Area and Perimeter of 2D Shapes | 二维图形的面积与周长

📚 Area and Perimeter of 2D Shapes | 二维图形的面积与周长

Area and perimeter are two fundamental measurements used to describe flat, two-dimensional figures. The perimeter tells us the total distance around the outside of a shape, while the area gives us the amount of space enclosed within it. Mastering these concepts in Key Stage 3 is essential, not only for geometry but also for solving real-world problems ranging from fencing a garden to tiling a floor.

面积和周长是描述平面二维图形的两个基本度量。周长告诉我们围绕图形外部的总距离，而面积则给出了其内部所包围的空间大小。在 Key Stage 3 掌握这些概念至关重要，这不仅对几何学本身，对解决从围花园到铺地砖等实际问题也同样重要。

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1. Understanding Perimeter | 理解周长

The perimeter of any closed 2D shape is the sum of all its side lengths. Think of it as the length of string you would need to go completely around the shape. It is measured in linear units such as centimetres (cm), metres (m) or kilometres (km).

任何闭合二维图形的周长是其所有边长的总和。可以把它想象成你需要用来完全围绕图形一圈的绳子长度。它以线性单位来度量，如厘米（cm）、米（m）或千米（km）。

To find a perimeter, we simply add up the given side lengths, taking care to use the same unit throughout. When sides are not directly labelled, we may need to deduce missing lengths from other given dimensions, especially in compound shapes.

要计算周长，我们只需将给出的边长相加，并注意全程使用相同单位。当有些边没有直接标注时，我们可能需要从其他给出的尺寸中推导出缺失的长度，特别是在组合图形中。

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2. Perimeter of Common Shapes | 常见图形的周长

For regular polygons, the perimeter can be found using a simple multiplication. A square of side length s has perimeter P = 4s. A regular pentagon with side length a has perimeter P = 5a. For a rectangle, we use P = 2(l + w), where l is the length and w is the width.

对于正多边形，周长可以用简单的乘法得出。边长为 s 的正方形，周长为 P = 4s。边长为 a 的正五边形，周长为 P = 5a。对于矩形，我们使用 P = 2(l + w)，其中 l 是长度，w 是宽度。

For circles, the perimeter is called the circumference. It is calculated using the formula C = 2πr or C = πd, where r is the radius and d is the diameter. The symbol π (pi) is approximately 3.14, and it represents the ratio of the circumference to the diameter of any circle.

对于圆，周长称为圆周。它用公式 C = 2πr 或 C = πd 来计算，其中 r 是半径，d 是直径。符号 π（圆周率）约等于 3.14，表示任何圆的周长与直径的比值。

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3. Understanding Area | 理解面积

Area measures the surface covered by a 2D shape. It is expressed in square units, such as cm², m² or km². One square centimetre (1 cm²) is the area of a square with sides of 1 cm. Visualising how many unit squares fit inside a shape is the foundation of area measurement.

面积度量的是二维图形所覆盖的表面。它以平方单位表示，如 cm²、m² 或 km²。1 平方厘米（1 cm²）就是边长为 1 厘米的正方形的面积。想象一个图形内部能容纳多少个单位正方形，这是面积度量的基础。

Unlike perimeter, which is a length, area is a two-dimensional measure. This means that converting between area units requires us to square the conversion factor. For example, 1 m = 100 cm, but 1 m² = 100 × 100 = 10 000 cm².

与作为长度的周长不同，面积是一种二维度量。这意味着在面积单位之间转换时，我们需要将换算因子平方。例如，1 m = 100 cm，但 1 m² = 100 × 100 = 10 000 cm²。

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4. Area of Rectangles and Squares | 矩形和正方形的面积

The area of a rectangle is found by multiplying its length by its width. The formula is A = l × w. For a square, since all sides are equal, we use A = s × s = s². These are the most basic area formulas and the building blocks for more complex shapes.

矩形的面积通过将其长度乘以宽度得出。公式为 A = l × w。对于正方形，因为所有边相等，我们使用 A = s × s = s²。这些是最基本的面积公式，也是更复杂图形的基础。

It is crucial that the length and width are in the same units before multiplying. If they are not, we must convert one so that both are, for example, in centimetres. The resulting area will be in square centimetres.

