本讲义涵盖主题 5,测量(Mensuration)(测量长度、面积和体积)。这里的核心和拓展内容几乎相同。在考试中,一些公式在公式表(List of formulas)里给出,但你仍应当把它们全部学会。
测量
IGCSE 数学 · 第 5 主题
7:47
Mensuration
The Great Pyramid of Giza has stood for four and a half thousand years. How much stone is inside it? About two and a half million cubic metres — and one…
英文讲解 · 内嵌中英文字幕
5.1
度量单位
大纲
| Subject content | Notes and examples |
|---|---|
| Use metric units of mass, length, area, volume and capacity in practical situations and convert quantities into larger or smaller units. | Units include: • mm, cm, m, km • $\text{mm}^2$, $\text{cm}^2$, $\text{m}^2$, $\text{km}^2$ • $\text{mm}^3$, $\text{cm}^3$, $\text{m}^3$ • ml, l • g, kg. Conversion between units includes: • between different units of area, e.g. $\text{cm}^2 \leftrightarrow \text{m}^2$ • between units of volume and capacity, e.g. $\text{m}^3 \leftrightarrow \text{litres}$. |
来源:剑桥国际大纲
我们对质量(mass)(g、kg)、长度(length)(mm、cm、m、km)、面积(area)、体积(volume)和容量(capacity)(ml、升——一个容器里面的空间)使用公制(metric)单位。
要换算单位,对平方和立方要小心:
- 长度:$1\text{ m} = 100\text{ cm}$。
- 面积:$1\text{ m}^{2} = 100^{2} = 10\,000\text{ cm}^{2}$。
- 体积:$1\text{ m}^{3} = 100^{3} = 1\,000\,000\text{ cm}^{3}$。
- 容量:$1\text{ litre} = 1000\text{ cm}^{3}$,所以 $1\text{ m}^{3} = 1000$ 升。
Worked example. 把 $3\text{ m}^{2}$ 换算成 $\text{cm}^{2}$。
$$3 \times 10\,000 = 30\,000\text{ cm}^{2}.$$
探索
Unit choice lab
Choose the unit that matches the measurement scale.
词汇表
训练
| 英文 | 中文 | 拼音 |
|---|---|---|
| metric/ˈmetrɪk/ | 公制 | gōng zhì |
| mass/mæs/ | 质量 | zhì liàng |
| length/leŋθ/ | 长度 | cháng dù |
| area/ˈeərɪə/ | 面积 | miàn jī |
| volume/ˈvɒljuːm/ | 体积 | tǐ jī |
| capacity/kəˈpæsɪti/ | 容量 | róng liàng |
5.2
面积与周长
大纲
| Subject content | Notes and examples |
|---|---|
| Carry out calculations involving the perimeter and area of a rectangle, triangle, parallelogram and trapezium. | Except for area of a triangle, formulas are not given. |
| Subject content | Notes and examples |
|---|---|
| Carry out calculations involving the perimeter and area of a rectangle, triangle, parallelogram and trapezium. | Except for the area of a triangle, formulas are not given. |
来源:剑桥国际大纲
周长(perimeter)是绕一个形状一整圈的距离。面积是它里面平坦空间的量。这里 $b$ 是底(base)而 $h$ 是垂直高(height)。
| 形状 | 面积 |
|---|---|
| 矩形(rectangle) | $\text{length} \times \text{width}$ |
| 三角形(triangle) | $\tfrac{1}{2} \times b \times h$ |
| 平行四边形(parallelogram) | $b \times h$ |
| 梯形(trapezium) | $\tfrac{1}{2}(a + b)h$,其中 $a$ 和 $b$ 是两条平行边 |
Worked example. 一个梯形有平行边 $6\text{ cm}$ 和 $10\text{ cm}$,以及高 $4\text{ cm}$。求它的面积。
$$\tfrac{1}{2}(6 + 10) \times 4 = \tfrac{1}{2} \times 16 \times 4 = 32\text{ cm}^{2}.$$
探索
Area scaling lab
area = side^2
Change side length and see why area grows quadratically.
