📚 Area of Rectangles and Triangles | 矩形与三角形的面积
Area is one of the most important concepts in geometry. Whether you are planning to tile a floor, paint a wall or design a garden, you need to know how much space a flat shape covers. In this article, we focus on the area of rectangles and triangles, which form the foundation for understanding all other 2D shapes. You will learn the formulas, see worked examples and discover how to apply these ideas to compound shapes and real-world problems.
面积是几何学中最重要的概念之一。无论是铺地砖、刷墙还是设计花园，你都需要知道平面图形覆盖了多少空间。本文重点介绍矩形和三角形的面积，这是理解所有其他二维图形的基础。你将学习公式，观摩例题，并了解如何将这些思想应用于复合图形和实际问题。
1. Understanding Area | 理解面积
Area measures the size of a surface. Imagine covering a shape with identical small squares of side 1 cm. The number of such squares that fit inside the shape, without gaps or overlaps, tells you its area in square centimetres (cm²).
面积衡量一个表面的大小。设想用边长为1厘米的相同小正方形覆盖一个图形，图形内部能放下的这些小正方形的数量（无空隙、无重叠）就是它的面积，以平方厘米（cm²）为单位。
One square centimetre (1 cm²) is the area of a square with sides of 1 cm. Larger areas are measured in square metres (m²) or square kilometres (km²). Understanding units is essential because mixing units, such as using cm for length but expecting an area in m², leads to serious mistakes.
1平方厘米（1 cm²）是边长为1厘米的正方形的面积。更大的面积用平方米（m²）或平方公里（km²）度量。理解单位至关重要，因为混淆单位（例如长度用厘米却期望得到平方米的面积）会导致严重错误。
The concept of area applies not only to rectangles and triangles but to all flat shapes. Once you master the area of these two, you can break down more complex figures into familiar pieces.
面积的概念不仅适用于矩形和三角形，也适用于所有平面图形。一旦掌握了这两种图形的面积，就能将更复杂的图形分解成熟悉的部分。
2. Area of a Rectangle | 矩形面积
A rectangle is a four‑sided shape with opposite sides equal and four right angles. To find its area, multiply the length by the width.
矩形是有四条边、对边相等且四个角均为直角的图形。要求矩形的面积，将长乘以宽即可。
Area of a rectangle = length × width
矩形面积 = 长 × 宽
width is 5 cm, then Area = 8 × 5 = 40 cm². Always include the square unit in your answer.
如果长为8厘米，宽为5厘米，那么面积 = 8 × 5 = 40 cm²。答案中务必带上平方单位。
The terms 'length' and 'width' can be interchanged – special rectangle with all sides equal, the formula becomes Area = side × side, or side².
“长”和“宽”可以互换——关键是要将两条互相垂直的边相乘。对于正方形这种四边相等的特殊矩形，公式变为 面积 = 边长 × 边长，或边长²。
| Shape (形状) | Dimensions (尺寸) | Area Calculation (面积计算) |
|---|---|---|
| Rectangle | length = 12 m, width = 3 m | 12 × 3 = 36 m² |
| Square | side = 9 cm | 9 × 9 = 81 cm² |
| Rectangle | length = 4.5 m, width = 2 m | 4.5 × 2 = 9 m² |
3. Worked Examples – Rectangles | 矩形例题
Example 1: A rectangle has length 15 mm and width 7 mm. Find its area.
例题1：一个矩形长15毫米，宽7毫米。求其面积。
Solution: Area = 15 × 7 = 105 mm². Always write the unit as mm², not just mm.
解：面积 = 15 × 7 = 105 mm²。单位务必写成mm²，不能只写mm。
Example 2: A square garden has a side of 20 m. What area of grass can be planted?
例题2：一个正方形花园边长为20米。可以种植草皮的面积是多少？
Solution: Area = 20 × 20 = 400 m².
解：面积 = 20 × 20 = 400 m²。
lated as Area = 2.5 × 4 = 10 m². If one tin of paint covers 2 m², how many tins are needed?
例题3：一堵墙高2.5米，宽4米。需要粉刷的面积为 2.5 × 4 = 10 m²。如果一罐油漆可刷2 m²，需要多少罐？
g area, remember to round up if you get a decimal, because you cannot buy a fraction of a tin in reality.
解：10 ÷ 2 = 5罐。在应用面积时，若得到小数要向上取整，因为实际生活中不能购买零头罐。
4. Area of a Triangle | 三角形面积
A triangle can be thought of as half of a rectangle. If you draw a rectangle and cut it along one diagonal, you get two identical triangles. This observation gives the triangle area formula.
三角形可以看作是矩形的一半。画出矩形并沿一条对角线剪开，就会得到两个完全相同的三角形。这一观察得出了三角形面积公式。
Area of a triangle = (1/2) × base × height
三角形面积 = (1/2) × 底 × 高
The base can be any side, but the height must be the perpendicular distance from the opposite vertex to that base. The height is not always a side of the triangle – it is the shortest distance from the base to the top, forming a right angle with the base.
