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 length of a rectangle is 45 inches and the width is x inches. The perimeter is at most 150 inches. Which inequality represents this situation?
原库参考解析
周长・面积与平面图形基础
A rectangle has a length of 100 centimeters and a width of 400 centimeters. What is the area, in square centimeters, of the rectangle?
原库参考解析
周长・面积与平面图形进阶
A polygon with 15 sides has a perimeter of 107 inches. The length of one side is 9 inches. The other 14 sides have equal lengths. What is the length, in inches, of one of the 14 sides with equal lengths?
原库参考解析
周长・面积与平面图形进阶
A rectangle has a perimeter of 48 inches. If the sum of the lengths of three sides of the rectangle is 31 inches, what is the length, in inches, of the fourth side?
原库参考解析
周长・面积与平面图形进阶
In triangle \( ABC \), side\( AC=10 \) cm. The area of triangle \( ABC \) is 250 square centimeters. What is the height h, in centimeters, of this triangle?
参考答案
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 45 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
The equation \( 66=2x+2y \) gives the perimeter of a rectangular garden that has length x, in feet, and width y, in feet. The width of the garden is 8 feet. What is the length, in feet, of the garden?
参考答案
周长・面积与平面图形进阶
Square A has side lengths that are 72 times the side lengths of square B. The area of square A is k times the area of square B. What is the value of k?
参考答案
周长・面积与平面图形进阶
The length of each side of a square is 68 inches. Which expression gives the area of the square?
参考答案
周长・面积与平面图形进阶
The area of a rectangle is 70 square inches. The length of the longest side of the rectangle is 14 inches. What is the length, in inches, of the shortest side of this rectangle?
参考答案
周长・面积与平面图形挑战
The figure shows a rectangular pool surrounded by a concrete path that is x feet (ft) wide on all sides. The pool is 21 ft long and 11 ft wide. The area of the concrete path is 144 ft2. What is the value of x?
参考答案
周长・面积与平面图形进阶
The area of a rectangle is 105 square inches. The length of the longest side of the rectangle is 15 inches. What is the length, in inches, of the shortest side of this rectangle?
参考答案
