This handout covers Topic 5, Mensuration (measuring length, area and volume). The Core and Extended content here is almost the same. In the exam, some formulas are given in the List of formulas, but you should still learn them all.
Mensuration
IGCSE Mathematics · Topic 5
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…
English narration · English + 中文 subtitles burned in
5.1
Units of measure
Syllabus
| 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. $\text{m}^3 \leftrightarrow \text{litres}$. |
Source: Cambridge International syllabus
We use metric 公制 units for mass 质量 (g, kg), length 长度 (mm, cm, m, km), area 面积, volume 体积 and capacity 容量 (ml, litres — the space inside a container).
To change units, be careful with squares and cubes:
- Length: $1\text{ m} = 100\text{ cm}$.
- Area: $1\text{ m}^{2} = 100^{2} = 10\,000\text{ cm}^{2}$.
- Volume: $1\text{ m}^{3} = 100^{3} = 1\,000\,000\text{ cm}^{3}$.
- Capacity: $1\text{ litre} = 1000\text{ cm}^{3}$, so $1\text{ m}^{3} = 1000$ litres.
Worked example. Convert $3\text{ m}^{2}$ to $\text{cm}^{2}$.
$$3 \times 10\,000 = 30\,000\text{ cm}^{2}.$$
Explore
Unit choice lab
Choose the unit that matches the measurement scale.
Vocabulary
Train
| English |
|---|
| metric/ˈmetrɪk/ |
| mass/mæs/ |
| length/leŋθ/ |
| area/ˈeərɪə/ |
| volume/ˈvɒljuːm/ |
| capacity/kəˈpæsɪti/ |
5.2
Perimeter and area of basic shapes
Syllabus
| 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. |
Source: Cambridge International syllabus
The perimeter 周长 is the distance all the way round a shape. The area is the amount of flat space inside it. Here $b$ is the base 底 and $h$ is the perpendicular height 高.
| Shape | Area |
|---|---|
| rectangle 矩形 | $\text{length} \times \text{width}$ |
| triangle 三角形 | $\tfrac{1}{2} \times b \times h$ |
| parallelogram 平行四边形 | $b \times h$ |
| trapezium 梯形 | $\tfrac{1}{2}(a + b)h$, where $a$ and $b$ are the two parallel sides |
Worked example. A trapezium has parallel sides $6\text{ cm}$ and $10\text{ cm}$, and height $4\text{ cm}$. Find its area.
$$\tfrac{1}{2}(6 + 10) \times 4 = \tfrac{1}{2} \times 16 \times 4 = 32\text{ cm}^{2}.$$
Explore
Area scaling lab
area = side^2
Change side length and see why area grows quadratically.
Vocabulary
Train
| English |
|---|
| perimeter/pəˈrɪmɪtə/ |
| base/beɪs/ |
| height/haɪt/ |
| rectangle/ˈrektæŋɡl/ |
| triangle/ˈtraɪæŋɡl/ |
| parallelogram/ˌpærəˈleləɡræm/ |
| trapezium/trəˈpiːzɪəm/ |
5.3
Circles
Syllabus
| 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. |
Source: Cambridge International syllabus
For a circle 圆 with radius 半径 $r$ (and diameter 直径 $d = 2r$):
$$\text{circumference} = 2\pi r = \pi d, \qquad \text{area} = \pi r^{2}.$$
The circumference 圆周 is the distance round the circle.
Worked example. A circle has radius $7\text{ cm}$. Find its circumference and area (leave $\pi$ in the answer).
$$\text{circumference} = 2\pi \times 7 = 14\pi\text{ cm}, \qquad \text{area} = \pi \times 7^{2} = 49\pi\text{ cm}^{2}.$$
Vocabulary
Train
| English |
|---|
| circle/ˈsɜːkl/ |
| radius/ˈreɪdɪəs/ |
| diameter/daɪˈæmɪtə/ |
| circumference/sɜːˈkʌmfrəns/ |
5.3
Arcs and sectors
Syllabus
| 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. |
Source: Cambridge International syllabus
An arc 弧 is part of the circumference. A sector 扇形 is a "pizza slice" between two radii. If the sector angle is $\theta$, the arc and sector are that fraction $\dfrac{\theta}{360}$ of the whole circle:
$$\text{arc length} = \frac{\theta}{360} \times 2\pi r, \qquad \text{sector area} = \frac{\theta}{360} \times \pi r^{2}.$$
A small slice is a minor sector 小扇形; the large rest is a major sector 大扇形.
Worked example. Find the arc length 弧长 and area of a sector with angle $90^{\circ}$ and radius $8\text{ cm}$.
