Class 6 Mathematics Chapter 6: Perimeter and Area

Neelima wants a fence

Neelima has been given a small patch behind her house for a vegetable bed. Goats keep wandering in. She wants a low wire fence all the way round the patch, and the shop sells the wire by the metre. How much should she ask for? Nothing about the soil or the plants matters here. Only the edge of the patch does.

What perimeter measures

Start at one corner of a closed figure. Walk along the boundary, never cutting across, until you arrive back where you began. The distance you covered is the perimeter of that figure. It is a journey, not a surface.

Adding the sides

When the boundary is made of straight sides, that walk breaks into short straight stretches. Each stretch is one side. So the perimeter is nothing more than all the side lengths added together. No figure is too odd for this rule, as long as it closes.

A rectangle, the slow way

Take a rectangle 9 cm long and 4 cm broad. Going round it you travel 9 cm, then 4 cm, then 9 cm, then 4 cm. Added together that is 26 cm. Every side has been used once, and none twice.

A rectangle, the quick way

Look at what you added. The 9 turned up twice and so did the 4. So one length plus one breadth is exactly half the journey. In this case 9 + 4 gives 13, and twice 13 is 26. That is where 2 x (length + breadth) comes from.

A square is easier still

Every side of a square matches the others, so there is nothing to sort out. A square of side 6 cm has a perimeter of 4 x 6 = 24 cm. Count the sides, multiply by the one measurement, and stop.

A triangle has no short cut

A triangle usually has three different sides, and no doubling will help. A triangle with sides 8 cm, 5 cm and 6 cm has a perimeter of 19 cm. You simply add, carefully, and check that you used each side once.

Worked example: ribbon for a board

Raghav is putting ribbon all round a rectangular notice board that is 7 m long and 3 m broad. He needs the perimeter. First add one length and one breadth: 7 + 3 gives 10. Now double it: in metres, 2 x 10 = 20. So he should buy 20 m of ribbon.

Worked example: rounds of a park

A park is square, with each side 45 m. Jaya jogs four full rounds of it. One round is the perimeter, and in metres 4 x 45 = 180. Four rounds come to 4 x 180 = 720. So Jaya has covered 720 m.

Working backwards on a rectangle

Now a harder question. A rectangle has a perimeter of 30 cm and a length of 9 cm. What is its breadth? Halve the perimeter first, because half the journey is one length plus one breadth. Half of 30 is 15. Take away the 9 and the breadth is 6 cm.

Working backwards on a square

A square frame has used 28 cm of beading. Here the four sides are equal, so sharing the total between them gives the side. Dividing 28 by 4 leaves 7. Each side of the frame measures 7 cm.

Working backwards on a triangle

A triangle has a perimeter of 24 cm, and two of its sides measure 9 cm and 7 cm. Those two together take up 16 cm of the total. Whatever is left must be the third side, so it measures 8 cm.

Write the unit, and write the right one

Every answer above ends in cm or m. That is not decoration. A perimeter is a distance, so it takes a length unit. Writing 26 square cm for a perimeter is not a slip of the pen. It says you measured the wrong thing.

Back to Neelima

She measures her patch and finds a rectangle 8 m by 5 m. One length and one breadth make 13. Doubling gives 26, so she asks the shop for 26 m of wire. If she had guessed, the goats would have found the gap.

Scenario:

A farmer wants to fence a rectangular plot 30 metres by 20. A class is asked how much wire he needs. Rohit multiplies and gets 600. Their teacher asks what 600 would be measuring. The class works out that multiplying gives the ground covered, not the edge. They walk the rectangle instead: 30, then 20, then 30, then 20. That comes to 100 metres. Rohit sees his mistake clearly. He had used the operation for the inside when the question asked about the boundary, and the two answers differ by a factor of six here.

Analysis:

Both numbers are correct arithmetic and only one answers the question. Walking the edge is what separates the boundary from the surface.

Activity:

Measure a rectangular table. Work out the length of tape to go round its edge and the paper to cover its top.