在相乘之前，长度和宽度必须使用相同单位，这一点至关重要。如果单位不同，我们必须转换其中一个，使两者都变为，例如，厘米。得出的面积单位将是平方厘米。

• Example: A rectangle has length 5 cm and width 3 cm. Area = 5 × 3 = 15 cm².
• 示例： 一个矩形长 5 厘米，宽 3 厘米。面积 = 5 × 3 = 15 平方厘米。

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5. Area of Triangles | 三角形的面积

A triangle can be thought of as half of a rectangle or parallelogram. Hence, the area of a triangle is given by A = ½ × base × height, often written as A = ½bh. The base can be any side, but the height must be the perpendicular distance from the opposite vertex to that base.

三角形可以被看成是矩形或平行四边形的一半。因此，三角形的面积公式为 A = ½ × 底 × 高，常写作 A = ½bh。底可以是任意一边，但高必须是从对角顶点到该底的垂直距离。

It is important to note that the height is not necessarily a side of the triangle unless the triangle is right-angled. In non-right triangles, the height is shown by a dashed perpendicular line from the vertex to the base or its extension.

需要注意的是，高并不一定是三角形的一边，除非该三角形是直角三角形。在非直角三角形中，高用从顶点到底边或其延长线的虚线垂直线段表示。

• Example: Triangle with base 8 m and perpendicular height 5 m. Area = ½ × 8 × 5 = 20 m².
• 示例： 底为 8 米，垂直高为 5 米的三角形。面积 = ½ × 8 × 5 = 20 平方米。

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6. Area of Parallelograms | 平行四边形的面积

A parallelogram is a quadrilateral with two pairs of parallel sides. Its area is calculated using the formula A = b × h, where b is the base length and h is the perpendicular height between the base and the opposite side. This is very similar to the rectangle formula, but the height is not the slanted side length.

平行四边形是有两组平行边的四边形。它的面积使用公式 A = b × h 来计算，其中 b 是底边长，h 是底边与对边之间的垂直高度。这与矩形公式非常相似，但高不是斜边的长度。

To avoid confusion, remember that the slanted side length is not used in the area calculation. If you are given the length of the slanted side, first identify the perpendicular height, often drawn inside the parallelogram.

为了避免混淆，请记住斜边长度不用于面积计算。如果给出了斜边的长度，首先要确定垂直高度，它通常画在平行四边形内部。

A(parallelogram) = base × perpendicular height

平行四边形面积 = 底 × 垂直高

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7. Area of Trapeziums | 梯形的面积

A trapezium (or trapezoid) has exactly one pair of parallel sides. Its area formula involves the average of the two parallel sides multiplied by the perpendicular distance between them: A = ½(a + b)h, where a and b are the lengths of the parallel sides and h is the height.

梯形有且仅有一组平行边。它的面积公式涉及两条平行边的平均值乘以它们之间的垂直距离：A = ½(a + b)h，其中 a 和 b 是平行边的长度，h 是高。

This formula works because a trapezium can be split into two triangles or rearranged into a parallelogram of base (a+b)/2. Always check that the height is perpendicular to the parallel sides, not the non-parallel ones.

这个公式之所以有效，是因为梯形可以分割成两个三角形，或重新排列成一个底为 (a+b)/2 的平行四边形。务必检查高是否垂直于平行边，而不是垂直于不平行的边。

• Example: Parallel sides 6 cm and 10 cm, height 4 cm. Area = ½ × (6+10) × 4 = ½ × 16 × 4 = 32 cm².
• 示例： 平行边长分别为 6 厘米和 10 厘米，高为 4 厘米。面积 = ½ × (6+10) × 4 = ½ × 16 × 4 = 32 平方厘米。

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8. Area of Composite Shapes | 组合图形的面积

Composite shapes are made up of two or more simple shapes combined together. To find the total area, we split the shape into rectangles, triangles, parallelograms, or trapeziums, calculate the area of each part separately, and then add them up. Alternatively, we can calculate the area of a larger enclosing rectangle and subtract the areas of cut-out sections.

组合图形由两个或更多简单图形组合而成。要计算总面积，我们将图形分割成矩形、三角形、平行四边形或梯形，分别计算每部分的面积，然后将它们相加。或者，我们可以计算一个较大的包围矩形的面积，再减去被切掉部分的面积。

A systematic approach is essential. First, sketch the shape and label all known lengths. Then, draw dashed lines to show how you will split the figure. Write down the area of each sub-shape before finding the total. Finally, check that no sub-shape has been missed or counted twice.

