life, whether it's while fencing a garden, laying carpet in a room, or even when you're playing in a park. So, pay attention and let's begin our journey into the world of measurements!
Let us start with Section 6.1, which is about Perimeter. Now, students, do you remember what the perimeter of a closed plane figure is? Let me refresh your understanding! The perimeter of any closed plane figure is the distance covered along its boundary when you go around it once. Imagine you are walking around a park, and you walk along the entire boundary of that park and come back to where you started. The total distance you walked is the perimeter of that park. For a polygon, which is a closed plane figure made up of line segments, the perimeter is simply the sum of the lengths of all its sides. So, we can say that the perimeter of a polygon equals the sum of the lengths of all its sides. That is the total distance along its outer boundary. Very simple, isn't it?
Now, let us revise the formulas for the perimeter of rectangles, squares, and triangles. These are the most common shapes we encounter, and you must remember their formulas.
First, let us talk about the perimeter of a rectangle. Consider a rectangle ABCD whose length is 12 cm and breadth is 8 cm. What will be its perimeter? Let me show you how to calculate it. The perimeter of a rectangle is the sum of the lengths of its four sides. So, we add AB plus BC plus CD plus DA. Now, in a rectangle, opposite sides are equal, so AB equals CD, and BC equals DA. Therefore, the perimeter becomes AB plus BC plus AB plus BC, which is 2 times AB plus 2 times BC. This can be written as 2 times the sum of length and breadth. So, substituting the values, we have 2 times (12 cm plus 8 cm), which is 2 times 20 cm, which equals 40 cm. So students, the perimeter of this rectangle is 40 cm.
From this example, we can see that the perimeter of a rectangle equals length plus breadth plus length plus breadth. In other words, it is 2 times the sum of its length and breadth. So, remember this formula: Perimeter of a rectangle equals 2 multiplied by (length plus breadth). The perimeter of a rectangle is twice the sum of its length and breadth. This is a very important formula, and you will use it throughout your life.
Now, let us look at the perimeter of a square. Suppose Debojeet wants to put colored tape all around a square photo frame of side 1 meter. What will be the length of the colored tape he requires? Since he wants to put the tape all around the square photo frame, he needs to find the perimeter of the photo frame. The length of the tape required equals the perimeter of the square, which is the sum of the lengths of all four sides of the square. So, we add 1 meter plus 1 meter plus 1 meter plus 1 meter, which equals 4 meters. Now, we know that all four sides of a square are equal in length. Therefore, instead of adding the lengths of each side, we can simply multiply the length of one side by 4. So, the length of the tape required equals 4 multiplied by 1 meter, which is 4 meters. From this example, we can see that the perimeter of a square equals 4 times the length of a side. In other words, the perimeter of a square is quadruple the length of its side. So, remember: Perimeter of a square equals 4 multiplied by the side length.
Now, let us look at the perimeter of a triangle. Consider a triangle having three given sides of lengths 4 cm, 5 cm, and 7 cm. Find its perimeter. This is very simple. The perimeter of a triangle is the sum of the lengths of its three sides. So, we add 4 cm plus 5 cm plus 7 cm, which equals 16 cm. So, the perimeter of this triangle is 16 cm. Remember: Perimeter of a triangle equals the sum of the lengths of its three sides.
Now students, let me give you some worked examples so that you understand how to apply these formulas in real life.
Here is Example 1: Akshi wants to put lace all around a rectangular tablecloth that is 3 meters long and 2 meters wide. Find the length of the lace required. Now, the length of the rectangular table cover is 3 meters, and the breadth is 2 meters. Akshi wants to put lace all around the tablecloth. Therefore, the length of the lace required will be the perimeter of the rectangular tablecloth. Now, the perimeter of the rectangular tablecloth equals 2 times (length plus breadth), which is 2 times (3 m plus 2 m), which is 2 times 5 m, which equals 10 m. Hence, the length of the lace required is 10 meters. Simple, isn't it?
