Mensuration
IGCSE Mathematics Topic 5 7:47 English narration · English + 中文 subtitles burned in
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The Great Pyramid of Giza has stood for four and a half thousand years.
吉萨大金字塔已经矗立了四千五百年。
How much stone is inside it?
它里面有多少石头？
About two and a half million cubic metres — and one formula from this topic will tell you.
大约两百五十万立方米—— 本专题里的一个公式就能告诉你。
A drinks can, a shipping container, a slice of pizza: each is a length, an area or a volume.
一罐饮料、一个集装箱、一块披萨： 每一个都是长度、面积或体积。
Topic Five is Mensuration: measuring length, area and volume.
第五个专题是度量：测量长度、面积和体积。
Core and Extended are almost the same here, and some formulas are printed in the exam paper.
这里的核心课程和扩展课程几乎相同， 有些公式印在试卷上。
But learn them all — you will use them all.
但要全部记熟——你全都会用到。
Here is the plan.
本节的安排是这样的。
First units, and the trap hiding inside square and cubic units.
首先是单位，以及平方单位和立方单位里藏着的陷阱。
Then the four flat shapes you must know by heart.
然后是必须背熟的四种平面图形。
Then circles, and arcs and sectors — the fraction questions.
接着是圆，还有弧与扇形——那些求分数的题。
Then solids: volume and surface area.
再然后是立体：体积和表面积。
And last, compound shapes, where a hard shape is split into easy ones.
最后是组合图形，把复杂的图形拆成简单的。
Start with units.
先看单位。
We use metric units for mass, length, area, volume and capacity.
我们用公制单位表示质量、长度、面积、体积和容量。
Going down to a smaller unit, you multiply.
换成更小的单位时要乘。
One kilometre is a thousand metres. One metre is a hundred centimetres. One centimetre is ten millimetres.
一公里等于一千米，一米等于一百厘米， 一厘米等于十毫米。
Going the other way, up to a bigger unit, you divide.
反过来，换成更大的单位时要除。
Say it out loud: smaller unit, multiply; bigger unit, divide.
大声说一遍：小单位用乘，大单位用除。
Now the trap that costs marks every year.
现在说那个每年都在丢分的陷阱。
One metre is a hundred centimetres — but one square metre is not a hundred square centimetres.
一米等于一百厘米——但一平方米并不等于一百平方厘米。
Square the conversion factor: a hundred squared is ten thousand.
要把换算倍数平方：一百的平方是一万。
So three square metres is thirty thousand square centimetres.
所以三平方米等于三万平方厘米。
For volume, cube it: one cubic metre is a million cubic centimetres.
体积则要立方：一立方米等于一百万立方厘米。
And a litre is a thousand cubic centimetres.
而一升等于一千立方厘米。
Four flat shapes, four formulas.
四种平面图形，四个公式。
Perimeter is the distance all the way round; area is the space inside.
周长是绕一圈的距离；面积是内部的空间。
A rectangle is length times width.
矩形是长乘宽。
A triangle is a half times base times height.
三角形是二分之一乘底乘高。
A parallelogram is base times height — no half.
平行四边形是底乘高——没有二分之一。
A trapezium is a half, times the sum of the two parallel sides, times the height.
梯形是二分之一，乘两条平行边之和，再乘高。
And in all four, the height is the perpendicular height: at right angles to the base, never the slanted side.
这四个里面，高都是垂直高：与底边成直角，绝不是那条斜边。
A trapezium has parallel sides of six centimetres and ten centimetres, and a height of four centimetres.
一个梯形的两条平行边是六厘米和十厘米，高是四厘米，求它的面积。
Find its area. Pause here and try it.
先暂停，自己算一算。
First, add the two parallel sides: six plus ten is sixteen.
第一步，把两条平行边相加：六加十等于十六。
Half of that is eight — the average width.
它的一半是八——也就是平均宽度。
Then multiply by the height: eight times four is thirty-two square centimetres.
然后乘以高：八乘四等于三十二平方厘米。
See the method: average the parallel sides, then treat it as a rectangle.
看清这个方法：先把平行边取平均，再当成矩形来算。
Circles.
圆。
Two formulas, and students swap them under pressure.
两个公式，学生一紧张就会弄反。
The circumference — the distance all the way round — is pi times the diameter.
周长——绕一圈的距离——等于派乘直径。
The area is pi times the radius squared.
面积等于派乘半径的平方。
Here is the check: area is the one with a square in it, because area is always in square units.
这里有个检查方法：带平方的那个是面积， 因为面积总是用平方单位。
And the diameter is twice the radius, so if you are given the diameter, halve it first.
而直径是半径的两倍，所以如果题目给的是直径，先把它除以二。
A sector is a slice of a circle, cut between two radii.
扇形是圆的一块，由两条半径切出来。
Start from the whole circle.
先从整个圆开始。
The angle at the centre gives the fraction: that angle divided by three hundred and sixty.
圆心处的角给出那个分数： 用这个角除以三百六十。
The curved edge is called the arc, and its length is that fraction of the circumference.
弯曲的那条边叫做弧，它的长度就是圆周的这个分数倍。
The sector area is the same fraction of the whole area.
扇形的面积，是整个圆面积的同一个分数倍。
A small slice is a minor sector; the rest is the major sector.
小的一块叫小扇形，剩下的叫大扇形。
Now the exam version.
现在看考试里的样子。
A sector has an angle of ninety degrees and a radius of eight centimetres. Find the arc length and the sector area.
