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・面积与平面图形基础
The area A, in square centimeters, of a rectangular cutting board can be represented by the expression w(w + 9), where w is the width, in centimeters, of the cutting board. Which expression represents the length, in centimeters, of the cutting board?
原库参考解析
周长・面积与平面图形进阶
Square X has a side length of 12 centimeters. The perimeter of square Y is 2 times the perimeter of square X. What is the length, in centimeters, of one side of square Y?
原库参考解析
周长・面积与平面图形基础
Each side of a square has a length of 45. What is the perimeter of this square?
原库参考解析
周长・面积与平面图形挑战
In the figure shown, AB=\( \sqrt{34} \) units, AC= 3 units, and CE= 2l units. What is the area, in square units, of triangle ADE?
原库参考解析
周长・面积与平面图形挑战
For an electric field passing through a flat surface perpendicular to it, the electric flux of the electric field through the surface is the product of the electric field's strength and the area of the surface. A certain flat surface consists of two adjacent squares, where the side length, in meters, of the larger square is 3 times the side length, in meters, of the smaller square. An electric field with strength 29.00 volts per meter passes uniformly through this surface, which is perpendicular to the electric field. If the total electric flux of the electric field through this surface is 4,640 volts · meters, what is the electric flux, in volts · meters, of the electric field through the larger 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 36 feet. Each of the two congruent sides of the triangle has a length of 10 feet. What is the length, in feet, of the third side?
原库参考解析
周长・面积与平面图形基础
Rectangle P has an area of 72 square inches. If a rectangle with an area of 20 square inches is removed from rectangle P, what is the area, in square inches, of the resulting figure?
原库参考解析
周长・面积与平面图形挑战
The area of a rectangular banner is 2,661 square inches. The banner's length x, in inches, is 24 inches longer than its width, in inches. Which equation represents this situation?
原库参考解析
周长・面积与平面图形基础
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
周长・面积与平面图形基础
The lengths of two sides of a triangle are 4 centimeters and 6 centimeters. If the perimeter of the triangle is 18 centimeters, what is the length, in centimeters, of the third side of this triangle?
原库参考解析
周长・面积与平面图形挑战
In triangle \( XYZ \) , angle \( Y \) is a right angle, the measure of angle \( Z \) is 33°, and the length of \( YZ \) is 26 units. If the area, in square units, of triangle \( XYZ \) can be represented by the expression k tan33°, where k is a constant, what is the value of k?
原库参考解析
周长・面积与平面图形基础
The area \( A \), in square centimeters, of a rectangular cutting board can be represented by the expression \( w\left(w+9\right) \), where w is the width, in centimeters, of the cutting board. Which expression represents the length, in centimeters, of the cutting board?
原库参考解析