Title:

Price the fencing for your own room

What You Notice:

The two answers land close to each other without being identical, because a pace is never exactly the same twice. The pacing also makes it clear that a perimeter really is a walk. Doorways and cupboards do not change the answer, because the boundary of the floor is what you measured.

Shopkeepers who sell fencing wire, picture framing and floor skirting all price their goods by the metre, so every order they take is really a perimeter.

The word perimeter comes from two Greek word parts meaning around and measure, so the name says exactly what the job is.

A running track in a stadium is marked in lanes because the outer lanes have longer perimeters, and runners would otherwise cover different distances.

Which formula to reach for — Four the same, so times four — A square gives you four equal sides, so multiply. A rectangle gives you two matching pairs, so add one of each and double. A triangle gives you nothing matching, so add all three.

The unit for perimeter — A walk is a length — You could pace out a perimeter with your feet. A walk is measured in cm or m, so a perimeter is too.

Who ran further?

Zoya and Imran train on two rectangular tracks in the same ground. Zoya uses the outer track, 60 m long and 35 m broad, and finishes 6 rounds. Imran uses the inner track, 50 m long and 25 m broad, and finishes 8 rounds. Zoya is sure she ran further, because her track is the bigger one. Guessing will not settle this.

Work out one round first

Each track is a rectangle, so one round is its perimeter. For Zoya the length and breadth add to 95, and in metres 2 x 95 = 190. For Imran they add to 75, and 2 x 75 = 150. So one of Zoya's rounds really is longer than one of Imran's.

Now multiply by the rounds

Zoya's six rounds come to 6 x 190 = 1140 m. Imran's eight rounds come to 8 x 150 = 1200 m. Imran has run 60 m further, on the smaller track. A bigger track is not the same thing as a longer run.

Why the answer surprises people

The eye compares the two rectangles and stops there. But the question asked about distance run, and that depends on two things, not one. Imran made up for his shorter round by doing two more of them. Whenever two quantities are multiplied, you cannot judge the product from one of them.

Edges that slope

Draw a figure on a square dot grid where some edges run along the grid and others cut across a square. Take a ruler to both. The sloping edge is clearly longer. Calling them both one unit would be untrue.

Counting two kinds of unit

So we count them separately. An edge from one dot to the next alongside it is one straight unit. An edge across a square from corner to corner is one diagonal unit. A figure with five of the first and two of the second has a perimeter of 5s + 2d units, and that is a complete answer.

Regular polygons

Some figures have all their sides alike and all their corners alike as well. These are called regular polygons. You already know two of them. A square is the regular polygon with four sides, and an equilateral triangle is the one with three.

One rule for all of them

If every side is the same, counting and multiplying is all that is needed. The perimeter of a regular polygon is the number of sides multiplied by the length of one side. The square rule was never special; it was this rule with a four in it.

Trying the rule out

A regular pentagon of side 7 cm has a perimeter of 5 x 7 = 35 cm. A regular hexagon of side 5 cm gives 6 x 5 = 30 cm. A regular octagon of side 4 cm gives 8 x 4 = 32 cm. Every one of them is a single multiplication.

The equilateral triangle

An equilateral triangle has three equal sides, so its perimeter is 3 x side. One with sides of 9 cm has a perimeter of 27 cm. Notice that an ordinary triangle still has to be added up, because its sides do not match.

Bending one wire into different shapes

Kiran has 48 cm of wire. Bent into a square it gives four sides of 12 cm each. Straightened and bent into an equilateral triangle, the same wire gives three sides of 16 cm. Bent into a regular hexagon it gives six sides of 8 cm. The perimeter never changed, because the wire never changed.

Cutting a rectangle in two

Now the opposite experiment. Anup cuts a paper rectangle 10 cm by 4 cm straight down the middle, making two pieces of 5 cm by 4 cm. Each piece on its own has a perimeter of 18 cm, so the two pieces together carry 36 cm of edge.

Joining them back, two ways

Put the two 4 cm edges together and the original 10 cm by 4 cm rectangle comes back, with a perimeter of 28 cm. Put the two 5 cm edges together instead and you get a 5 cm by 8 cm rectangle, whose perimeter is only 26 cm. Same paper, different walk around.

