Mensuration

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).

Changing length units: multiply going down, divide going up

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 $\text{cm}^{2}$.

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.

Area of the basic shapes; $b$ is the base, $h$ the perpendicular height, and $a$, $b$ the two parallel sides of a trapezium.

Worked example. A trapezium has parallel sides $6\text{ cm}$ and $10\text{ cm}$, and height $4\text{ cm}$. Find its area.

For a circle 圆 with radius 半径 $r$ (and diameter 直径 $d = 2r$):

A circle: circumference = pi d and area = pi r squared

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).

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:

A sector is the fraction $\tfrac{\theta}{360}$ of the whole circle, so its arc and area are that fraction of the circumference and the area.

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

The pyramids of Giza are square-based pyramids — a 3-D solid.

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):

Common solids and the dimensions ($r$, $h$, $\ell$, slant $l$) used in their volume and surface-area formulas.

Worked example. A cylinder has radius $5\text{ cm}$ and height $10\text{ cm}$. Find its volume and total surface area (in terms of $\pi$).

Stacked containers are cuboids; volume is length times width times height.

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

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.

Split a compound shape into parts you know — here a rectangle plus a semicircle — then add the areas.

The semicircle has radius $2.5\text{ cm}$:

• 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.