Hello, and welcome to today's mathematics lesson! Today, we are going to explore perimeter and area of plane figures. By the end of this lesson, you will understand what perimeter and area mean, how to calculate them for rectangles, squares, triangles, and other shapes, and how these concepts connect to real-life situations.
Let us begin with some basics. In mathematics, we often need to measure things. When you buy cloth for a shirt, you measure by length. When you buy a plot of land, you measure by area. When you buy milk or petrol, you measure by volume. These measurements all fall under a branch called mensuration.
First, let us understand what a closed figure is. Any plane shape that is bounded by lines — whether straight or curved — is called a plane closed figure. Think of a rectangle, a triangle, a circle, or even a pentagon. All of these are closed figures because they completely enclose a space.
Now, when we talk about the region of a figure, we mean the interior of that figure plus its boundary. Imagine a rectangle drawn on paper. The space inside the rectangle is its interior. When we include the edges — the boundary — along with that interior space, we call this the rectangular region. Similarly, for a triangle, the shaded interior plus its three sides form the triangular region.
Let us move to our first major concept: perimeter.
Imagine you are walking around the edge of a garden, starting at one corner and walking all the way around until you return to where you started. The total distance you walked is the perimeter of that garden.
Here is the precise definition: the perimeter of a closed figure is the length of its boundary. The unit of perimeter is the same as the unit of length — centimetres, metres, or kilometres.
For example, consider a triangle with sides measuring 3.5 centimetres, 4 centimetres, and 4.5 centimetres. To find its perimeter, simply add all three sides: 3.5 plus 4 plus 4.5, which equals 12 centimetres.
Or take a four-sided figure with sides 5 centimetres, 7 centimetres, 6 centimetres, and 8 centimetres. Its perimeter is 5 plus 7 plus 6 plus 8, which equals 26 centimetres.
One important tip: always convert all measurements to the same unit before adding. If one side is in metres and another in centimetres, convert them first. Remember, 1 metre equals 100 centimetres.
Now let us look at specific formulas for common shapes.
For a rectangle, the perimeter is twice the sum of its length and breadth. If we call length l and breadth b, then the formula is: perimeter equals 2(l + b). This makes sense because a rectangle has two lengths and two breadths.
From this formula, we can also find the length if we know the perimeter and breadth: length equals perimeter divided by 2, minus breadth. Similarly, breadth equals perimeter divided by 2, minus length.
For a square, since all four sides are equal, the perimeter is simply 4 times the side. If each side is l, then perimeter equals 4l. Conversely, if you know the perimeter, you can find the side by dividing the perimeter by 4.
For an equilateral triangle — where all three sides are equal — the perimeter is 3 times the side. This pattern continues: a regular pentagon has perimeter equal to 5 times its side, and a regular hexagon has perimeter equal to 6 times its side.
Let us work through some examples together.
Suppose a rectangle has length 50 centimetres and breadth 40 centimetres. Its perimeter equals 2(50 + 40), which is 2 times 90, giving us 180 centimetres, or 1.8 metres.
What if the units differ? Say length is 6 metres and breadth is 80 centimetres. First, convert 80 centimetres to metres: 80 divided by 100 equals 0.8 metres. Then perimeter equals 2(6 + 0.8), which is 2 times 6.8, giving 13.6 metres.
Here is a practical problem: a rectangular field has length 200 metres and breadth 160 metres. First, the perimeter is 2(200 + 160), which equals 720 metres. This is also the length of fence needed to enclose the field. If fencing costs 50 rupees per metre, the total cost is 720 times 50, which equals 36,000 rupees.
Now consider this interesting situation: a square field has each side 80 metres, while a rectangular field has length 120 metres and breadth 60 metres. The square has perimeter 4 times 80, which is 320 metres. The rectangle has perimeter 2(120 + 60), which is 360 metres. So the rectangular field has the greater perimeter, by 40 metres.
Here is something fascinating: different shapes can have the same perimeter.
Imagine an equilateral triangle with each side 12 centimetres. Its perimeter is 3 times 12, which is 36 centimetres. Now imagine a rectangle with length 13 centimetres and breadth 5 centimetres. Its perimeter is 2(13 + 5), which is also 36 centimetres. And a square with side 9 centimetres has perimeter 4 times 9, which is again 36 centimetres.
Three completely different shapes, yet the same perimeter! This shows that perimeter depends only on the total boundary length, not on the shape itself.
