Area

Why This Matters

Think about colouring a rangoli. You spread coloured powder evenly over a shape on the floor. A bigger shape needs more powder. A smaller shape needs less.

How much powder you need depends on how much flat space the shape covers. That flat space is what we call area.

Area is everywhere once you look. How much paint to cover a wall. How much cloth to stitch a kurta. How much grass seed for a park. How much tile to lay on a floor. A shopkeeper buying a piece of land pays by its area. A farmer thinks about the area of a field. Even the screen you are reading this on has an area.

For a simple rectangle, finding area is easy — you may already know length × width. But real shapes are not always neat rectangles. Fields are slanted. Plots have odd shapes. A roof might be a triangle or a trapezium.

This chapter does something powerful. It shows you how to find the area of any flat shape — parallelogram, triangle, trapezium, rhombus, or a five-sided plot — and, more importantly, why each formula is true. You will not just memorise formulas. You will see where they come from, by cutting shapes up and rearranging the pieces. That way you will remember them for life, not just for the exam.

The Big Idea

Area is the amount of flat space a shape covers. We measure it by counting how many unit squares (squares of side 1 cm) fit inside the shape. For a rectangle, that count is simply length × width. The clever idea of this whole chapter is that every other shape can be turned into a rectangle or built from triangles by cutting and rearranging its pieces — and rearranging never changes the area. From this one idea, all the area formulas follow: parallelogram = base × height, triangle = ½ × base × height, trapezium = ½ × (a + b) × h, and any polygon = a sum of triangles.

Let’s Break It Down

First, what does “area” really mean?

Before any formulas, let us be clear what we are measuring. We will lean on this everywhere, so here is a quick refresher.

Figure 7.1 shows this packing of unit squares for a 5 cm by 3 cm rectangle.

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One quick warning, because students mix these up. A bigger perimeter does not always mean a bigger area. You can have two shapes where one has a longer boundary but covers less flat space. Perimeter measures the edge; area measures the inside. They are different things.

The area of a triangle — half of a rectangle

Here is the first big jump. Take any rectangle and draw one diagonal across it. The diagonal cuts the rectangle into two equal triangles. They are exactly the same — one is just the other flipped over.

So each triangle covers exactly half the rectangle. If the rectangle is base wide and height tall, its area is base × height. Half of that is the area of one triangle.

This is the single most useful area fact in the chapter, so let us see why it works for every triangle, not just the neat right-angled one inside a rectangle.

The trick is to take a second copy of the triangle, turn it upside down, and fit it against the first. The two copies always join to form a parallelogram. Figure 7.2 shows this.

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So no matter what the triangle looks like — tall, thin, slanted, or wide — two copies make a parallelogram, and one triangle is half of it.

Area of a triangle = ½ × base × height

The base is any one side you choose. The height is the perpendicular distance from that side up to the opposite corner.

The word perpendicular is important. The height must go straight across at a right angle to the base, not along a slanted side. We will come back to this — it is the number-one mistake students make.

Let us use the formula.

Sometimes you can run the formula backwards to find a missing length. Since the area of a triangle can be written using any side as the base, two different base-and-height pairs give the same area. That lets us find an unknown height.

The area of a parallelogram — cut and slide into a rectangle

A parallelogram is a four-sided shape with both pairs of opposite sides parallel. It looks like a rectangle that has been pushed over to one side, so it leans.

Because it leans, we cannot just multiply two sides. But we can do something clever: cut a piece off one end and slide it to the other end to make a rectangle. Since we only moved a piece — we did not add or remove anything — the area stays the same. This cutting-and-rearranging move has a name: dissection.

Here is the cut. Drop a perpendicular line from the top corner straight down to the base. This cuts off a right-angled triangle. Slide that triangle across to the other side, and the parallelogram becomes a neat rectangle. Figure 7.3 shows the whole move.

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The cut piece fits the gap on the other side perfectly. (You can prove the two triangles are identical using the RHS rule for congruent triangles, but you can also just try it with a paper cut-out and see.) So:

Area of a parallelogram = base × height

Here base is one side, and height is the perpendicular distance from that side across to the opposite parallel side.

Notice you can pick either pair of parallel sides as the base, as long as you use the matching perpendicular height. Both choices give the same area.

Watch out: height is the perpendicular distance, not the slant side

This is the trap that catches the most students, so let us look at it carefully. A parallelogram has a base, and it also has a slanted side. The slanted side is not the height.

The height is the perpendicular distance — the straight-across distance between the two parallel sides, measured at a right angle. The slanted side is longer than this perpendicular distance, because it goes across and up at the same time. Figure 7.4 makes the difference clear.

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So whenever a problem gives you both the slant side and the perpendicular height, use the height. The slant side is often given just to test whether you know the difference.