What the join explains

The shared edge stops being boundary on both pieces, so it is lost twice. With a 4 cm join, 36 loses 8 and leaves 28. With a 5 cm join, 36 loses 10 and leaves 26. A longer join hides more edge, and the perimeter shrinks.

The lesson to carry forward

The paper was never added to or taken away. Only the arrangement changed, and the perimeter moved anyway. Hold on to that, because the next topic asks whether the surface inside changed as well.

Scenario:

Two farmers each have 36 metres of fencing. One makes a rectangle 16 by 2 and the other makes a square 9 by 9. A class checks the fencing and both use exactly 36 metres. Then they work out the ground enclosed. The long thin plot holds 32 square metres and the square holds 81. Nita is surprised, since the fences are the same length. Their teacher asks the class what changed. Only the shape. The children see that perimeter does not fix area, and that for a given length of fence the squarest shape encloses the most.

Analysis:

Equal fencing and more than twice the land. Shape is the free variable once the perimeter is fixed.

Activity:

Take a loop of string 36 centimetres round. Make three different rectangles and work out the area of each.

Title:

One rectangle, two perimeters

Safety:

Ask an adult before using scissors, and cut away from your fingers.

What You Notice:

The paper never changed, yet the two answers differ by 2 cm. The longer join gives the smaller perimeter every time, because a shared edge is lost from both pieces at once. Your subtraction rule agrees with your ruler, which is a good sign that the reasoning is sound.

Stadium tracks are marked with staggered starts precisely because the outer lanes have longer perimeters, so a common finish line needs different starting points.

A honeycomb is built from regular hexagons. Bees get the most floor for the least wax, because a hexagon encloses a lot of room for a short perimeter.

Nut and bolt heads are usually regular hexagons, so a spanner can grip them from six different positions instead of only one.

Perimeter of a regular polygon — Count the sides, then multiply — Three sides give 3 x side, five give 5 x side, eight give 8 x side. The square rule was only this rule wearing a four.

Joining two pieces — Long join, short walk — The longer the edge the two pieces share, the more boundary disappears inside, and the smaller the perimeter of the joined figure becomes.

Devi is buying tiles

Devi's family is laying tiles in a small store room. The shopkeeper asks how many square metres she wants. Last week she bought skirting board for the same room, and that was sold by the metre. She realises the two questions are not the same question at all. One was about the edge. This one is about the floor.

What area measures

Area tells you how much surface sits inside a shut figure. Imagine carpeting that inside with small squares, laid flat, so that none overlap and no bare patch is left. The number of squares you need is the area. The boundary plays no part in this count.

Counting squares

Take a rectangle 8 cm long and 3 cm broad and lay centimetre squares inside it. They settle into three neat rows with eight squares in each row. Counting them one by one gives 24, so the area is 24 square centimetres.

The rule falls out of the rows

Nobody wants to count 24 squares, let alone 2400. Look at how they sat. The length told you how many squares fit in a row, and the breadth told you how many rows there were. So 8 x 3 = 24 without counting. Multiplying the two measurements of a rectangle gives its area.

A square is a rectangle in disguise

In a square the length and the breadth are the same number. So the rule becomes side times side. A square of side 5 cm covers 5 x 5 = 25 square centimetres.

Why the unit carries the word square

The counting unit is itself a square, one centimetre along each side. That is why the answer is 24 square centimetres and not 24 centimetres. The word is not decoration; it names what was counted.

The unit mistake, and why it matters

Suppose a child writes that a table top has an area of 6000 cm. That is not untidy writing. It says a distance was measured when a surface was asked for. A perimeter takes cm or m. An area takes square cm or square m. Getting this wrong changes the meaning of the answer.

Worked example: a floor and a mat

A room is 6 m long and 5 m broad, so its floor covers 6 x 5 = 30 square metres. A square mat of side 2 m is spread on it, covering 2 x 2 = 4 square metres. The floor left uncovered is therefore 30 take away 4, which is 26 square metres.

