这部分需要会什么？
- 正确使用常见平面图形的周长与面积公式
- 将复合图形拆分或补全
- 区分长度倍数和面积倍数
核心概念与公式
长方形面积 \(lw\)，三角形面积 \(bh/2\)，平行四边形面积 \(bh\)，梯形面积 \((a+b)h/2\)。这里高度都必须垂直于对应的底。圆面积 \(\pi r^2\)，周长 \(2\pi r\)。
圆心角为 \(\theta\) 度的扇形面积为 \((\theta/360)\pi r^2\)；若 \(\theta\) 用弧度表示，则为 \(r^2\theta/2\)。复合图形可用大图形减去空缺，或把互不重叠的部分相加。相似图形长度倍率为 \(k\)，面积倍率为 \(k^2\)。
做题时怎样使用
- 判断所问是边界长度还是覆盖面积，统一单位
- 画出必要的垂线，把复合图形分成可计算的部分
- 列式相加或相减，检查是否重复计算区域或漏掉边界
容易混淆的地方
斜边长度通常不是高。求带缺口图形的周长时，新增的内凹边也属于边界；求面积时单位必须平方。
用一个例题串起方法
知识点教学例题一个 \(10\times8\) 的长方形内部挖去半径为 2 的圆，剩余面积是多少？
展开解题思路 ＋
长方形面积为 80，圆面积为 \(4\pi\)。剩余面积为 \(80-4\pi\) 平方单位。
本知识点题库
先独立作答，再查看答案与解析。
周长・面积与平面图形挑战
The expression \( k·cosx^{∘} \) represents the area of triangle ABC shown. What is the value of k?
参考答案
周长・面积与平面图形挑战
The expression \( 26+k·cosx^{∘} \) represents the perimeter of triangle ABC shown. What is the value of k?
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周长・面积与平面图形进阶
The length of a side of square X is 8 centimeters. The area of rectangle Y is 14 square centimeters. What is the total area, in square centimeters, of square X and rectangle Y?
参考答案
周长・面积与平面图形进阶
A space probe uses a square-shaped solar sail to move through space. The side length of the solar sail is 8.61w meters, where w is the width, in meters, of the space probe. Which equation gives the area A, in square meters, of the solar sail?
参考答案
周长・面积与平面图形进阶
A rectangle has a length of 7 centimeters and a width of 5 centimeters. What is the perimeter, in centimeters, of this rectangle?
参考答案
周长・面积与平面图形基础
What is the area, in square centimeters, of a rectangle with a length of 41 centimeters and a width of 39 centimeters?
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周长・面积与平面图形进阶
A space probe uses a square-shaped solar sail to move through space. The side length of the solar sail is 8.76w meters, where w is the width, in meters, of the space probe. Which equation gives the area A, in square meters, of the solar sail?
参考答案
周长・面积与平面图形进阶
A rectangle has a length of 8 centimeters and a width of 5 centimeters. What is the perimeter, in centimeters, of this rectangle?
参考答案
周长・面积与平面图形基础
What is the area, in square centimeters, of a rectangle with a length of 49 centimeters and a width of 46 centimeters?
参考答案
周长・面积与平面图形进阶
The area A, in square centimeters, of a rectangular painting can be represented by the expression w(w + 78), where w is the width, in centimeters, of the painting. Which expression represents the length, in centimeters, of the painting?
参考答案
周长・面积与平面图形进阶
A space probe uses a square-shaped solar sail to move through space. The side length of the solar sail is 8.63w meters, where w is the width, in meters, of the space probe. Which equation gives the area A, in square meters, of the solar sail?
参考答案
周长・面积与平面图形进阶
The length of the base of a triangle is 4 inches, and the height of the triangle is 14 inches. What is the area, in square inches, of the triangle?
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