Volume and Surface Area of Prisms | 棱柱的体积与表面积

📚 Volume and Surface Area of Prisms | 棱柱的体积与表面积

In KS3 Cambridge Mathematics, page 246 introduces students to the important concepts of calculating the volume and surface area of prisms. Understanding these 3D shapes is essential for solving real-world problems and builds a strong foundation for geometry. Let’s explore the key formulas and techniques step by step.

在剑桥KS3数学课程中，第246页向学生介绍了计算棱柱体积和表面积的重要概念。理解这些立体图形对于解决实际问题至关重要，并为几何学习打下坚实基础。让我们逐步探索关键公式与技巧。

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1. What is a Prism? | 什么是棱柱？

A prism is a 3D shape with a constant cross-section along its length. The two ends are identical polygons, and the side faces are parallelograms (often rectangles). Common examples include triangular prisms, rectangular prisms (cuboids), and cylinders. Although a cylinder has a circular cross-section, it is treated as a special type of prism.

棱柱是一种沿长度方向横截面保持不变的立体图形。两端为全等的多边形，侧面为平行四边形（通常为矩形）。常见的例子包括三角柱、长方体（矩形棱柱）和圆柱体。虽然圆柱体的横截面是圆形，但它也被视为一种特殊的棱柱。

The cross-section is the shape you see when you cut straight through the prism, perpendicular to its length. Identifying the cross-section correctly is the first and most important step in both volume and surface area calculations.

横截面是指沿着垂直于棱柱长度的方向切割时所看到的形状。正确识别横截面是计算体积和表面积时最重要的一步。

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2. Understanding Volume of a Prism | 理解棱柱的体积

Volume measures the amount of space inside a 3D object. For prisms, we imagine filling the shape with unit cubes. The volume is found by multiplying the area of the cross-section by the length (or height) of the prism. This rule works for all prisms, regardless of the shape of the base.

体积用于衡量三维物体内部的空间大小。对于棱柱，我们可以想象用单位立方体将图形填满。体积等于横截面面积乘以棱柱的长度（或高度）。无论底面形状如何，这一规则适用于所有棱柱。

Volume = Area of cross-section × length

In symbols: V = A × l, where A is the cross-sectional area and l is the length (sometimes height h is used instead). Always remember that A is an area, so it has square units, and l is a length.

用符号表示为：V = A × l，其中 A 为横截面面积，l 为长度（有时也用高 h 表示）。请记住 A 是一个面积，所以带平方单位，而 l 是长度。

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3. Volume of a Cuboid | 长方体的体积

A cuboid (or rectangular prism) has a rectangular cross-section. Its volume can be found quickly using length × width × height. This is actually a special case of the prism formula: the area of the rectangle (length × width) multiplied by the third dimension (height).

长方体（即矩形棱柱）的横截面是矩形。其体积可以直接用长 × 宽 × 高来计算。这其实是棱柱公式的一个特例：矩形面积（长 × 宽）乘以第三维度（高）。

Volume of cuboid = l × w × h

For example, a box with length 5 cm, width 3 cm, and height 2 cm has a volume of 5 × 3 × 2 = 30 cm³.

例如，一个长5 cm、宽3 cm、高2 cm的盒子，其体积为5 × 3 × 2 = 30 cm³。

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4. Volume of a Triangular Prism | 三角柱的体积

For a triangular prism, the cross-section is a triangle. The first step is to calculate the area of that triangle using the formula: ½ × base × perpendicular height of the triangle. Then multiply this area by the length of the prism.

对于三角柱，横截面是一个三角形。第一步是用公式 ½ × 底 × 三角形的高 来计算三角形的面积。然后将此面积乘以棱柱的长度。

Volume = (½ × b × h_triangle) × L

It is very important to use the perpendicular height of the triangle, not the sloping side edge. Mixing these up is a common mistake.

使用三角形的垂直高而非斜边长度至关重要，将两者混淆是常见错误。

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5. Volume of a Cylinder | 圆柱体的体积

A cylinder has a circular cross-section. The area of a circle is given by π × r², where r is the radius. The volume of the cylinder is then that area multiplied by the height h (which is the length of the prism).

圆柱体具有圆形横截面。圆的面积为 π × r²，其中 r 是半径。圆柱体的体积等于该面积乘以高 h（即棱柱的长度）。

Volume = π × r² × h

If the radius is 3 cm and the height is 10 cm, the volume is π × 3² × 10 ≈ 3.14 × 9 × 10 = 282.6 cm³. Answers may be left in terms of π or given as a decimal.