底可以是任意一边，但高必须是从对角顶点向该底边所作的垂直距离。高不一定是三角形的一条边——它是从底边到顶点最短的距离，与底边构成直角。
For a right‑ the base and the other as the height. The formula still gives Area = (1/2) × leg₁ × leg₂.
对于直角三角形，两条直角边互相垂直，因此可以直接将一条直角边作为底，另一条作为高。公式仍给出 面积 = (1/2) × 直角边₁ × 直角边₂。
5. Worked Examples – Triangles | 三角形例题
Example 1: A triangle has base 10 cm and perpendicular height 6 cm. Find its area.
例题1：三角形底边长10厘米，高6厘米。求其面积。
30 cm². Notice the use of ½ instead of dividing by 2 at the end; both methods are correct.
解：面积 = ½ × 10 × 6 = 30 cm²。注意使用½而不是最后除以2；两种方法均正确。
Example 2: A right‑angled triangle has legs of length 9 m and 12 m. What is its area?
例题2：一个直角三角形的两条直角边长分别为9米和12米。它的面积是多少？
Solution: Area = ½ × 9 × 12 = 54 m².
解：面积 = ½ × 9 × 12 = 54 m²。
ht 5 cm. Area = ½ × 8.4 × 5 = 21 cm². Even with decimal lengths, the process is identical.
例题3：一个三角形底长8.4厘米，高5厘米。面积 = ½ × 8.4 × 5 = 21 cm²。即使长度有小数，计算过程完全相同。
Watch out for the common error of using a slant side instead of the perpendicular height. Always draw or imagine a dashed line to show the true height.
注意一个常见错误：用斜边代替垂直高。务必画出或想象一条虚线来表示真实的高。
6. Finding Missing Lengths | 求缺失长度
Area formulas can be rearranged to find a missing dimension. If you know the area and one side of a rectangle, you divide the area by the known side.
面积公式可以变形来求缺失的尺寸。如果已知矩形面积和一条边，用面积除以已知边长。
Area ÷ length. For a triangle: if the area and base are known, height = (2 × Area) ÷ base.
对于矩形：宽 = 面积 ÷ 长。对于三角形：若已知面积和底，高 = (2 × 面积) ÷ 底。
Example: The area of a rectangle is 72 cm² and its length is 9 cm. Find the width.
例题：矩形面积为72 cm²，长9厘米。求宽。
Width = 72 ÷ 9 = 8 cm. Check: 9 × 8 = 72, correct.
宽 = 72 ÷ 9 = 8 cm。验证：9 × 8 = 72，正确。
Example: A triangle has area 35 m² and base 10 m. Find its height.
例题：三角形面积为35 m²，底为10米。求高。
Height = (2 × 35) ÷ 10 = 70 ÷ 10 = 7 m. Check: ½ × 10 × 7 = 35, correct.
高 = (2 × 35) ÷ 10 = 70 ÷ 10 = 7米。验证：½ × 10 × 7 = 35，正确。
7. Area of Compound Shapes | 复合形状面积
Compound shapes (or composite figures) are made by joining two or more basic shapes. To find the total area, split the shape into rectangles and triangles, calculate each area separately, and then add or subtract as needed.
复合形状（复合图形）由两个或更多基本图形拼接而成。要求总面积，可以将图形拆分成矩形和三角形，分别计算面积，再按需相加或相减。
Strategy: look for horizontal or vertical lines to divide the figure into parts with right angles. Label the dimensions you know and deduce any missing lengths using addition or subtraction.
策略：寻找水平或竖直线条将图形分割成含有直角的若干部分。标注已知尺寸，并通过加减法推算出缺失的长度。
For example, an L‑shaped figure can be split into two rectangles. A parallelogram can be cut into a rectangle and two right triangles. Always double‑check that the sum of the parts gives the original shape.
例如，一个L形图形可以分割成两个矩形。平行四边形可以切割成一个矩形和两个直角三角形。务必复核各部分之和能否还原为原图形。
Sometimes it is easier to calculate the area of a larger rectangle and then subtract the area of a cut‑out piece. Choose the method that makes the numbers simplest.
有时，先计算一个大矩形的面积再减去挖去部分的面积会更为简便。选择让计算数字最简单的方法。
8. Real-Life Applications | 实际应用
Area calculations appear in countless everyday situations. When buying carpet, you measure the floor area in m². When sizing a screen or a photograph, you might talk about the area in cm². Architects, engineers, and designers use area to plan spaces and materials.
面积计算出现在无数日常情境中。购买地毯时，要测量地板面积（m²）；谈及屏幕尺寸或照片大小时，可能会用到cm²。建筑师、工程师和设计师用面积来规划空间和材料。
Farmers estimate the area of fields to calculate the amount of seed or fertiliser needed. Painters use wall area to decide how many litres of paint to buy. These applications rely on accurate area formulas and careful unit conversion.
农民估算田地面积以计算所需种子或肥料量；油漆工利用墙壁面积决定购买多少升涂料。这些应用都依赖于准确的面积公式和细致的单位换算。
A triangle‑shaped sail has an area that affects the boat's performance. Understanding area helps you appreciate why shapes are designed in particular ways.
