Compound shapes and parts of shapes · 组合图形与部分图形
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| compound shapes/ˈkɒmpaʊnd ʃeɪps/ | 复合图形 | fù hé tú xíng |
| hemisphere/ˈhemɪsfɪə/ | 半球 | bàn qiú |
| frustum/ˈfrʌstəm/ | 平截头体 | píng jié tóu tǐ |
Jigsaw shapes
- A patio is a rectangle with a semicircular flower bed on one end. How much paving do you need?
- Compound shapes 复合图形 are built from simple ones. Split, calculate, then add or subtract.
拼图形状
- 一个露台是一个矩形,一端有一个半圆形花坛。你需要多少铺路材料?
- 复合形状(compound shapes)由简单的形状构建。分割、计算,然后相加或相减。
Compound shape route · 组合图形路线
Break a compound shape into simple parts, then recombine. · 将组合图形分解为简单部分，然后重新组合。
Strategy: split into known parts
- Break the shape into rectangles, triangles, circles — anything you have a formula for.
- Calculate each area separately, then add them together (or subtract if a piece is cut out).
A rectangle $8 \times 5$ with a semicircle (diameter $5$) on one end: Area $= 40 + \dfrac{1}{2}\pi(2.5)^2 = 40 + 3.125\pi \approx 49.8\text{ cm}^2$.
Watch for cut-outs. If a shape has a hole, you subtract the hole's area. A rectangle with a circular hole: total area $=$ rectangle area $-$ circle area.
Split a compound shape into parts you know, here a rectangle plus a semicircle, then add the areas
策略:分割成已知的部分
- 把形状分成矩形、三角形、圆——任何你有公式的东西。
- 分别计算每个面积,然后把它们相加(或如果一块被切掉则相减)。
一个 $8 \times 5$ 的矩形,一端有一个半圆(直径 $5$): 面积 $= 40 + \dfrac{1}{2}\pi(2.5)^2 = 40 + 3.125\pi \approx 49.8\text{ cm}^2$。
注意切口。 如果一个形状有一个洞,你减去洞的面积。一个有圆形洞的矩形:总面积 $=$ 矩形面积 $-$ 圆面积。
把一个复合形状分成你知道的部分,这里是一个矩形加一个半圆,然后把面积相加
To find the area of a rectangle with a semicircle on one end, you: · 要计算一端带有半圆的矩形的面积，你应该：
The shape is the rectangle plus the semicircle, so add the two areas. · 该形状是矩形加上半圆，因此将两个面积相加。
A compound shape is a rectangle 8 cm by 5 cm plus a semicircle on the end. What is the rectangle part of the area (cm²)? · 一个组合图形由一个 8 cm × 5 cm 的矩形和末端的一个半圆组成。矩形部分的面积是多少（cm²）？
Rectangle area = 8 × 5 = 40 cm² (then add the semicircle). · 矩形面积 = 8 × 5 = 40 cm²（然后加上半圆）。
If a shape has a circular hole cut out, you subtract the circle area from the outer shape area. · 如果图形中挖去了一个圆形孔洞，应从外图形的面积中减去该圆的面积。
Cut-outs are subtracted: total = outer area − hole area. · 挖去部分需从总面积中扣除：总面积 = 外图形面积 − 孔洞面积。
Parts of solids
- A hemisphere 半球 is half a sphere: volume $= \dfrac{1}{2} \times \dfrac{4}{3}\pi r^3 = \dfrac{2}{3}\pi r^3$.
- A frustum 平截头体 is a cone or pyramid with the top sliced off parallel to the base.
- Find it by subtracting: whole cone minus small top cone.
立体的部分
- 一个半球(hemisphere)是半个球:体积 $= \dfrac{1}{2} \times \dfrac{4}{3}\pi r^3 = \dfrac{2}{3}\pi r^3$。
- 一个截头锥(frustum)是一个顶部被平行于底切掉的圆锥或棱锥。
- 通过相减找到它:整个圆锥减小的顶部圆锥。
The volume of a hemisphere of radius r is: · 半径为 r 的半球体体积为：
Half of (4/3)πr³ is (2/3)πr³. · (4/3)πr³ 的一半是 (2/3)πr³。
A hemisphere has radius 6 cm. Its volume is kπ cm³. What is k? · 一个半球体的半径为 6 cm。其体积为 kπ cm³。求 k 的值？
(2/3) × π × 6³ = (2/3) × 216π = 144π, so k = 144. · (2/3) × π × 6³ = (2/3) × 216π = 144π，因此 k = 144。
A frustum is found by subtracting the small top cone from the ______ cone. · 圆台是通过从大圆锥中减去顶部的小圆锥得到的。
Frustum = whole cone − small top cone (cut parallel to the base). · 圆台 = 完整圆锥 − 平行于底面切下的顶部小圆锥。
Worked example — hemisphere
- Hemisphere of radius $6\text{ cm}$:
- Volume $= \dfrac{2}{3}\pi(6)^3 = \dfrac{2}{3}\pi(216) = 144\pi\text{ cm}^3$.
Compound shapes are built from simple ones — here two squares of different sizes, like a rectangle made of two parts.
例题——半球
- 半径 $6\text{ cm}$ 的半球:
- 体积 $= \dfrac{2}{3}\pi(6)^3 = \dfrac{2}{3}\pi(216) = 144\pi\text{ cm}^3$。
复合形状由简单的形状构建——这里是两个不同大小的正方形,像一个由两部分组成的矩形。
Why the frustum matters
- The frustum is the shape of most lampshades, buckets, and drinking glasses. Engineers and architects use the formula constantly.
为什么截头锥重要
- 截头锥是大多数灯罩、水桶和饮用杯的形状。工程师和建筑师不断使用这个公式。
Exposed area at a join
- Two cubes of edge 3 cm join face to face. Their volume adds: $V=2(3^3)=54\text{ cm}^3$. Their area does not simply add: the two touching faces disappear from the outside, so $S=12(3^2)-2(3^2)=90\text{ cm}^2$.
- Half a sphere of radius 6 cm has curved area $2\pi r^2=72\pi\text{ cm}^2$; including the flat circular base gives total $3\pi r^2=108\pi\text{ cm}^2$. State whether the flat base is exposed.
拼接处的暴露面积
- 两个边长为 3 cm 的正方体面贴面拼接。体积相加：$V=2(3^3)=54\text{ cm}^3$。表面积不能简单相加：因为两个接触的面不再暴露在外，所以 $S=12(3^2)-2(3^2)=90\text{ cm}^2$。
- 半径为 6 cm 的半球曲面面积为 $2\pi r^2=72\pi\text{ cm}^2$；若包含平坦的圆形底面，则总表面积为 $3\pi r^2=108\pi\text{ cm}^2$。请说明平坦底面是否暴露。
Two 3 cm cubes join face to face. Find the exposed area in cm². · 两个边长为3 cm的立方体面相连接。求暴露在外部的表面积，单位为cm²。
Ten square faces remain exposed: 10 × 9 = 90 cm². · 剩余十个正方形面暴露在外：10 × 9 = 90 cm²。
You've got it
- split a compound shape into known parts, then add or subtract
- hemisphere volume $= \dfrac{2}{3}\pi r^3$
- a frustum = whole cone minus the small top cone
你掌握了
- 把一个复合形状分成已知的部分,然后相加或相减
- 半球体积 $= \dfrac{2}{3}\pi r^3$
- 一个截头锥 = 整个圆锥减小的顶部圆锥