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
A total of 2 squares each have side length r. A total of 6 equilateral triangles each have side length t. None of these squares and triangles shares a side. The sum of the perimeters of all these squares and triangles is 210. Which equation represents this situation?
原库参考解析
周长・面积与平面图形进阶
A rectangle has an area of 63 square meters and a length of 9 meters. What is the width, in meters, of the rectangle?
原库参考解析
周长・面积与平面图形挑战
f(x) =2x +244
The given function f represents the perimeter, in centimeters (cm), of a rectangle with a length of x cm and a fixed width. What is the width, in cm, of the rectangle?
原库参考解析
周长・面积与平面图形基础
The area of a rectangle is 57 square inches. The length of the longest side of the rectangle is 19 inches. What is the length, in inches, of the shortest side of this rectangle?
原库参考解析
周长・面积与平面图形挑战
The perimeter of triangle \( ABC \) is 17 inches, the length of side \( AB \) is 4 inches, and the length of side \( AC \) is 7 inches. What is the length, in inches, of side \( BC \)?
原库参考解析
周长・面积与平面图形挑战
A graphic designer is creating a logo for a company. The logo is shown in the figure above. The logo is in the shape of a trapezoid and consists of three congruent equilateral triangles. If the perimeter of the logo is 20 centimeters, what is the combined area of the shaded regions, in square centimeters, of the logo?
原库参考解析
周长・面积与平面图形挑战
The perimeter of an equilateral triangle is 624 centimeters. The height of this triangle is k\( \sqrt{3} \) centimeters, where k is a constant. What is the value of k ?
原库参考解析
