📚 Composite Shapes: Area and Perimeter | 组合图形的面积与周长
Composite shapes are figures made from two or more simple shapes such as rectangles, triangles, circles and semicircles. In the Cambridge KS3 mathematics curriculum, you need to find both the total area and the outer perimeter of these combined figures by splitting them into parts or by subtracting one shape from another.
组合图形是由两个或两个以上的简单图形（如矩形、三角形、圆和半圆）组成的图形。在剑桥 KS3 数学课程中，你需要通过将组合图形拆分成多个部分、或从一个图形中减去另一个图形，来求出总面积和外周长。
1. Understanding Composite Shapes | 认识组合图形
A composite shape is formed when two or more basic shapes are joined together or overlap. You may see an L-shape made from two rectangles, a flag made from a triangle and a rectangle, or a running track made from two semicircles and a rectangle. The goal is to use known formulas for simple shapes to calculate the area and perimeter of the whole figure.
组合图形由两个或多个基本图形拼接或重叠而成。你可能会看到由两个矩形组成的 L 形、由三角形和矩形组成的旗帜，或由两个半圆和一个矩形组成的跑道。目标是利用简单图形的已知公式来计算整个图形的面积和周长。
It is important to identify the basic shapes before you calculate. Draw dashed lines to show how you split the composite shape, and label every side or radius that you know. This visual step reduces mistakes and helps you decide whether to add or subtract areas.
在计算之前，识别基本图形非常重要。用虚线标出你如何拆分组合图形，并标出每一条已知的边或半径。这个可视化步骤能减少错误，并帮助你决定是相加还是相减面积。
2. Key Formulas and Units | 关键公式与单位
Before working with composite shapes, you must be confident with these simple shape formulas. For a rectangle, the area is length times width and the perimeter is twice the sum of length and width.
在处理组合图形之前，你必须熟练掌握以下简单图形的公式。对于矩形，面积等于长乘宽，周长等于长加宽之和的两倍。
For a triangle, the area is one half of the base times the perpendicular height. The perimeter is the sum of all three side lengths. For a circle, the area is π times the radius squared and the circumference is 2π times the radius or π times the diameter.
对于三角形，面积等于底乘垂直高度的一半。周长是三条边长之和。对于圆，面积等于 π 乘半径的平方，周长等于 2π 乘半径，或 π 乘直径。
For a semicircle, remember to take half of the circle area and half of the circle circumference, then add the straight diameter to get the full outer perimeter. Always check whether the question wants an exact answer in terms of π or a decimal rounded to a given number of decimal places.
对于半圆，记住面积取圆面积的一半，弧长取圆周长的一半，然后再加上直边直径才能得到完整的外周长。始终要检查题目是要求用 π 表示的精确答案，还是要求四舍五入到指定小数位的近似值。
Rectangle: A = lw, P = 2(l + w) | Triangle: A = ½bh | Circle: A = πr², C = 2πr
矩形：A = lw，P = 2(l + w) | 三角形：A = ½bh | 圆：A = πr²，C = 2πr
3. Splitting Method for Area | 拆分法求面积
The most common way to find the area of a composite shape is to split it into rectangles, triangles or circles. Find the area of each part separately and then add the results together. This method works well for L-shapes, T-shapes and figures that can be divided into non-overlapping regions.
求组合图形面积最常用的方法是把它拆分成矩形、三角形或圆。分别求出每一部分的面积，然后把结果相加。这种方法适用于 L 形、T 形以及可以分割成不重叠区域的图形。
For example, an L-shape can be divided into two rectangles by drawing one vertical or horizontal cut. If the first rectangle is 6 cm by 4 cm and the second is 3 cm by 2 cm, the total area is 24 cm² + 6 cm² = 30 cm².
例如，一个 L 形可以通过画一条垂直或水平分割线分成两个矩形。如果第一个矩形是 6 cm × 4 cm，第二个是 3 cm × 2 cm，那么总面积就是 24 cm² + 6 cm² = 30 cm²。
Always label the dimensions of each small shape before calculating. If a side length is missing, use the fact that opposite sides of a rectangle are equal, or subtract known lengths from a total length.
在计算之前，一定要标出每个小图形的边长。如果缺少某条边长，可以利用矩形对边相等这一性质，或者用总长度减去已知长度。
4. Triangles in Composite Figures | 组合图形中的三角形
When a composite shape includes a triangle, the area is found by multiplying the base by the perpendicular height and then dividing by two. The perpendicular height is the length measured at a right angle to the base, not the sloping side length.