一个扇形的圆心角是九十度，半径是八厘米，求弧长和扇形面积。
Start with the fraction: ninety over three hundred and sixty is one quarter.
先求分数：九十除以三百六十等于四分之一。
Now the whole circle.
再看整个圆。
Its circumference is two pi times eight, sixteen pi, so a quarter of that is four pi centimetres.
它的周长是二派乘八，也就是十六派，四分之一就是四派厘米。
Its area is pi times eight squared, sixty-four pi, so a quarter of that is sixteen pi square centimetres.
它的面积是派乘八的平方，也就是六十四派，四分之一就是十六派平方厘米。
One fraction, used twice.
一个分数，用两次。
Now into three dimensions.
现在进入三维。
Learn these four solids and the letters on them.
记住这四种立体，以及上面的字母。
A cuboid has a length, a width and a height.
长方体有长、宽、高。
A cylinder has a radius and a height.
圆柱有半径和高。
A sphere has only a radius.
球只有半径。
And look carefully at the cone: it has two heights.
再仔细看圆锥：它有两个高。
The vertical height goes straight up the middle; the slant height runs down the sloping side.
竖直高从中间直上去，斜高沿着斜面下来。
Volume uses the vertical one; curved surface area uses the slant one.
体积用竖直的那个，侧面积用斜的那个。
Here are the volumes.
下面是体积公式。
A prism is any solid with the same cross-section all the way through: find that area, then multiply by the length.
棱柱是指从头到尾横截面都一样的立体：先求出那个面积，再乘以长度。
A cylinder is just a prism with a circular cross-section, so its volume is pi r squared times the height.
圆柱其实就是横截面为圆的棱柱，所以体积是派乘半径的平方再乘高。
A cone is one third of the cylinder around it.
圆锥是外面那个圆柱的三分之一。
And a sphere is four thirds pi r cubed.
球是三分之四派乘半径的立方。
Most are printed in the formula list — the cuboid is not.
大多数都印在公式表上——长方体没有。
Now a full exam question.
现在来一道完整的考题。
A cylinder has radius five centimetres and height ten centimetres. Find its volume and its total surface area.
一个圆柱半径五厘米，高十厘米，求它的体积和总表面积。
Pause here and try it.
先暂停，自己算一算。
Volume first: pi times five squared, times ten.
先求体积：派乘五的平方，再乘十。
Five squared is twenty-five, times ten is two hundred and fifty pi cubic centimetres.
五的平方是二十五，乘十等于二百五十派立方厘米。
Now the surface area, where marks are lost.
再求表面积，这里最容易丢分。
Unroll the cylinder.
把圆柱展开。
The curved part opens into a rectangle: its width is the circumference and its height is ten, and that gives a hundred pi.
侧面展开成一个长方形：它的宽是圆周长，高是十，于是得到一百派。
Then add the two circular ends, fifty pi.
再加上两个圆形的底面，五十派。
Total: a hundred and fifty pi square centimetres.
合计：一百五十派平方厘米。
Compound shapes are not hard.
组合图形并不难。
Split the shape into parts you already know.
把它拆成你已经会算的部分。
Here it is a rectangle, eight by five, with a semicircle on the end.
这里是一个八乘五的矩形， 末端接了一个半圆。
Then read the numbers carefully.
然后要仔细读数字。
The flat side of the semicircle is the end of the rectangle, so its diameter is five — which makes the radius two point five, not five.
半圆的那条直边就是矩形的一端， 所以它的直径是五——半径因此是二点五，而不是五。
That step is where the marks are won or lost.
这一步正是分数得失的地方。
So find the total area.
那就来求总面积。
The rectangle is eight times five, forty square centimetres.
矩形是八乘五，等于四十平方厘米。
The semicircle is half of pi times two point five squared, about nine point eight.
半圆是二分之一乘派乘二点五的平方，约等于九点八。
Add them: forty plus nine point eight is about forty-nine point eight square centimetres.
相加：四十加九点八，约等于四十九点八平方厘米。
And if the semicircle had been cut out of the rectangle instead of stuck on, everything is the same — you just subtract.
如果那个半圆不是接上去，而是从矩形里挖掉，做法完全一样——只是把加改成减。
The same idea works in three dimensions.
同样的想法在三维里也成立。
A hemisphere is half a sphere, so halve the sphere formula.
半球是球的一半，所以把球的公式除以二。
A frustum is a cone with its top cut off: work out the whole cone, then subtract the small cone you removed.
平截头体是一个被切掉顶部的圆锥：先算整个圆锥，再减去切掉的那个小圆锥。
For a solid made of two parts, add the volumes.
如果一个立体由两部分组成，就把体积相加。
But careful with surface area: the faces where the parts meet are inside now, not part of the surface.
但表面积要小心： 两部分接合的那些面现在在内部，不属于表面。
Four habits before you go.
结束之前，四个习惯。
First, match the formula to the shape: area has the square in it, circumference does not.
第一，把公式和图形对上：带平方的是面积，周长没有平方。
Second, convert the units before you calculate — and square the factor for area, cube it for volume.
第二，先换算单位再计算——面积把倍数平方，体积把倍数立方。
Third, for an arc or a sector, write the fraction down first.
第三，遇到弧或扇形，先把那个分数写下来。
Fourth, surface area means every face; if unsure, draw the net.
第四，表面积是指每一个面； 不确定就画出展开图。
Get those right, and Topic Five is yours.
做到这些，第五个专题就是你的了。