This leads to practical problems involving wires bent into different shapes. Suppose a wire is bent into a square of side 25 centimetres. The wire length is 4 times 25, which is 100 centimetres. If the same wire is bent into a rectangle with length 30 centimetres, what is its breadth? Since the perimeter must still be 100 centimetres, we have 100 cm = 2(30 cm + breadth). This gives 100 cm = 60 cm + 2 × breadth, so 40 cm = 2 × breadth, giving breadth = 20 cm.
Now we turn to our second major concept: area.
While perimeter tells us about the boundary, area tells us about the space inside. The area of a plane figure is the measure of the region enclosed by that figure.
For a rectangle, the area is found by multiplying length by breadth. So area equals length times breadth, or A = l × b. The unit of area is always a square unit — square centimetres or square metres — because you are multiplying two lengths together.
For a square, since length equals breadth, the area is simply side squared. Area equals side squared, or A = (side)². If you know the area, you can find the side using: side equals square root of area.
From the area formula, we can also derive: length equals area divided by breadth, that is l = A/b, and breadth equals area divided by length, that is b = A/l.
Let us work through some area examples.
A rectangle has length 1.4 metres and breadth 0.5 metres. Its area is 1.4 times 0.5, which equals 0.70 square metres, written as 0.70 m².
A square has side 2.4 centimetres. Its area is 2.4 times 2.4, which equals 5.76 square centimetres, written as 5.76 cm².
If a rectangle has area 45 square centimetres and breadth 10 centimetres, its length is 45 cm²/10 cm, which equals 4.5 centimetres.
Here is a combined problem: a square field has perimeter 112 metres. First, find its side: since perimeter equals 4 times side, we have 4 × side = 112 m, so side equals 112/4 metres equals 28 metres. Then its area is 28 squared, which is 28 times 28, giving 784 m².
Conversely, if a square field has area 1600 square metres, its side is the square root of 1600, which is 40 metres. Its perimeter is then 4 times 40, which equals 160 metres.
Let us explore how changing dimensions affects area.
If both length and breadth of a rectangle are doubled, what happens to the area? Suppose original length is l and breadth is b. Original area is l × b. New length is 2l and new breadth is 2b. New area is 2l × 2b, which equals 4 × l × b, or 4A — four times the original area.
What if length is tripled and breadth is doubled? New area would be 3l × 2b, which equals 6 × l × b, or 6A — six times the original area. The increase in area is 6A − A, which equals 5A, or five times the original area.
For a square with original side s and area s², if each side is tripled, the new area is (3s)², which equals 9s² — nine times the original area. If each side is halved, the new area is (s/2)², which equals s²/4 — one-fourth of the original area.
Finally, let us look at a practical tiling problem.
A square hall has side 8 metres. Its area is 8 squared, which equals 64 square metres. We want to cover this with square tiles of side 40 centimetres. First, convert 40 centimetres to 0.4 metres. Each tile has area 0.4 times 0.4, which equals 0.16 m². The number of tiles needed is 64 divided by 0.16, which equals 64 × 100/16, giving 400 tiles.
Alternatively, work entirely in centimetres: the hall is 800 centimetres by 800 centimetres, and each tile is 40 centimetres by 40 centimetres. Along each side, we need 800 divided by 40, which is 20 tiles. So total tiles needed is 20 times 20, which equals 400 tiles.
Let us recap the key takeaways from today's lesson.
First, the perimeter of a closed figure is the length of its boundary, measured in centimetres or metres. For a rectangle, perimeter equals 2(l + b). For a square, perimeter equals 4 × side, and side equals perimeter divided by 4. For an equilateral triangle, perimeter equals 3 × side, and this pattern continues for regular polygons — a regular pentagon has perimeter 5 times side, a regular hexagon 6 times side.
Second, different shapes can have the same perimeter — this helps solve problems where a wire is bent into different shapes.
Third, the area of a rectangle equals length times breadth, or A = l × b, measured in square units. The area of a square equals side squared, or A = (side)². From area, you can find length as area divided by breadth, or breadth as area divided by length.
Fourth, perimeter and area are connected — knowing perimeter lets you find side lengths, which then give you area, and vice versa.
Fifth, area changes faster than perimeter — doubling both length and breadth quadruples the area.
Sixth, always convert to the same unit before calculating, and remember that area uses square units while perimeter uses linear units.
You have done wonderfully today! You now understand how to measure the boundary and the space inside plane figures — skills you will use throughout your mathematical journey and in everyday life. Keep practising these formulas, and remember: mathematics is all around you, from the fields you walk around to the floors you tile. Until next time, keep exploring and stay curious!