当组合图形中包含三角形时，三角形面积等于底乘垂直高再除以二。垂直高是指到底边的垂直距离，而不是斜边的长度。
Some figures combine a rectangle and a triangle on top, like a house shape. If the rectangle is 10 m by 5 m and the triangle has a base of 10 m and a height of 4 m, the total area is 50 m² + 20 m² = 70 m².
有些图形像房屋形状，由底部矩形和顶部三角形组成。如果矩形是 10 m × 5 m，三角形底为 10 m，高为 4 m，那么总面积是 50 m² + 20 m² = 70 m²。
For perimeter, do not forget to include the two sloping sides of the triangle. You may need to use Pythagoras’ theorem if the sloping side is unknown, but in KS3 most questions give the side lengths directly or can be found by simple addition.
计算周长时，不要忘记加上三角形的两条斜边。如果斜边未知，你可能需要使用勾股定理，但在 KS3 阶段，大多数题目会直接给出边长，或通过简单加法即可求出。
5. Circles and Semicircles | 圆与半圆
Composite shapes often include full circles, semicircles or quarter circles. For a full circle, use A = πr² for area and C = 2πr for circumference. For a semicircle, the area is half of a full circle, and the curved arc length is half of the circumference.
组合图形常常包含整圆、半圆或四分之一圆。对于整圆，面积使用 A = πr²，周长使用 C = 2πr。对于半圆，面积是整圆的一半，弧长是圆周长的一半。
If a semicircle has diameter 8 cm, its radius is 4 cm. The area of the semicircle is ½ × π × 4² = 8π cm². The curved arc length is ½ × 2π × 4 = 4π cm. To find the outer perimeter of the semicircle, add the straight diameter: 4π + 8 cm.
如果一个半圆的直径是 8 cm，那么它的半径是 4 cm。半圆面积是 ½ × π × 4² = 8π cm²。弧长是 ½ × 2π × 4 = 4π cm。求半圆的外周长时，还要加上直边直径：4π + 8 cm。
Use the diameter correctly: the radius is half the diameter. A common error is to use the diameter in place of the radius when calculating area, which gives an answer four times too large.
要正确使用直径：半径是直径的一半。一个常见错误是在计算面积时把直径当作半径使用，这样得到的答案会大四倍。
6. Subtraction Method for Shaded Regions | 阴影区域的减法策略
Sometimes a composite shape is shown as a shaded region inside a larger shape. Instead of splitting the shaded part into small pieces, it is often easier to find the area of the large shape and subtract the area of the unshaded shape.
有时组合图形表示为一个大图形内部的阴影区域。与其把阴影部分拆分成小块，通常更简单的方法是求出大图形的面积，再减去未阴影图形的面积。
For example, a rectangle is 12 cm by 9 cm, and a smaller rectangle of 5 cm by 4 cm is removed from the centre. The shaded area is 108 cm² − 20 cm² = 88 cm².
例如，一个矩形是 12 cm × 9 cm，中心去掉一个 5 cm × 4 cm 的小矩形。阴影面积就是 108 cm² − 20 cm² = 88 cm²。
This method also works when a triangle or circle is cut out of a rectangle. Always write down the two separate areas before subtracting, and keep the units the same throughout.
当从矩形中挖去三角形或圆时，这个方法同样适用。一定要先分别写出两个面积再相减，并且全程保持单位一致。
7. Perimeter Strategy: Finding Missing Sides | 周长策略：求缺失边长
Perimeter means the total distance around the outside of a shape. When a composite shape has a notch or step, some side lengths may not be labelled. You can find these missing lengths by comparing opposite sides or by using the total width and height.
周长是指围绕图形外部的总距离。当组合图形有凹口或台阶时，某些边可能没有标出长度。你可以通过比较对边，或利用总宽度和总高度来求出这些缺失的边长。
Suppose an L-shape has total width 8 m and total height 5 m. A rectangular notch is cut out with width 3 m and height 2 m. The missing vertical side is 5 m − 2 m = 3 m, and the missing horizontal side is 8 m − 3 m = 5 m.
假设一个 L 形的总宽度为 8 m，总高度为 5 m。切掉一个宽 3 m、高 2 m 的矩形凹口。缺失的竖直边是 5 m − 2 m = 3 m，缺失的水平边是 8 m − 3 m = 5 m。
When adding the outer sides for perimeter, ignore any internal dashed lines used for splitting. Internal lines are not part of the outside edge and must never be counted.