Here is another example: Find the distance traveled by Usha if she takes three rounds of a square park of side 75 m. First, we find the perimeter of the square park, which is 4 times the length of a side, so 4 times 75 m equals 300 m. This is the distance covered by Usha in one round. Therefore, the total distance traveled by Usha in three rounds is 3 times 300 m, which equals 900 m. So, Usha travels 900 meters in total.
Now students, let me tell you about regular polygons. Like squares, closed figures that have all sides and all angles equal are called regular polygons. We studied the sequence of regular polygons in Chapter 1. Examples of regular polygons are the equilateral triangle, where all three sides and all three angles are equal, and the regular pentagon, where all five sides and all five angles are equal, and so on.
Now, what is the perimeter of an equilateral triangle? We know that for any triangle, its perimeter is the sum of all three sides. Using this understanding, we can find the perimeter of an equilateral triangle. Since all three sides are equal, the perimeter of an equilateral triangle equals 3 times the length of one side. So, remember: Perimeter of an equilateral triangle equals 3 multiplied by the length of a side.
Now, can you think of a similarity between a square and an equilateral triangle? Well, both are regular polygons! A square has 4 equal sides and 4 equal angles, while an equilateral triangle has 3 equal sides and 3 equal angles. Both are examples of regular polygons.
Now, students, I want you to remember a general formula for the perimeter of any regular polygon. Since all sides are equal in a regular polygon, the perimeter simply equals the number of sides multiplied by the length of one side. So, if you have a regular pentagon with 5 sides, its perimeter is 5 times the side length. If you have a regular hexagon with 6 sides, its perimeter is 6 times the side length, and so on. This is a very useful generalization.
Now, let us move on to Section 6.2, which is about Area. Students, we have studied the areas of closed figures in previous grades. Let me recall some key points for you.
The amount of region enclosed by a closed figure is called its area. Think of it like this: if you have a piece of land and you want to know how much space that land covers, that is its area. It is the measure of the region inside the boundary of the figure.
In previous grades, we arrived at the formula for the area of a rectangle and a square using square grid paper. Do you remember? Let me remind you. The area of a square equals side multiplied by side, or side squared. And the area of a rectangle equals length multiplied by breadth. We measure area in square units, like square centimeters, square meters, and so on.
Now, let us look at some real-life problems related to these ideas.
Here is an example: A floor is 5 meters long and 4 meters wide. A square carpet of sides 3 meters is laid on the floor. Find the area of the floor that is not carpeted. Now, the length of the floor is 5 m, and the width is 4 m. So, the area of the floor is length times width, which is 5 m times 4 m, which equals 20 square meters. The length of the square carpet is 3 m, so its area is 3 m times 3 m, which equals 9 square meters. Hence, the area of the floor laid with carpet is 9 square meters. Therefore, the area of the floor that is not carpeted is the area of the floor minus the area of the floor laid with carpet, which is 20 sq m minus 9 sq m, which equals 11 sq m. So, 11 square meters of the floor is not carpeted.
Here is another example: Four square flower beds each of side 4 m are in four corners on a piece of land 12 m long and 10 m wide. Find the area of the remaining part of the land. Now, the length of the land is 12 m, and the width is 10 m. So, the area of the whole land is length times width, which is 12 m times 10 m, which equals 120 square meters. The side length of each of the four square flower beds is 4 m. So, the area of one flower bed is 4 m times 4 m, which is 16 square meters. Hence, the area of the four flower beds is 4 times 16 sq m, which equals 64 sq m. Therefore, the area of the remaining part of the land is the area of the complete land minus the area of all four flower beds, which is 120 sq m minus 64 sq m, which equals 56 sq m. So, the remaining area is 56 square meters.