若半径为3 cm，高为10 cm，则体积为 π × 3² × 10 ≈ 3.14 × 9 × 10 = 282.6 cm³。答案可以用含π的式子表示，也可写成小数。

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6. Units of Volume | 体积单位

When lengths are given in centimetres, the volume is in cubic centimetres (cm³). If lengths are in metres, volume is in cubic metres (m³). The unit always carries a power of 3 because volume is three-dimensional. Sometimes you need to convert between units, e.g. 1 m³ = 1,000,000 cm³.

当长度单位为厘米时，体积单位为立方厘米（cm³）。若长度单位为米，体积则为立方米（m³）。因为体积是三维量，所以单位总是带有三次方。有时需要在单位之间转换，例如 1 m³ = 1,000,000 cm³。

Always write the unit symbol with a superscript 3, for example cm³. Avoid writing ‘cubic cm’ in formal work; use the symbol.

书写时务必用上标 3 表示单位，如 cm³。正式作答时尽量避免写“cubic cm”，而应使用符号。

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7. Surface Area of Prisms | 棱柱的表面积

Surface area is the total area of all the faces of a prism. To find it efficiently, we often draw a net – a 2D layout that shows all faces connected by their edges. The surface area is the sum of the areas of every individual face.

表面积是指棱柱所有面的总面积。为了高效计算，我们通常会画出展开图（net）——一种显示所有面及其连接边的二维排列方式。表面积即为各个面面积之和。

A key point: every face must be counted exactly once. Opposite faces in a prism are congruent, so you can use symmetry to save time.

关键点：每个面恰好计算一次。棱柱中相对的面全等，因此可以利用对称性来节省时间。

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8. Surface Area of a Cuboid | 长方体的表面积

A cuboid has 6 rectangular faces, arranged in 3 pairs of opposite faces. If the dimensions are length l, width w, and height h, then the three pairs have areas lw, lh, and wh. The total surface area formula is:

长方体有6个矩形面，组成3对相对的面。若尺寸为长 l、宽 w、高 h，则三对面的面积分别为 lw、lh 和 wh。总表面积公式为：

Surface area = 2(lw + lh + wh)

For a cuboid measuring 4 cm by 3 cm by 5 cm, the surface area is 2(4×3 + 4×5 + 3×5) = 2(12 + 20 + 15) = 94 cm².

对于一个尺寸为 4 cm × 3 cm × 5 cm 的长方体，表面积为 2(4×3 + 4×5 + 3×5) = 2(12 + 20 + 15) = 94 cm²。

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9. Surface Area of a Triangular Prism | 三角柱的表面积

A triangular prism has 5 faces: 2 triangular ends and 3 rectangular sides. To calculate the surface area, find the area of each triangle (½ × base × perpendicular height) and the area of each rectangle (length of prism × side length of the triangle). Then add all five areas together.

三角柱有5个面：2个三角形端面和3个矩形侧面。计算表面积时，分别求出每个三角形的面积（½ × 底 × 垂直高）和每个矩形的面积（棱柱长度 × 三角形的对应边长），然后将这五块面积相加。

A quicker method: the lateral area (the three rectangles) can be found by multiplying the prism length by the perimeter of the triangular cross-section. Then add twice the area of the triangle.

一种更快捷的方法：侧面积（即三个矩形）可以通过棱柱长度乘以三角形横截面的周长来求得，然后加上两倍的三角形面积。

Remember: if the triangle is not a right-angled triangle, you still need the perpendicular height to find its area. Do not use the slant edges in the area formula for the triangle itself.

切记：若三角形不是直角三角形，仍需用其垂直高来求面积。切勿在三角形面积公式中使用斜边长度。

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10. Step-by-Step Strategy and Common Mistakes | 解题步骤与常见错误

To solve problems with confidence, follow these steps:

• Identify the type of prism and its cross-section.
• For volume: calculate the area of the cross-section, then multiply by the length.
• For surface area: list all faces, mark dimensions, find each area, and sum them up.
• Check units: area in square units, volume in cubic units.

解题时可按以下步骤进行：

• 辨识棱柱类型及其横截面。
• 体积：计算横截面面积，然后乘以长度。
• 表面积：列出所有面，标注尺寸，分别计算面积后求和。
• 检查单位：面积为平方单位，体积为立方单位。

Common mistakes include forgetting to use the perpendicular height of a triangle, confusing the slant edge with the height, mixing up area and volume units, and omitting a face when calculating surface area. Always double-check your net or sketch, and verify that your final answer is reasonable for the shape’s size.

常见错误包括：忘记使用三角形的垂直高、将斜边与高混淆、混淆面积与体积单位、以及在计算表面积时遗漏某个面。请务必检查展开图或草图，并结合图形大小来判断最终答案是否合理。

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Published by TutorHao | Mathematics Revision Series | aleveler.com

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