系统的方法是必不可少的。首先，画出图形草图并标注所有已知长度。然后，画虚线表示你将如何分割图形。在求总面积之前，写出每个子图形的面积。最后，检查是否有遗漏或重复计算的子图形。

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9. Problem Solving with Area and Perimeter | 面积与周长的解题应用

Many KS3 problems test understanding by mixing area and perimeter, or by giving one quantity and asking for another. For example, you might be told the perimeter of a rectangle and its length, and asked to find the width and then the area. This requires rearranging formulas and careful algebraic reasoning.

许多 KS3 题目通过混合面积和周长，或给出一个量而要求另一个量来检验理解。例如，你可能被告知一个矩形的周长和长度，并被要求求出宽度，然后求面积。这需要重新排列公式并仔细进行代数推理。

Another common type of problem involves a border of a fixed width around a shape. To find the area of the border alone, you can calculate the area of the outer shape and subtract the area of the inner shape. This “difference of areas” method is very powerful.

另一类常见问题涉及图形周围固定宽度的边框。要单独求出边框的面积，你可以计算外部图形的面积并减去内部图形的面积。这种“面积差”方法非常有效。

• Worked example: A rectangular lawn is 12 m by 8 m. It has a concrete path 1 m wide all around it. Area of path = area of outer rectangle (14 m × 10 m) – area of lawn (12 m × 8 m) = 140 – 96 = 44 m².
• 解题示例： 一个矩形草坪长 12 米、宽 8 米。它周围有一条 1 米宽的混凝土小路。小路的面积 = 外部矩形面积 (14 m × 10 m) – 草坪面积 (12 m × 8 m) = 140 – 96 = 44 平方米。

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10. Real-life Applications | 实际应用

Area and perimeter calculations are used in many everyday situations. When painting a wall, you need the wall area to buy the right amount of paint. When installing skirting boards, you need the perimeter of the room. Gardeners use perimeter for fencing and area for seeding lawns.

面积和周长的计算用于许多日常情境中。粉刷墙壁时，你需要墙壁面积来购买适量的油漆。安装踢脚板时，你需要房间的周长。园丁用周长来设置围栏，用面积来播撒草坪种子。

In construction and design, cost estimation frequently depends on area. For instance, tiling a floor requires knowing how many tiles are needed, which is found by dividing the floor area by the area of one tile. Always allow a small extra percentage for wastage.

在建筑和设计中，成本估算往往取决于面积。例如，铺地砖需要知道需要多少块砖，这可以通过将地板面积除以一块砖的面积得出。通常要额外增加一小比例以考虑损耗。

Scenario 场景	Measurement 所需度量	Formula Used 所用公式
Fencing a field 围田地	Perimeter 周长	Add all side lengths 所有边长相加
Carpeting a room 铺地毯	Area 面积	A = l × w
Framing a picture 装裱画框	Perimeter 周长	Sum of frame edges 边框边长之和
Painting a triangular gable 油漆三角形山墙	Area 面积	A = ½bh

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11. Key Formulas Summary | 关键公式总结

Bringing all the formulas together in one place helps in revision and problem-solving. The most common ones for KS3 are listed below. Remember that height always means perpendicular height, and units must be consistent.

将所有公式集中在一起有助于复习和解题。以下是 KS3 最常用的公式。请记住，高始终指垂直高度，且单位必须保持一致。

• Square: P = 4s, A = s²
• 正方形： P = 4s, A = s²
• Rectangle: P = 2(l + w), A = lw
• 矩形： P = 2(l + w), A = lw
• Triangle: A = ½bh
• 三角形： A = ½bh
• Parallelogram: A = bh
• 平行四边形： A = bh
• Trapezium: A = ½(a + b)h
• 梯形： A = ½(a + b)h
• Circle: C = 2πr = πd, A = πr²
• 圆： C = 2πr = πd, A = πr²

Applying these formulas confidently requires extensive practice. Start with straightforward substitution before moving to multi-step problems and real-life contexts. Draw clear diagrams and write down your steps to avoid mistakes.

自信地应用这些公式需要大量练习。从简单的代入开始，然后再处理多步骤问题和实际情境。绘制清晰的图表并写下解题步骤，以避免错误。

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Published by TutorHao | Mathematics Revision Series | aleveler.com

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