在把外部边相加求周长时，忽略所有用于拆分的内部虚线。内部线不属于外部边缘，绝不能计入周长。
8. Worked Example: Rectangle with Semicircle | 分步例题：矩形加半圆
A shape is made from a rectangle of width 12 cm and height 5 cm, with a semicircle of diameter 12 cm sitting on top. Find the total area and the outer perimeter. Give exact answers in terms of π.
一个图形由宽 12 cm、高 5 cm 的矩形和一个直径为 12 cm 的半圆组成，半圆位于矩形顶部。求总面积和外周长。答案用 π 表示。
Step 1: Find the radius. The diameter is 12 cm, so the radius is 6 cm. Step 2: Calculate the rectangle area: 12 × 5 = 60 cm².
第 1 步：求半径。直径是 12 cm，所以半径是 6 cm。第 2 步：计算矩形面积：12 × 5 = 60 cm²。
Step 3: Calculate the semicircle area: ½ × π × 6² = ½ × 36π = 18π cm². Step 4: Total area = 60 + 18π cm².
第 3 步：计算半圆面积：½ × π × 6² = ½ × 36π = 18π cm²。第 4 步：总面积 = 60 + 18π cm²。
Step 5: Calculate the outer perimeter. The curved arc length is ½ × 2π × 6 = 6π cm. The straight outer edges are the bottom side 12 cm and the two vertical sides 5 cm and 5 cm.
第 5 步：计算外周长。弧长为 ½ × 2π × 6 = 6π cm。外部直边是底边 12 cm 和两条竖直边 5 cm、5 cm。
Step 6: Add them: P = 6π + 12 + 5 + 5 = 6π + 22 cm. Therefore total area = 60 + 18π cm² and perimeter = 22 + 6π cm.
第 6 步：相加：P = 6π + 12 + 5 + 5 = 6π + 22 cm。因此总面积 = 60 + 18π cm²，外周长 = 22 + 6π cm。
9. Common Errors and How to Avoid Them | 常见错误与避免方法
One common mistake is forgetting to halve the area of a circle when working with a semicircle. If you use πr² for a semicircle, the area is double the correct value. Always multiply by ½.
一个常见错误是在计算半圆时忘记把圆面积除以二。如果半圆面积用了 πr²，结果就是正确值的两倍。一定要乘以 ½。
Another mistake is counting the internal shared side when calculating perimeter. For example, when a triangle sits on top of a rectangle, the top side of the rectangle is not part of the outer edge if the triangle covers it. Only count sides on the outside.
另一个错误是在计算周长时把内部共用边也算进去。例如，当三角形位于矩形顶部时，如果三角形覆盖了矩形的上边，那么这条上边就不再是外部边缘。只计算外部的边。
Students also mix up radius and diameter. If a circle has diameter d, the radius is d ÷ 2. Check whether the question gives radius or diameter before substituting into A = πr² or C = 2πr.
学生们还经常混淆半径和直径。如果圆的直径为 d，那么半径是 d ÷ 2。在代入 A = πr² 或 C = 2πr 之前，先检查题目给出的是半径还是直径。
Finally, always include units for every answer. Area is measured in square units such as cm² or m², while perimeter is measured in length units such as cm or m.
最后，每个答案都要写上单位。面积用平方单位，如 cm² 或 m²；周长用长度单位，如 cm 或 m。
10. Exam-Style Practice Tips | 考试型练习技巧
In exam questions, start by sketching the shape and adding any missing labels. Show all steps clearly, because method marks are often awarded even if the final answer is slightly wrong.
在考试题中，先画出图形并补上缺失的标注。清晰写出所有步骤，因为即使最终答案有轻微错误，步骤分通常仍然可以获得。
When a question asks for an answer in terms of π, do not replace π with 3.14. Leave the exact form such as 18π. If a decimal answer is required, state the rounding rule, for example to one decimal place.
当题目要求答案用 π 表示时，不要把 π 替换为 3.14。保留精确形式，如 18π。如果要求给出小数答案，要说明舍入规则，例如保留一位小数。
For perimeter problems, trace your finger around the outer edge and list every side once. This prevents you from missing a short side or counting an internal line twice. For area problems, draw split lines and label each part before calculating.
对于周长问题，用手指沿着外边缘描一遍，并列出每一条边一次。这样可以避免漏掉短边或把内部线计算两次。对于面积问题，先画分割线并在计算前标出每一部分。
Practise with past questions that combine rectangles, triangles and semicircles. Being confident with exact values of π and place value will help you move quickly and accurately through multi-step problems.
多练习包含矩形、三角形和半圆的历年真题。熟练掌握 π 的精确值和数位，可以帮助你快速准确地完成多步问题。
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