三角形船帆的面积会影响船只的性能。理解面积有助于你明白为什么形状会以特定方式设计。
9. Units of Area | 面积单位
Length units can be converted, but area unit conversion is different: because area involves two multiplied lengths, the conversion factor must be squared.
长度单位可以换算，但面积单位的换算不同：由于面积涉及两个长度相乘，换算因数必须平方。
| Length conversion (长度换算) | Area conversion (面积换算) |
|---|---|
| 1 m = 100 cm | 1 m² = 100 × 100 = 10,000 cm² |
| 1 cm = 10 mm | 1 cm² = 10 × 10 = 100 mm² |
| 1 km = 1000 m | 1 km² = 1000 × 1000 = 1,000,000 m² |
ch is wrong. Remember that a square metre contains 100 × 100 little squares of 1 cm² each.
一个常见错误是说 1 m² = 100 cm²，这是错的。记住：1平方米含有100 × 100个1 cm²的小方格。
Alternatively, change both to centimetres: 250 cm × 80 cm = 20,000 cm², which equals 2 m².
当计算中混用厘米和米时，应先将所有长度转换为相同单位。例如，求 2.5 m × 80 cm 矩形的面积，可把80 cm化成0.8 m，则面积 = 2.5 × 0.8 = 2 m²。也可以都换成厘米：250 cm × 80 cm = 20,000 cm²，等于2 m²。
10. Common Mistakes to Avoid | 常见错误避免
Mistake 1: Forgetting to square the unit when writing the area. Writing 'cm' instead of 'cm²' loses marks in exams. The unit must reflect two‑dimensional measure.
错误一：书写面积时忘记平方单位。把“cm”写成“cm²”在考试中会丢分。单位必须反映二维度量。
Mistake 2: Using the slant side as the height of a triangle. The height must be perpendicular to the base. If no perpendicular height is given, you may need to construct it or use coordinates.
错误二：将斜边当作三角形的高。高必须与底边垂直。如果未给出垂直高度，可能需要作图或利用坐标来求。
Mistake 3: Confusing perimeter and area. Perimeter is the distance around a shape, while area is the surface enclosed. They share no simple relationship, so do not mix the formulas.
错误三：混淆周长和面积。周长是图形一周的长度，面积是围成的面的大小。两者没有简单的关系，不要混用公式。
Mistake 4: Forgetting to divide by 2 for a triangle. Even when you know the base and height, missing the half factor changes the answer completely.
错误四：三角形计算时忘记除以2。即使知道了底和高，漏掉1/2因子会完全改变答案。
Mistake 5: Using inconsistent units without converting first. Always make sure all measurements are in the same unit before multiplying.
错误五：单位不一致而未提前换算。相乘前务必确保所有量度单位相同。
11. Practice Questions | 练习题
Test your understanding with these questions. Answers should include correct units and be checked for reasonableness.
用以下题目测试你的理解。答案应包含正确的单位，并检查其合理性。
A rectangle measures 11 cm by 4 cm. Find its area.
一个矩形尺寸为 11 cm × 4 cm。求其面积。A square has side length 1.2 m. Calculate its area in m².
一个正方形边长为1.2 m。用m²计算其面积。A triangle has base 14 mm and perpendicular height 8 mm. What is its area?
一个三角形底14 mm，垂直高8 mm。它的面积是多少？The area of a right‑angled triangle is 60 cm² and one leg is 15 cm. Find the length of the other leg.
一个直角三角形面积为60 cm²，其中一条直角边长15 cm。求另一条直角边的长度。An L‑shaped floor is made of a 6 m by 4 m rectangle with a 2 m by 2 m square removed from one corner. Find the floor area.
一个L形地板由一个6 m × 4 m的矩形去掉一角的一个2 m × 2 m正方形构成。求地板面积。A rectangular garden 8 m long and 5 m wide has a triangular flower bed of base 3 m and height 2 m inside it. What area of lawn remains?
一个长8 m、宽5 m的矩形花园内有一个底3 m、高2 m的三角形花坛。剩下的草坪面积是多少？
12. Summary and Key Points | 总结与关键点
rea of a triangle is half the product of base and perpendicular height: ½ × base × height.
面积衡量图形内部的空间。矩形面积 = 长 × 宽。正方形是特殊的矩形，面积 = 边长²。三角形面积是底与垂直高乘积的一半：½ × 底 × 高。
Always express area in square units. Convert all lengths to the same unit before multiplying. For compound shapes, split into rectangles and triangles, calculate separately, and then add or subtract. Rearranging the formula lets you find a missing length when the area is known.
始终用平方单位表示面积。相乘前将所有长度化为相同单位。对于复合图形，拆分成矩形和三角形分别计算，再相加或相减。当已知面积时，变形公式可以求出缺失的长度。
Visualising the shapes and drawing the height clearly will help you avoid common pitfalls. Practise regularly, and soon these area calculations will become second nature.
想象图形并清晰地画出高可以帮助你避开常见陷阱。定期练习，这些面积计算很快会变得得心应手。
Published by TutorHao | Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图