Worked example: working backwards

A rectangular garden is 20 m long and covers 240 square metres. How broad is it? The area came from the length multiplied by the breadth, so undoing it means dividing. Sharing 240 between 20 gives 12, so the garden is 12 m broad.

Shapes that are not straight sided

A leaf or a footprint has no formula. Trace it onto squared paper and count. Some squares will lie fully inside, and some will only be caught in part, so a few agreed rules are needed.

The counting rules

A square that is fully inside counts as one square unit. A square more than half covered also counts as one. A square less than half covered is ignored. A square covered exactly half way counts as half a unit. The answer is an estimate, and a good one.

Why squares and not circles

You might ask why the unit is a square at all. Try covering a page with coins. They leave curved gaps everywhere, and the gaps are hard to account for. Squares sit edge to edge with nothing left over, which is exactly what a counting unit has to do.

The same walk, two different floors

Now the most important idea in this chapter. Take a rectangle 8 cm by 2 cm. Its perimeter is 20 cm and its area is 16 square centimetres. Now take a square of side 5 cm. Its perimeter is also 20 cm, but its area is 25 square centimetres.

So a perimeter does not fix an area

Both walks measure 20 cm. Yet one shape holds nine more square centimetres than the other. A fence of a fixed length can be bent to enclose a great deal or very little. Perimeter and area answer different questions, and one does not settle the other.

Back to Devi

Her store room is 4 m by 3 m, so she needs 12 square metres of tiles. The skirting board she bought last week was 14 m. Two different numbers, two different units, and both of them right.

Scenario:

The pupils of a village school in Wardha are given a rectangular patch behind the kitchen, 12 m long and 5 m broad. They want to grow vegetables for the midday meal. Two jobs must be paid for. A wire fence has to run right round the patch to keep goats out, and the soil inside has to be turned over and manured. The fencing shop charges by the metre. The man who turns soil charges by the square metre. Kavita, who keeps the accounts, says she needs two different numbers before she can ask either of them for a price. Her friend Bala thinks one measurement of the patch should be enough for both.

Analysis:

One job follows the boundary and the other covers the surface, so the patch needs a perimeter and an area. Both come from the same two measurements.

Activity:

Measure a patch of ground or a room at home and work out both its perimeter and its area, writing each with the right unit.

Floor tiles are sold by the square metre and skirting board by the metre, so one shop bill can carry an area and a perimeter side by side.

Farm land in India is often measured in bighas or acres, which are square units too, even though the words hide it.

Squares, equilateral triangles and regular hexagons are the only regular shapes that tile a floor with no gaps at all.

Which measurement is which — Fence or floor — A fence runs round the edge, so that is perimeter in metres. A floor is the surface inside, so that is area in square metres. Ask which one the question is buying.

Why the rectangle rule works — Rows of squares — Squares inside a rectangle line up in rows. The length says how many are in a row and the breadth says how many rows there are.

Farida cuts a card

Farida has a rectangular piece of card. She rules a line from one corner to the opposite corner and cuts along it. Two triangles fall onto the table. They look different from each other, because one is turned round. She wants to know whether they are really the same size.

The test that settles it

She picks one triangle up and turns it over onto the other. Every edge lands on an edge and every corner on a corner. Nothing sticks out. When one figure covers another exactly, the two enclose the same surface, so the areas are equal.

What follows at once

The two triangles together were the whole rectangle, and they are equal. So each of them covers exactly half of the rectangle. That is the first fact worth keeping from this topic.

Putting a number on it

Suppose the card was 10 cm long and 6 cm broad. Its area is 10 x 6 = 60 square centimetres. Each triangle takes half of that, so each covers 30 square centimetres. No new formula was needed.

A triangle that is not a half card

Now a harder case. Draw a triangle whose top corner sits somewhere along the top of a rectangle rather than above a bottom corner. It leans. It is clearly not one of a matching pair, so the argument above does not fit.