词汇表
训练
| 英文 | 中文 | 拼音 |
|---|---|---|
| perimeter/pəˈrɪmɪtə/ | 周长 | zhōu cháng |
| base/beɪs/ | 底 | dǐ |
| height/haɪt/ | 高 | gāo |
| rectangle/ˈrektæŋɡl/ | 矩形 | jǔ xíng |
| triangle/ˈtraɪæŋɡl/ | 三角形 | sān jiǎo xíng |
| parallelogram/ˌpærəˈleləɡræm/ | 平行四边形 | píng xíng sì biān xíng |
| trapezium/trəˈpiːzɪəm/ | 梯形 | tī xíng |
5.3
圆、弧与扇形
大纲
| Subject content | Notes and examples |
|---|---|
| 1 Carry out calculations involving the circumference and area of a circle. | Answers may be asked for in terms of $\pi$. |
| 2 Carry out calculations involving arc length and sector area as fractions of the circumference and area of a circle, where the sector angle is a factor of $360^\circ$. | Formulas are given in the List of formulas. |
| Subject content | Notes and examples |
|---|---|
| 1 Carry out calculations involving the circumference and area of a circle. | Answers may be asked for in terms of $\pi$. Formulas are given in the List of formulas. |
| 2 Carry out calculations involving arc length and sector area as fractions of the circumference and area of a circle. | Includes minor and major sectors. |
来源:剑桥国际大纲
对于一个有半径(radius)$r$(和直径(diameter)$d = 2r$)的圆(circle):
$$\text{circumference} = 2\pi r = \pi d, \qquad \text{area} = \pi r^{2}.$$
圆周(circumference)是绕圆一圈的距离。
Worked example. 一个圆有半径 $7\text{ cm}$。求它的圆周和面积(在答案里保留 $\pi$)。
$$\text{circumference} = 2\pi \times 7 = 14\pi\text{ cm}, \qquad \text{area} = \pi \times 7^{2} = 49\pi\text{ cm}^{2}.$$
词汇表
训练
| 英文 | 中文 | 拼音 |
|---|---|---|
| circle/ˈsɜːkl/ | 圆 | yuán |
| radius/ˈreɪdɪəs/ | 半径 | bàn jìng |
| diameter/daɪˈæmɪtə/ | 直径 | zhí jìng |
| circumference/sɜːˈkʌmfrəns/ | 圆周 | yuán zhōu |
5.3
圆、弧与扇形
大纲
| Subject content | Notes and examples |
|---|---|
| 1 Carry out calculations involving the circumference and area of a circle. | Answers may be asked for in terms of $\pi$. |
| 2 Carry out calculations involving arc length and sector area as fractions of the circumference and area of a circle, where the sector angle is a factor of $360^\circ$. | Formulas are given in the List of formulas. |
| Subject content | Notes and examples |
|---|---|
| 1 Carry out calculations involving the circumference and area of a circle. | Answers may be asked for in terms of $\pi$. Formulas are given in the List of formulas. |
| 2 Carry out calculations involving arc length and sector area as fractions of the circumference and area of a circle. | Includes minor and major sectors. |
来源:剑桥国际大纲
一个弧(arc)是圆周的一部分。一个扇形(sector)是两条半径之间的一个"披萨片"。若扇形角是 $\theta$,弧和扇形是整个圆的那个分数 $\dfrac{\theta}{360}$:
$$\text{arc length} = \frac{\theta}{360} \times 2\pi r, \qquad \text{sector area} = \frac{\theta}{360} \times \pi r^{2}.$$
一个小的片是一个小扇形(minor sector);大的其余是一个大扇形(major sector)。
Worked example. 求一个角 $90^{\circ}$、半径 $8\text{ cm}$ 的扇形的弧长(arc length)和面积。
分数是 $\dfrac{90}{360} = \dfrac{1}{4}$,所以
$$\text{arc} = \tfrac{1}{4} \times 2\pi \times 8 = 4\pi\text{ cm}, \qquad \text{area} = \tfrac{1}{4} \times \pi \times 8^{2} = 16\pi\text{ cm}^{2}.$$
探索
Arcs & sectors
s = rθ · A = ½r²θ
A bigger angle or radius means a longer arc and larger sector area.
探索
Arc length and sector area
Change the angle and radius and read off the arc length and sector area — a fraction of the whole circle.
词汇表
训练
| 英文 | 中文 | 拼音 |
|---|---|---|
| arc/ɑːk/ | 弧 | hú |
| sector/ˈsektə/ | 扇形 | shàn xíng |
| minor sector/ˈmaɪnə ˈsektə/ | 小扇形 | xiǎo shàn xíng |
| major sector/ˈmeɪdʒə ˈsektə/ | 大扇形 | dà shàn xíng |
| arc length/ɑːk leŋθ/ | 弧长 | hú zhǎng |
5.4
表面积与体积
大纲
| Subject content | Notes and examples |
|---|---|
| Carry out calculations and solve problems involving the surface area and volume of a: • cuboid • prism • cylinder • sphere • pyramid • cone. | Answers may be asked for in terms of $\pi$. The following formulas are given in the List of formulas: • curved surface area of a cylinder • curved surface area of a cone • surface area of a sphere • volume of a prism • volume of a pyramid • volume of a cylinder • volume of a cone • volume of a sphere. The term prism refers to any solid with a uniform cross-section, e.g. a cylindrical sector. |
来源:剑桥国际大纲
表面积(surface area)是所有外部面的总面积。体积是里面的空间。对于这些立体($r$ = 半径,$h$ = 高):
| 立体 | 体积 | 表面积 |
|---|---|---|
| 长方体(cuboid) | $\text{length} \times \text{width} \times \text{height}$ | 把六个面相加 |
| 棱柱(prism) | (横截面(cross-section)面积) $\times$ 长度 | — |
| 圆柱(cylinder) | $\pi r^{2} h$ | $2\pi r h$(侧面积(curved surface area)) $+\, 2\pi r^{2}$ |
| 棱锥(pyramid) | $\tfrac{1}{3} \times \text{base area} \times h$ | — |
| 圆锥(cone) | $\tfrac{1}{3}\pi r^{2} h$ | $\pi r l$(侧) $+\, \pi r^{2}$,其中 $l$ 是斜高(slant height) |
| 球(sphere) | $\tfrac{4}{3}\pi r^{3}$ | $4\pi r^{2}$ |
Worked example. 一个圆柱有半径 $5\text{ cm}$ 和高 $10\text{ cm}$。求它的体积和总表面积(以 $\pi$ 表示)。
$$\text{volume} = \pi \times 5^{2} \times 10 = 250\pi\text{ cm}^{3}.$$
$$\text{surface area} = 2\pi(5)(10) + 2\pi(5)^{2} = 100\pi + 50\pi = 150\pi\text{ cm}^{2}.$$
探索
Volume scaling lab
surface area grows with scale^2
Change length scale and see volume grow faster than surface area.