The area of a trapezium — split it into pieces you know

A trapezium is a four-sided shape with exactly one pair of parallel sides. Think of the cross-section of a dam wall, or a handbag shape: a wide bottom, a narrower top, and two slanted sides joining them.

The two parallel sides are usually different lengths. Call them a (the shorter one) and b (the longer one). The perpendicular distance between them is the height h.

How do we find the area? We use the same trick as always: split it into shapes we already know. Drop a perpendicular from each end of the top side straight down to the bottom side. This carves the trapezium into a rectangle in the middle and two right triangles on the sides. Figure 7.5 shows the split.

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Now add the three pieces. The rectangle has width a (same as the top side) and height h, so its area is a × h. Each triangle has height h. If the two triangle bases are x and y, their areas are ½ × x × h and ½ × y × h.

Add them up:

Area = ½ × x × h + a × h + ½ × y × h

Now here is the neat finish. The bottom side b is made of the two triangle bases x and y plus the rectangle width a. So x + y + a = b, which means x + y = b − a. Substitute that in and tidy up, and the messy expression collapses into one clean formula:

Area of a trapezium = ½ × (sum of the parallel sides) × height = ½ × (a + b) × h

A lovely way to remember this: it is the average of the two parallel sides, (a + b) ÷ 2, multiplied by the height h. The trapezium behaves like a rectangle whose width is the average of its top and bottom.

Let us use it.

The area of a rhombus — half the product of the diagonals

A rhombus is a special parallelogram where all four sides are equal — like a diamond on a playing card. Since it is a parallelogram, base × height still works for it.

But a rhombus has a neat extra feature: its two diagonals cross each other at right angles and cut each other exactly in half. This gives a second, often easier, formula based on the diagonals.

The diagonals split the rhombus into four small right-angled triangles. Rearranging them (another dissection) shows that the rhombus has the same area as a rectangle whose sides are one full diagonal and half the other diagonal. Multiplying those gives:

Area of a rhombus = ½ × (product of its diagonals) = ½ × d₁ × d₂

where d₁ and d₂ are the lengths of the two diagonals.

The area of any polygon — break it into triangles

Now the most powerful idea of all. What about a shape that is none of the above — a five-sided plot of land, or some odd many-sided figure?

The answer is simple and always works: break it into triangles. From one corner of the polygon, draw straight lines (diagonals) to every other corner. These lines cut the whole shape into triangles. Find the area of each triangle and add them all up. Figure 7.6 shows this for a pentagon.

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So you never get stuck. Even the strangest straight-sided shape is just a collection of triangles in disguise.

The same idea works for composite shapes made of familiar pieces — say a rectangle with a triangle on top (like a simple house shape). You do not need a single formula. Just split the shape into a rectangle and a triangle, find each area, and add.

Common Mistakes

These are the slips students make most often with area. Read them once and you will avoid them.

Quick Check

Try these. Each one checks an idea from the chapter.

Practice Problems

Try each one yourself first. Only then tap to see the full solution.

Easy

Medium

Challenge

Summary

• Area is the flat space a shape covers. We measure it by counting unit squares (squares of side 1 cm), and it is always in square units like cm² or m².
• Area of a rectangle = length × width. A square is a rectangle with equal sides, so its area is side × side.
• Area of a triangle = ½ × base × height. This is true because two copies of any triangle join to make a parallelogram, so one triangle is half of it.
• Area of a parallelogram = base × height. We get this by cutting a triangle off one end and sliding it across to make a rectangle (a dissection).
• The height always means the perpendicular distance — measured straight across at a right angle — not the slanted side. Using the slant side gives an answer that is too big.
• Area of a trapezium = ½ × (a + b) × h, where a and b are the two parallel sides and h is the perpendicular distance between them. It is the average of the parallel sides times the height.
• Area of a rhombus = ½ × (product of its diagonals) = ½ × d₁ × d₂.
• Any polygon can be split into triangles (and any composite shape into known pieces). Find each piece’s area and add them up.

What’s Next

That brings Ganita Prakash Part II — and your whole Class 8 maths journey — to a close. Look at how far you have come. You explored squares, cubes and powers; you learned the story of numbers and how we write them; you mastered quadrilaterals, number play, and the magic of distributing and factorising; you built up proportional reasoning; you uncovered fractions hiding in disguise; and now you can find the area of any flat shape and explain why every formula is true.

The thread running through all of it is the habit you just practised here: never accept a formula as a given. Take it apart, see where it comes from, and you will own it for life. That is exactly the thinking that the next stage of maths — Class 9 and beyond — will reward.

Want to revisit any chapter or move to another subject? Head back to your full list of topics here: Class 8 Maths. Well done finishing the book — you have earned it.