Perimeter and Area

Why This Matters

Imagine you want to put a fence around your garden. How much wire do you need to buy? That is a perimeter question.

Now imagine you want to put grass inside that same garden. How much grass do you need to cover the ground? That is an area question.

These two ideas are everywhere. The lace around a tablecloth, the tape around a photo frame, the running track around a field — all of these are perimeter. The carpet on a floor, the tiles in a bathroom, the paint on a wall — all of these are area.

In this chapter you will learn two simple things. First, how to measure the distance around a shape (perimeter). Second, how to measure the space inside a shape (area). They sound similar, but they are very different. By the end, you will never mix them up again.

The Big Idea

Perimeter is how far you walk if you go once all the way around the edge of a shape. You just add up the lengths of all its sides. Area is how much flat space the shape covers. You measure it by counting how many little squares fit inside. Perimeter is about the border. Area is about the inside.

Let’s Break It Down

Perimeter — the distance around

The perimeter of a shape is the total distance around its edge.

Here is an easy way to picture it. Suppose a tiny ant walks along the boundary of a shape and comes back to where it started. The total distance the ant walked is the perimeter.

For any shape made of straight sides, finding the perimeter is simple. You just add up the length of every side.

Perimeter of a rectangle

A rectangle is a four-sided shape where opposite sides are equal and all corners are square (a perfect “L” corner). Its longer side is the length and its shorter side is the breadth (breadth just means width).

Let’s find the perimeter of a rectangle that is 12 cm long and 8 cm wide. We walk around it and add every side. The picture below shows the ant’s walk.

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So we added 12 + 8 + 12 + 8 = 40 cm.

Notice something. There are two lengths (12 and 12) and two breadths (8 and 8). So instead of adding four numbers, we can take one length plus one breadth, and then double it.

That gives us a quick formula.

Perimeter of a rectangle = length + breadth + length + breadth

Perimeter of a rectangle = 2 × (length + breadth)

Let’s check it: 2 × (12 + 8) = 2 × 20 = 40 cm. Same answer.

Perimeter of a square

A square is a special rectangle where all four sides are equal.

Say a square photo frame has each side 5 cm. To put tape all around it, we add all four sides: 5 + 5 + 5 + 5 = 20 cm. The picture below shows this.

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Because all four sides are the same, adding the side four times is the same as multiplying by 4.

Perimeter of a square = side + side + side + side

Perimeter of a square = 4 × side

Perimeter of a triangle

A triangle has three sides. There is no special formula needed. You just add the three sides.

Perimeter of a triangle = side + side + side

For a triangle with sides 4 cm, 5 cm and 7 cm, the perimeter is 4 + 5 + 7 = 16 cm.

Before we do a worked example, here is a quick reminder of one small idea we will use a lot.

Let’s use the perimeter ideas on a real problem.

Here is one more, with a square.

Area — counting unit squares

The area of a shape is the amount of flat space it covers.

To measure length we use a unit like the centimetre (cm). But area is space spread out in two directions — across and down. So we measure it with a square, not a line.

The square we use is called a unit square. A unit square is a square with each side 1 unit long. If each side is 1 cm, that square is “1 square centimetre”. We write this as 1 cm² (the small 2 means “square”).

Here is the simple idea: the area of a shape is the number of unit squares that fit inside it.

If 15 unit squares of size 1 cm each fit inside a shape, its area is 15 cm². That is all area means — counting squares.

Area of a rectangle and square

We could count the squares one by one. But there is a much faster way for a rectangle. And there is a beautiful reason why it works.

Look at Figure 6.3. We have a rectangle that is 5 cm long and 3 cm wide, drawn on grid paper where every little square is 1 cm².

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Look carefully at the grid. Each row holds 5 little squares, because the length is 5 cm. And there are 3 such rows, because the breadth is 3 cm.

So the total number of squares is 3 rows of 5 squares each. That is 3 × 5 = 15 squares. So the area is 15 cm².

Do you see the pattern? The length told us how many squares fit in one row. The breadth told us how many rows there are. Multiply them and you get the total squares.

That is why this works.

Area of a rectangle = length × breadth

For a square, the length and the breadth are the same (call it the side). So:

Area of a square = side × side

Let’s use the formula on a real floor problem.

Area of irregular shapes by counting squares

Not every shape is a neat rectangle. A leaf, your handprint, or a lake on a map are all “irregular” — they have curvy or uneven edges. There is no simple formula for these.

But we can still estimate the area. We place the shape on squared paper (graph paper) and count the squares it covers. Figure 6.4 shows how.

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Here are the simple rules we follow when counting:

A square fully inside the shape counts as 1 whole square.

A square that is more than half covered also counts as 1.

A square that is exactly half covered counts as ½.

A square that is less than half covered is ignored (counts as 0).

So you count the full squares, then add ½ for each half square. The total is your estimate of the area.

This will not be perfectly exact — it is an estimate. But for a curvy shape, it is a very good way to get close.

Perimeter vs Area

This is the part students mix up the most. So let’s make it crystal clear.

Perimeter is the distance around the border. It is a length, so it uses plain units like cm or m.

Area is the space inside the shape. It uses square units like cm² or m².

Here is the surprising part: two shapes can have the same perimeter but different areas. Figure 6.5 shows two such shapes.

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Both shapes have a perimeter of 12 units. But Shape A holds 8 squares and Shape B holds 9 squares. Same border, different space inside. That is the clearest proof that perimeter and area are not the same thing.

Here is a side-by-side comparison to lock it in.

Common Mistakes

Quick Check

Practice Problems

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Medium

Challenge

Summary

• Perimeter is the total distance around the edge of a shape. You find it by adding the lengths of all the sides.
• Perimeter of a rectangle = 2 × (length + breadth). Perimeter of a square = 4 × side. Perimeter of a triangle = sum of its three sides.
• Area is the amount of flat space a shape covers. We measure it by counting how many unit squares fit inside.
• Area of a rectangle = length × breadth, because the length is how many squares fit in one row and the breadth is how many rows there are. Area of a square = side × side.
• For curvy (irregular) shapes, estimate the area by counting squares: count full squares as 1, more-than-half as 1, exactly-half as ½, and ignore squares less than half covered.
• Perimeter uses plain units (cm, m). Area uses square units (cm², m²). Do not mix them up.
• Two shapes can have the same perimeter but different areas. The border and the inside are two separate measurements.

What’s Next

You now know how to measure the edge and the inside of a shape. Next, in Chapter 7 — Fractions, you will learn how to handle “parts of a whole” — like half a chapati, a quarter of a pizza, or three-fifths of a glass of juice. Fractions help you describe and compare amounts that are not whole numbers. And they connect right back to this chapter: half a square (½) was already a fraction you used when counting the area of curvy shapes!