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
A space probe uses a square-shaped solar sail to move through space. The side length of the solar sail is 8.83w meters, where w is the width, in meters, of the space probe.Which equation gives the area A, in square meters, of the solar sail?
原库参考解析
周长・面积与平面图形进阶
The length of a rectangle is 20 inches and the width is x inches. The perimeter is at most 100 inches. Which inequality represents this situation?
原库参考解析
周长・面积与平面图形进阶
A circle has a circumference of 48π centimeters. What is the diameter in centimeters, of the circle?
原库参考解析
周长・面积与平面图形挑战
A total of 3 squares each have side length r. A total of 6 equilateral triangles each have side length t. None of these squares and triangles shares a side. The sum of the perimeters of all these squares and triangles is 60. Which equation represents this situation?
原库参考解析
周长・面积与平面图形挑战
In the given figure, \( NP \) is the diameter of semicircle O. Angles Q, R, and S are right angles, and point P is the midpoint of \( NQ \) . If PQ = 52 feet and NS = 18 feet, the area of the figure can be expressed as \( aπ+b \) square feet, where a and b are integers. What is the value of \( a−b \) ?
原库参考解析
周长・面积与平面图形进阶
The area of a triangle is equal to \( x^2 \) square centimeters. The length of the base of the triangle is \( 2x+12.5 \) centimeters, and the height of the triangle is \( x−5 \)centimeters. What is the value of x?
原库参考解析
周长・面积与平面图形挑战
If the perimeter of an equilateral triangle with a height of 46 units is \( a\sqrt{3} \) units, what is the value of a?
原库参考解析
周长・面积与平面图形进阶
A circle has a radius of 2.9 inches. The area of the circle is bπ square inches, where b is a constant. What is the value of b?
原库参考解析
周长・面积与平面图形挑战
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
The area of triangle ABC is 230 square centimeters. What is the height h, in centimeters, of this triangle?
原库参考解析
周长・面积与平面图形挑战
In the figure shown,\( AB=\sqrt{61} \)units,\( AC=6 \) units, and \( CE=18 \) units. What is the area, in square units, of triangle \( ADE \)?
原库参考解析
周长・面积与平面图形进阶
A rectangular region of land is divided into 88 square lots of equal area R, in square units. The length of the region is 5.5 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