Box it and split the box

Draw the smallest rectangle that just holds the triangle, say 8 cm along the bottom and 4 cm tall. Now rule a line straight down from the top corner of the triangle to the bottom. This cuts the rectangle into two smaller rectangles, and it cuts the triangle into two right angled pieces.

Halve each part

If the top corner sits 3 cm from the left, the left rectangle is 3 cm by 4 cm with an area of 12 square centimetres. The piece of triangle inside it is one half of a rectangle, so it covers 6 square centimetres. The right rectangle is 5 cm by 4 cm, with an area of 20, so its triangle piece covers 10.

Add the halves back

The whole triangle covers 6 + 10, which is 16 square centimetres. The whole rectangle covered 8 x 4 = 32 square centimetres. Sixteen is exactly half of thirty two, so the leaning triangle behaved no differently from the upright one.

The statement to remember

So here it is. A triangle covers half the area of the rectangle that just encloses it. This is true whether the triangle leans or stands upright, and it saves learning a separate rule for every kind of triangle.

Trying it on a grid

Draw a triangle on squared paper with a base of 6 units and a height of 4 units. The enclosing rectangle covers 24 square units. Half of that is 12, so the triangle covers 12 square units. Counting the squares by hand agrees.

Figures that are neither

Many real outlines are neither rectangle nor triangle. A house shaped figure, for instance, has a rectangular body with a triangular roof. Cut it along the line where the roof meets the wall and treat the two pieces separately.

Adding the pieces up

Suppose the body is 5 units by 3 units and the roof has a base of 5 units and a height of 4 units. The body covers 15 square units. The roof is half of a 5 by 4 rectangle, so it covers 10. Together the figure covers 25 square units.

The other half of the big idea

Earlier you saw two figures share a perimeter and hold different areas. Now try it the other way round. Lay out twelve unit squares as a block three across and four down, and again as a strip twelve long and one wide.

Same surface, different walks

Both arrangements cover twelve square units, because twelve squares were used each time. But the block has a perimeter of 14 units and the strip has a perimeter of 26 units. Equal areas, and the walks round them are nothing like each other.

What this chapter leaves you with

Perimeter measures a journey round an edge and takes cm or m. Area measures a surface and takes square cm or square m. Neither number can be worked out from the other, in either direction. Keep them apart and this chapter holds no traps at all.

Scenario:

A triangular corner plot has a base of 12 metres and a height of 8. A class is asked its area. Imran multiplies 12 by 8 and gets 96. Their teacher asks the class to draw the triangle inside a rectangle of the same base and height. The rectangle holds 96 square metres and the triangle clearly fills about half of it. The class cuts the drawing and fits the two offcuts together, and they match the triangle exactly. So the area is 48. Imran sees why the half appears. He had found the rectangle that encloses the plot, not the plot.

Analysis:

The 96 was not wrong, it was the wrong figure. Cutting the rectangle shows the triangle is exactly half of it.

Activity:

Draw a triangle inside a rectangle of the same base and height. Cut and rearrange the offcuts to check the halving.

Title:

Cut a rectangle and prove the halving

Safety:

Ask an adult before using scissors, and cut away from your fingers.

What You Notice:

In the first test the two triangles match, so each covers 30 of the 60 squares. In the second the two scraps together cover the leaning triangle exactly, so that triangle is half of its rectangle as well. Leaning made no difference at all to the answer.

Tailors cutting cloth on the diagonal call it cutting on the bias, and a diagonal cut across a rectangle of cloth always gives two matching triangles.

A roof gable, a slice of a samosa and a set square are all triangles whose areas would be found by boxing them and halving.

Land surveyors still divide an awkward plot into triangles, because a triangle is the easiest shape whose area can be trusted.

Area of a triangle — Box it, then halve it — Draw the smallest rectangle that just holds the triangle. Work out that rectangle's area and take half of it. This works for a leaning triangle as well as an upright one.

Mixed figures — Cut, solve, add — Cut the figure into rectangles and triangles, solve each piece on its own, and add the answers. Check that the pieces neither overlap nor leave a gap.