PrepShorts · Study sheet · Class 8 Mathematics · Chapter 7, Area
Chapter 7 · Area
Units for real areas, and why converting between them squares the length factor
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One square inch is 6.4516 square centimetres. An inch is only 2.54 centimetres — and the gap between those two facts is the whole topic.
The idea
A unit of area is a square, so changing the unit of length acts on both of its sides at once. Multiply every length by some factor and each unit square is replaced by one that is that many times wider and that many times taller, so the count of them changes by the factor squared and never by the factor itself. That single sentence is the whole of the chapter's conversion: one square inch is 6.4516 square centimetres because 2.54 has been squared, not applied. It is also why a bigger unit always returns a smaller number, why converting the other way is a division, and why doubling a plot's sides quadruples the land.
What you should be able to do
- Compute the area of a stated rectangular object from its two sidelengths, with the unit
- Convert a length between inches, feet and centimetres using the two printed relations
- Explain why converting an area between two units uses the square of the length factor, and show the square as a picture rather than as a rule
- Compute the number of square centimetres in one square inch, and in a stated number of square inches
- Convert an area from square centimetres back to square inches, and say why the operation is division
- State how many square inches are in a square foot, how many square centimetres in a square metre and how many square metres in a square kilometre, each with its reason
- Name several units used for land in India, and say which of them are not squares of a length unit
- Estimate the area of a room, a school and a settlement, choose an appropriate unit for each, and check the estimate against real data
- Say what happens to an area when every length in a figure is doubled, and by how much each part increases
Words to know
| Term | Definition in one line | First introduced |
|---|---|---|
| area | the number of unit squares whose material would exactly fill a region | printed in this chapter from its title onward (Part II p.148) |
| unit square | the square whose side is one unit of length, used as the thing being counted | printed in this chapter, Part II §7.1 (Part II p.149) |
| sidelength | the length of one side of a figure, set as a single word by this book | printed in this chapter, Part II §7.1 (Part II pp.149, 170) |
| acre | a unit of land area, given in this chapter as 43,560 square feet | printed in this chapter, Part II §7.1 under the subheading "Areas in Real Life" (Part II p.171) |
| bigha | one of the regional Indian land units the chapter lists | printed in this chapter, Part II §7.1 (Part II p.171) |
| gaj | another of the regional land units listed | printed in this chapter, Part II §7.1 (Part II p.171) |
| katha | another of the regional land units listed | printed in this chapter, Part II §7.1 (Part II p.171) |
| dhur | another of the regional land units listed | printed in this chapter, Part II §7.1 (Part II p.171) |
| cent | another of the regional land units listed, used in southern India | printed in this chapter, Part II §7.1 (Part II p.171) |
| ankanam | another of the regional land units listed | printed in this chapter, Part II §7.1 (Part II p.171) |
| length factor | the number every length gets multiplied by when the unit of length changes | an added term, and the organising idea of the whole topic; not printed |
| squaring the factor | that the count of unit squares changes by the length factor multiplied by itself | an added phrasing; the chapter performs the squaring on Part II p.170 and gives it no name |
Where people slip up
- "One square metre is a hundred square centimetres." It is ten thousand. This is the single most common area error in the grade, and the reason to spend a whole section on the square rather than on the rule.
- "One square foot is twelve square inches." It is 144. Draw the twelve-by-twelve grid once and the error does not come back.
- "To convert an area, multiply by the length factor." Square it. Say the word square while pointing at a square.
- "A bigger unit gives a bigger number." Fewer big squares fit, so the number falls. The 48 m² classroom and its 517 ft² are the pair to show.
- "Converting back means multiplying by 6.4516 again." It means dividing. The chapter's phrasing on Part II p.171 — every 6.4516 cm² supplies one square inch — is the sentence that makes division feel like counting rather than like undoing.
- "Doubling the sides doubles the area." It quadruples it, which is the Part II p.152 item and also the conversion law with a factor of 2.
- "An acre is a length." It is an area, and it is not the square of any unit of length. Neither are most of the regional units.
- "Every area unit is a square, so the squaring rule converts all of them." The squaring rule works when the area unit is built from a length unit. For an acre or a bigha you have to be told the number, which is exactly why the chapter states 43,560 rather than deriving it.
- "cm² is an abstract notation." It is a square you can draw with a ruler. If the explanation never draws it, the whole topic reduces to arithmetic.
- "A town ten times a school's size is ten times as wide." It is about three times as wide. Area ratios and length ratios are not the same ratio, and the chapter's last question walks straight into this.
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Worked answers to this chapter’s exercises · this video explains Figure it Out · 7.1 Q5
Transcript1,316 words
How big is this desk? Not in any unit yet - just how big. Here is one honest way to answer. Take a sheet of paper, lay it down, and count how many it takes to cover the top. This sheet is 21 centimetres by 29.7, so it covers 623.7 square centimetres. The desk is 60 by 45, which is 2700. Four sheets is not enough. Five is more than enough. The desk is somewhere between the two - about four and a third.
A dining table, 150 by 90, takes about twenty-two of them. That is exactly five desks, which sounds right, and it is. So measuring area is counting copies of something. Everything that follows is about what you choose to count. Nobody actually measures in sheets of paper, and the reason is not that the sheet is a bad size. It is that a sheet is a rectangle with two different sides, so you have to remember both of them, and they do not line up with any ruler you own.
