这部分需要会什么？
- 区分表面积与体积
- 使用柱体、锥体和球的体积公式
- 处理相似立体的平方与立方倍率
核心概念与公式
直棱柱或圆柱的体积为底面积乘高 \(V=Bh\)；棱锥或圆锥为 \(V=Bh/3\)；球为 \(V=4\pi r^3/3\)。长方体表面积为 \(2(lw+lh+wh)\)，封闭圆柱表面积为 \(2\pi r^2+2\pi rh\)，球表面积为 \(4\pi r^2\)。
相似立体的长度倍率为 \(k\) 时，表面积倍率为 \(k^2\)，体积倍率为 \(k^3\)。圆锥体积中的高是垂直高，侧面积 \(\pi r\ell\) 中的 \(\ell\) 则是母线长。
做题时怎样使用
- 先确认形体、开口情况及所求是表面积还是体积
- 统一长度单位，找出底面积和垂直高
- 相似问题先求长度倍率，再根据面积或体积选择平方或立方
容易混淆的地方
无盖容器的表面积不含上底。将体积倍率直接当作半径倍率，会把立方关系误当成线性关系。
用一个例题串起方法
知识点教学例题两个相似圆柱的体积比为 \(27:8\)，求其半径比与表面积比。
展开解题思路 ＋
长度比是体积比的立方根，为 \(3:2\)，因此半径比为 \(3:2\)。表面积比是长度比的平方，为 \(9:4\)。
本知识点题库
先独立作答，再查看答案与解析。
表面积・体积与相似立体挑战
The figure shown is a right circular cylinder with a radius of r and height of h. A second right circular cylinder (not shown) has a volume that is 392 times as large as the volume of the cylinder shown. Which of the following could represent the radius R, in terms of r, and the height H, in terms of h, of the second cylinder?
原库参考解析
表面积・体积与相似立体挑战
A cube has an edge length of 68 inches. A solid sphere with a radius of 34 inches is inside the cube, such that the sphere touches the center of each face of the cube. To the nearest cubic inch, what is the volume of the space in the cube not taken up by the sphere?
原库参考解析
表面积・体积与相似立体挑战
A cube has a volume of 474,552 cubic units. What is the surface area, in square units, of the cube?
原库参考解析
表面积・体积与相似立体挑战
A right circular cone has a height of 22 centimeters cm and a base with a diameter of 6 cm. The volume of this cone is
nπ cm3. What is the value of n?
原库参考解析
表面积・体积与相似立体挑战
A manufacturer determined that right cylindrical containers with a height that is 4 inches longer than the radius offer the optimal number of containers to be displayed on a shelf. Which of the following expresses the volume, V, in cubic inches, of such containers, where r is the radius, in inches?
原库参考解析
表面积・体积与相似立体挑战
A right rectangular prism has a length of 28 centimeters cm, a width of 15 cm, and a height of 16 cm. What is the surface area, in cm2, of the right rectangular prism?
原库参考解析
表面积・体积与相似立体挑战
A right circular cone has a volume of \( \frac{1}{3}π \) cubic feet and a height of 9 feet. What is the radius, in feet, of the base of the cone?
原库参考解析
表面积・体积与相似立体挑战
A right rectangular prism has a base area of \( 24t \) square centimeters (cm²). The length of the base of the rectangular prism is \( \frac{8}{3} \) cm, and the height of the rectangular prism is 15 cm. Which expression represents the surface area, in cm², of the right rectangular prism?
原库参考解析
表面积・体积与相似立体进阶
A cylinder has a diameter of 8 inches and a height of 12 inches.What is the volume, in cubic inches, of the cylinder?
原库参考解析
表面积・体积与相似立体进阶
A cube has an edge length of 41 inches. What is the volume, in cubic inches, of the cube?
原库参考解析
表面积・体积与相似立体挑战
A cube has a volume of 474 552 cubic units. What is the surface area, in square units, of the cube?
原库参考解析
表面积・体积与相似立体挑战
Two identical rectangular prisms each have a height of 90 centimeters (cm). The base of each prism is a square, and the surface area of each prism is K cm2. If the prisms are glued together along a square base, the resulting prism has a surface area of \(\frac{92}{47}\)K cm2. What is the side length, in cm, of each square base?
原库参考解析
