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・面积与平面图形进阶
The perimeter of a square can be expressed as 4(76) units. What is the length, in units, of a side of this square?
原库参考解析
周长・面积与平面图形进阶
The perimeter of an obtuse triangle is 27 meters. The lengths of two sides of this triangle are 5 meters and 9 meters. What is the length, in meters, of the third side of this triangle?
原库参考解析
周长・面积与平面图形挑战
A circle has a radius of 19 meters. What is the area, in square meters, of the circle?
原库参考解析
周长・面积与平面图形挑战
Circle N has a radius of 6 millimeters (mm). Circle M has an area of 121π mm². What is the total area, in mm², of circles N and M ?
原库参考解析
周长・面积与平面图形进阶
A circle has a circumference of 89π centimeters. What is the diameter, in centimeters, of the circle?
原库参考解析
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 \( w(w+79) \), where w is the width, in centimeters, of the painting. Which expression represents the length, in centimeters, of the painting?
原库参考解析
周长・面积与平面图形进阶
\( h\left(x\right)=\frac{1}{2}\left(x\right)\left(89\right) \)
The given function \( h \) represents the area, in square inches, of a triangle with a base of \( x \) inches and a fixed height. What is the height, in inches, 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 69 inches. Which equation represents this situation?
原库参考解析
周长・面积与平面图形挑战
Rectangle X has a length of 24 millimeters (mm) and a width of 17.5 mm. Right triangle Y has a base of 20 mm. The area of rectangle X is 3 times the area of right triangle Y. What is the height, in mm, of right triangle Y?
原库参考解析
周长・面积与平面图形进阶
The width of a rectangle is 9 centimeters. The length of the rectangle is 50 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
The width of a rectangle is 9 centimeters. The length of the rectangle is 50 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
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?
原库参考解析
