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 rectangular region of land is divided into 40 square lots of equal area R, in square units. The length of the region is 1.6 times the width of the region. If the width of the region is \( x\sqrt{R} \) units, what is the value of x?
原库参考解析
周长・面积与平面图形挑战
The triangle inequality theorem states that the sum of any two sides of a triangle must be greater than the length of the third side. If a triangle has side lengths of 7 and 10, which inequality represents the possible lengths, x, of the third side of the triangle?
原库参考解析
周长・面积与平面图形进阶
The width of a rectangle is 5 centimeters. The length of the rectangle is 30 centimeters longer than the width. What is the area, in square centimeters, of this rectangle?
原库参考解析
周长・面积与平面图形进阶
Circle N has a radius of 9 millimeters (mm). Circle M has an area of \( 25π \) mm². What is the total area, in mm², of circles N and M?
原库参考解析
周长・面积与平面图形挑战
The triangle inequality theorem states that the sum of any two sides of a triangle must be greater than the length of the third side. If a triangle has side lengths of 10 and 12. Which inequality represents the possible lengths, x, of the third side of the triangle?
原库参考解析
周长・面积与平面图形进阶
The triangle shown has one side with a length of x inches and two sides each with a length of y inches. The perimeter of the triangle is 47 inches. Which equation represents this situation?
原库参考解析
周长・面积与平面图形进阶
Circle T has a radius of 35 millimeters (mm). What is the area of circle T, in mm²?
原库参考解析
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 13 ft wide. The area of the concrete path is 240 ft2 . What is the value of x?
原库参考解析
周长・面积与平面图形进阶
The area of a rectangle is 108 square inches. The length of the longest side of the rectangle is 18 inches. What is the length, in inches, of the shortest side of this rectangle?
原库参考解析
周长・面积与平面图形进阶
Circle T has a radius of 25 millimeters (mm). What is the area of circle T, in mm²?
原库参考解析
周长・面积与平面图形进阶
A rectangular banner has an area of 2,300 square inches. A copy of the banner is made in which the length and width of the original banner are each increased by 20%. What is the area of the copy, in square inches?
原库参考解析
周长・面积与平面图形进阶
A rectangular poster has an area of 330 square inches. A copy of the poster is made in which the length and width of the original poster are each increased by 50%. What is the area of the copy, in square inches?
原库参考解析