So we agree on a square instead. One centimetre along each side, and it is called one square centimetre. Hold on to that word. A unit of area is a SQUARE. It has two sides, and they are the same length, and that is the whole reason this topic is not obvious. The desk is 2700 of those squares. The sheet is 623.7 of them. Now bring in a second ruler.
One inch is 2.54 centimetres. That is a definition, not a measurement - it is exact. So five inches is 12.7 centimetres. And 7.4 inches is 18.796. Multiply by 2.54, every time. Backwards, you divide. 5.08 centimetres is exactly two inches. 11.43 is exactly four and a half. Now here is one square inch. I am going to lay centimetre squares inside it, and I want you to watch what the square itself does.
Nothing. The square does not move. It does not change size. The ruler changed. The square is exactly where it was. Count the centimetre squares that fit. Across the bottom, 2.54 of them. Up the side, 2.54 of them. It is a grid. Rows times columns. So one square inch is 2.54 times 2.54, which is 6.4516 square centimetres. Not 2.54. That is the mistake, and it is worth naming out loud.
The factor is SQUARED. It is not applied. The difference is 3.9116 square centimetres, sitting inside a square the size of your thumbnail. And you can see exactly where it comes from. Changing the ruler changed the width. It also changed the height. A square has both. One thing here does stay simple, and it is worth separating from the rest. Ten square inches is ten times 6.4516, which is 64.516 square centimetres.
The TEN is not squared. Only the conversion factor is. Ten squares became ten squares. Each one got recounted in smaller pieces, and there were still ten of them. So squaring happens once, when the unit changes - not once per square you own. Which way does the number go? The centimetre is the smaller unit, so more of them fit. A bigger number, always. Which means going the other way is a division, and the same 6.4516 does the work.
Here is 161.29 square centimetres. Divide by 6.4516. Exactly 25 square inches. Nothing left over. Multiplying instead would have handed you a number over a thousand for a patch you can cover with one hand. That is the check worth keeping: ask which unit is bigger first, and then see whether your answer went the right way. The same argument, with a friendlier factor. Twelve inches make one foot. So how many square inches make one square foot?
Not twelve. Twelve rows of twelve. 144. And the picture is the thing to keep, not the number. You can see all 144 of them at once, and you can see why the answer could never have been twelve. A hundred centimetres make one metre. So one square metre is a hundred by a hundred. Ten thousand square centimetres. Not a hundred. That answer is a hundred times too small, and it is the commonest wrong answer in this entire topic.
A thousand metres make a kilometre. So one square kilometre is a thousand by a thousand. A million square metres. A thousand went in. A million came out. This is where squaring stops being a technicality you can ignore. A classroom, 8 metres by 6. That is 48 square metres. Somebody who works in feet will measure the same room and write down something near 517. 48 and 517. One room.
One metre is a bit over three feet, so a square metre is a bit over ten square feet - between ten and eleven of them. The foot is the smaller unit, so its number is the bigger one. Neither is more correct. Which is why a number on its own is not an area. 48 is not an area. 48 square metres is. Now the edge of the rule, which is more interesting than the rule.
An acre is 43,560 square feet. Where does that come from? Try to find the square. Try to find the length whose square is 43,560. There isn't one. 208 squared is under it. 209 squared is over it. No whole number of feet has an acre for its square. It is 66 feet by 660 - a long thin strip, which is what a team could plough in a day. It was never built from a length unit at all.
So the squaring argument has a condition on it, and the condition is the interesting part. It works when the unit of area IS the square of a unit of length. When it isn't, there is nothing to derive, and the number has to be told to you. Several land measures used in different regions are like this: they are areas from the start, and their sizes vary from place to place, so you look them up rather than work them out.
Last one, and no unit changes at all. Here is a square plot. Double every side. Not twice the land. Four times. Same rule. The factor is 2, and it acts on the width and on the height, so the area is multiplied by 4. And watch what that means for a piece of it. Cut the square with a diagonal, and cut again from a corner to the middle of that diagonal.
Three regions: a quarter, a quarter, and a half. They account for the whole square. Now double the sides. Each region is multiplied by four - so each one GAINS three times what it started with. The half gains one and a half whole original squares. The three gains together add up to three of them, which is the four minus the one you began with. Treble the sides and it is nine, not three.
One last consequence, read backwards. A town covers a hundred times the land a school does. How many times wider is it? Ten. Not a hundred. Because the hundred is the square of the width factor, so the width factor is what you square to get a hundred. Four times the land is twice across - which is the doubling we just did, said the other way round. And a town covering ten times the land? A bit over three times across. Three squared is 9, which is short. Three point two squared is over 10.
So ten times the area is nowhere near ten times the walk. That is the whole idea, and it fits in one line. A unit of area is a square, and a square has two sides. Change the ruler and both of them change. So the count changes by the factor times itself - never by the factor.
Where this fits
Taken from the notes each video was made from, not from the reading order — these are the ideas this one rests on and the ones that later rest on it.
Builds on
- Area as a count of unit squaresClass 8 · Ch 7, Area
Either side of this one
- Area of a trapezium, derived two different waysClass 8 · Ch 7, Area
- Where coordinates came from: grid cities, meridians, and the road to the Cartesian planeClass 9 · Ch 1, Orienting Yourself: The Use of Coordinates