词汇表
训练
| 英文 | 中文 | 拼音 |
|---|---|---|
| surface area/ˈsɜːfɪs ˈeərɪə/ | 表面积 | biǎo miàn jī |
| cuboid/ˈkjuːbɔɪd/ | 长方体 | cháng fāng tǐ |
| prism/ˈprɪzəm/ | 棱柱 | léng zhù |
| cross-section/krɒs ˈsekʃn/ | 横截面 | héng jié miàn |
| cylinder/ˈsɪlɪndə/ | 圆柱 | yuán zhù |
| curved surface area/kɜːvd ˈsɜːfɪs ˈeərɪə/ | 侧面积 | cè miàn jī |
| pyramid/ˈpɪrəmɪd/ | 棱锥 | léng zhuī |
| cone/kəʊn/ | 圆锥 | yuán zhuī |
| slant height/slænt haɪt/ | 斜高 | xié gāo |
| sphere/sfɪə/ | 球 | qiú |
5.5
复合图形与图形的部分
大纲
| Subject content | Notes and examples |
|---|---|
| 1 Carry out calculations and solve problems involving perimeters and areas of: • compound shapes • parts of shapes. | Answers may be asked for in terms of $\pi$. |
| 2 Carry out calculations and solve problems involving surface areas and volumes of: • compound solids • parts of solids. | e.g. find the volume of half of a sphere. |
| Subject content | Notes and examples |
|---|---|
| 1 Carry out calculations and solve problems involving perimeters and areas of: • compound shapes • parts of shapes. | Answers may be asked for in terms of $\pi$. |
| 2 Carry out calculations and solve problems involving surface areas and volumes of: • compound solids • parts of solids. | e.g. find the surface area and volume of a frustum. |
来源:剑桥国际大纲
一个组合图形(compound shape)通过连接或切割简单形状构成。把它分成你知道的部分,然后加或减。
对于一个圆或立体的"部分",取正确的分数。例如,一个半球(hemisphere)(半个球)有体积
$$\tfrac{1}{2} \times \tfrac{4}{3}\pi r^{3} = \tfrac{2}{3}\pi r^{3}.$$
一个平截头体(frustum)是一个顶部被切掉的圆锥或棱锥;通过从整个圆锥减去小的顶部圆锥来求它的体积。
Worked example. 求一个由一个矩形 $8\text{ cm} \times 5\text{ cm}$ 加一端一个直径 $5\text{ cm}$ 的半圆构成的形状的面积。
半圆有半径 $2.5\text{ cm}$:
$$\text{area} = 8 \times 5 + \tfrac{1}{2}\pi (2.5)^{2} = 40 + 3.125\pi \approx 49.8\text{ cm}^{2}.$$
探索
Compound shape route
Break a compound shape into simple parts, then recombine.
词汇表
训练
| 英文 | 中文 | 拼音 |
|---|---|---|
| compound shape/ˈkɒmpaʊnd ʃeɪp/ | 组合图形 | zǔ hé tú xíng |
| hemisphere/ˈhemɪsfɪə/ | 半球 | bàn qiú |
| frustum/ˈfrʌstəm/ | 平截头体 | píng jié tóu tǐ |
5.5
考试技巧
- 把公式匹配到形状:一个圆的面积是 $\pi r^2$、圆周是 $\pi d$(或 $2\pi r$)——不要把它们搞混。
- 保持单位一致,并记住面积单位是平方的而体积单位是立方的(例如 $1\text{ m}^2 = 10\,000\text{ cm}^2$)。
- 对于一个弧或扇形,取整个圆周或面积的分数 $\frac{\theta}{360}$。
- 表面积是所有面的总和——若你不确定就画展开图。只有在问题允许时才以 $\pi$ 表示保留一个答案。
本主题的互动课程
逐步学习,并即时检测练习。