The fraction is $\dfrac{90}{360} = \dfrac{1}{4}$, so
$$\text{arc} = \tfrac{1}{4} \times 2\pi \times 8 = 4\pi\text{ cm}, \qquad \text{area} = \tfrac{1}{4} \times \pi \times 8^{2} = 16\pi\text{ cm}^{2}.$$
Explore
Arcs & sectors
s = rθ · A = ½r²θ
A bigger angle or radius means a longer arc and larger sector area.
Explore
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.
Vocabulary
Train
| English |
|---|
| arc/ɑːk/ |
| sector/ˈsektə/ |
| minor sector/ˈmaɪnə ˈsektə/ |
| major sector/ˈmeɪdʒə ˈsektə/ |
| arc length/ɑːk leŋθ/ |
5.4
Surface area and volume of solids
Syllabus
| 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. |
Source: Cambridge International syllabus
The surface area 表面积 is the total area of all the outside faces. The volume is the space inside. For these solids ($r$ = radius, $h$ = height):
| Solid | Volume | Surface area |
|---|---|---|
| cuboid 长方体 | $\text{length} \times \text{width} \times \text{height}$ | add the six faces |
| prism 棱柱 | (cross-section 横截面 area) $\times$ length | — |
| cylinder 圆柱 | $\pi r^{2} h$ | $2\pi r h$ (curved surface area 侧面积) $+\, 2\pi r^{2}$ |
| pyramid 棱锥 | $\tfrac{1}{3} \times \text{base area} \times h$ | — |
| cone 圆锥 | $\tfrac{1}{3}\pi r^{2} h$ | $\pi r l$ (curved) $+\, \pi r^{2}$, where $l$ is the slant height 斜高 |
| sphere 球 | $\tfrac{4}{3}\pi r^{3}$ | $4\pi r^{2}$ |
Worked example. A cylinder has radius $5\text{ cm}$ and height $10\text{ cm}$. Find its volume and total surface area (in terms of $\pi$).
$$\text{volume} = \pi \times 5^{2} \times 10 = 250\pi\text{ cm}^{3}.$$
$$\text{surface area} = 2\pi(5)(10) + 2\pi(5)^{2} = 100\pi + 50\pi = 150\pi\text{ cm}^{2}.$$
Explore
Volume scaling lab
surface area grows with scale^2
Change length scale and see volume grow faster than surface area.
Vocabulary
Train
| English |
|---|
| surface area/ˈsɜːfɪs ˈeərɪə/ |
| cuboid/ˈkjuːbɔɪd/ |
| prism/ˈprɪzəm/ |
| cross-section/krɒs ˈsekʃn/ |
| cylinder/ˈsɪlɪndə/ |
| curved surface area/kɜːvd ˈsɜːfɪs ˈeərɪə/ |
| pyramid/ˈpɪrəmɪd/ |
| cone/kəʊn/ |
| slant height/slænt haɪt/ |
| sphere/sfɪə/ |
5.5
Compound shapes and parts of shapes
Syllabus
| 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. |
Source: Cambridge International syllabus
A compound shape 组合图形 is made by joining or cutting simple shapes. Split it into parts you know, then add or subtract.
For "parts" of a circle or solid, take the right fraction. For example, a hemisphere 半球 (half a sphere) has volume
$$\tfrac{1}{2} \times \tfrac{4}{3}\pi r^{3} = \tfrac{2}{3}\pi r^{3}.$$
A frustum 平截头体 is a cone or pyramid with its top cut off; find its volume by subtracting the small top cone from the whole cone.
Worked example. Find the area of a shape made of a rectangle $8\text{ cm} \times 5\text{ cm}$ with a semicircle of diameter $5\text{ cm}$ on one end.
The semicircle has radius $2.5\text{ cm}$:
$$\text{area} = 8 \times 5 + \tfrac{1}{2}\pi (2.5)^{2} = 40 + 3.125\pi \approx 49.8\text{ cm}^{2}.$$
Explore
Compound shape route
Break a compound shape into simple parts, then recombine.
Vocabulary
Train
| English |
|---|
| compound shape/ˈkɒmpaʊnd ʃeɪp/ |
| hemisphere/ˈhemɪsfɪə/ |
| frustum/ˈfrʌstəm/ |
5.5
Exam tips
- Match the formula to the shape: the area of a circle is $\pi r^2$, the circumference is $\pi d$ (or $2\pi r$) — do not mix them up.
- Keep units consistent, and remember area units are squared and volume units cubed (e.g. $1\text{ m}^2 = 10\,000\text{ cm}^2$).
- For an arc or sector, take the fraction $\frac{\theta}{360}$ of the whole circumference or area.
- Surface area is the total of all the faces — draw the net if you are unsure. Leave an answer in terms of $\pi$ only if the question allows it.
Interactive lessons on this topic
Work through it step by step, with instant-check exercises.