Now, students, let me tell you about estimating the area of irregular shapes. We can estimate the area of any simple closed shape by using a sheet of squared paper or graph paper, where every square measures 1 unit by 1 unit, or 1 square unit. To estimate the area, we can trace the shape onto a piece of transparent paper and place it on the squared paper. Then we follow these conventions: first, the area of one full small square is taken as 1 square unit. Second, we ignore portions of the area that are less than half a square. Third, if more than half of a square is in a region, we just count it as 1 square unit. Fourth, if exactly half the square is counted, we take its area as half a square unit. This is a very useful method for estimating the area of irregular shapes.
Now, students, let me tell you why area is generally measured using squares. If you try to draw a circle on a graph sheet and count the squares to estimate the area, you will see that circles cannot be packed tightly without gaps in between. So, it is difficult to get an accurate measurement of area using circles as units. The same rectangle, when packed with circles in different ways, gives different counts. But squares, on the other hand, can be arranged perfectly without any gaps. That is why squares are the best shape to use to measure area. They fit together perfectly and make calculation easy.
Now, let us move on to Section 6.3, which is about the Area of a Triangle. This is a very interesting section, and I want you to pay special attention here.
Draw a rectangle on a piece of paper and draw one of its diagonals. Now, cut the rectangle along that diagonal, and you will get two triangles. Check whether the two triangles overlap each other exactly. Do they have the same area? Yes, they do! Try this with more rectangles having different dimensions. You can check this for a square as well. You will find that when you cut a rectangle along its diagonal, you always get two triangles of equal area. This is a very important observation.
Now, look at the figures below. Is the area of the blue rectangle more or less than the area of the yellow triangle? Or is it the same? Can you see some relationship between the blue rectangle and the yellow triangle and their areas?
Let me explain this to you. When you draw a diagonal in a rectangle, you divide the rectangle into two triangles. Each of these triangles has exactly half the area of the rectangle. So, the area of a triangle is half the area of the rectangle that has the same base and height. This is a very important formula that you must remember: The area of a triangle equals half times base times height.
Now, let me give you another example. Consider triangle ABE, which is formed by taking half of one rectangle and half of another rectangle. The area of triangle ABE equals the area of triangle AEF plus the area of triangle BEF. Here, the area of triangle AEF is half the area of rectangle AFED. Similarly, the area of triangle BEF is half the area of rectangle BFEC. Thus, the area of triangle ABE equals half the area of rectangle AFED plus half the area of rectangle BFEC, which equals half the sum of the areas of the rectangles AFED and BFEC, which equals half the area of rectangle ABCD. So, the conclusion is that the area of any triangle formed by drawing a diagonal or by combining halves of rectangles is always half the area of the rectangle that contains it.
So, students, remember this very important relationship: The area of a triangle is always half the area of a rectangle that has the same base and the same height. This is why the formula for the area of a triangle is half times base times height.
Now, let me summarize what we have learned in this chapter.
First, we learned about perimeter. The perimeter of any closed plane figure is the distance covered along its boundary when you go around it once. For polygons, it is the sum of the lengths of all sides. Specifically, the perimeter of a rectangle is twice the sum of its length and breadth, the perimeter of a square is four times the length of any one of its sides, and the perimeter of a triangle is the sum of the lengths of its three sides. For any regular polygon, the perimeter equals the number of sides multiplied by the length of one side.
Then, we learned about area. The area of a closed figure is the measure of the region enclosed by the figure. We measure area in square units. The area of a rectangle is its length times its width, and the area of a square is the length of any one of its sides multiplied by itself. We also learned how to estimate the area of irregular shapes using grid paper.
Finally, we learned about the area of a triangle. We discovered that when you cut a rectangle along its diagonal, you get two triangles of equal area. Therefore, the area of a triangle is half the area of a rectangle that has the same base and height. So, the formula for the area of a triangle is half times base times height.
We also learned an important concept: two closed figures can have the same area with different perimeters, or the same perimeter with different areas. This is a very interesting property of shapes.
Students, this is all for today. I hope you have understood all the concepts clearly. Remember to practice the formulas and solve problems to strengthen your understanding. Thank you for listening attentively, and I will see you in the next lesson!