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 area of triangle ABC above is at least 48 but no more than 60. If y is an integer, what is one possible value of x ?
原库参考解析
周长・面积与平面图形挑战
A landscaper is designing a rectangular garden. The length of the garden is to be 5 feet longer than the width. If the area of the garden will be 104 square feet, what will be the length, in feet, of the garden?
原库参考解析
周长・面积与平面图形挑战
Square P has a side length of x inches. Square Q has a perimeter that is 176 inches greater than the perimeter of square P. The function f gives the area of square Q, in square inches. Which of the following defines f ?
原库参考解析
周长・面积与平面图形进阶
The area of a square is 64 square inches. What is the side length, in inches, of this square?
原库参考解析
周长・面积与平面图形基础
The perimeter of an isosceles triangle is 83 inches. Each of the two congruent sides of the triangle has a length of 24 inches. What is the length, in inches, of the third side?
原库参考解析
周长・面积与平面图形基础
A rectangle has a length of 64 inches and a width of 32 inches. What is the area, in square inches, of the rectangle?
原库参考解析
周长・面积与平面图形基础
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 6 and 12, which inequality represents the possible lengths, x, of the third side of the triangle?
原库参考解析
周长・面积与平面图形挑战
The side length of a square is 55 centimeters (cm). What is the area, in cm2, of the square?
原库参考解析
周长・面积与平面图形进阶
A triangle has a base length of 40 centimeters and a height of 90 centimeters. What is the area, in square centimeters, of the triangle?
原库参考解析
周长・面积与平面图形基础
What is the area, in square inches, of a rectangle with a length of 7 inches and a width of 6 inches?
原库参考解析
周长・面积与平面图形基础
The equation \( 46=2x+2y \) gives the perimeter of a rectangular rug that has length x, in feet, and width y, in feet. The width of the rug is 8 feet. What is the length, in feet, of the rug?
原库参考解析
周长・面积与平面图形挑战
The length of a rectangle is 50 inches and the width is x inches. The perimeter is at most 210 inches. Which inequality represents this situation?
原库参考解析
