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・体积与相似立体进阶
For the right circular cone shown, B is a point on the circumference of the base, and the length of segment AB (not shown) is 88 centimeters. If the height of the cone is 43 centimeters and the volume of the cone is \( kπ \)cubic centimeters, what is the value of k?
参考答案
表面积・体积与相似立体挑战
A right square pyramid has a height of 12 centimeters (cm). A second right square pyramid has a height of 24 centimeters. The area of the bases of each of the two pyramids is 81 cm2. Which of the following is closest to the difference in the surface area of the second pyramid and the surface area of the first pyramid, in cm2?
参考答案
表面积・体积与相似立体进阶
A cube has an edge length of 6 inches. What is the volume, in cubic inches, of the cube?
原库参考解析
表面积・体积与相似立体进阶
The figure shown is a right rectangular pyramid, where \( l=16 \) units, \( w=8 \) units, and \( h=18 \) units. What is the surface area, in square units, of the pyramid?
原库参考解析
表面积・体积与相似立体挑战
In the figure, a hollow steel pipe is in the shape of a right circular cylinder. The steel pipe has an outside diameter of 76 inches, a wall thickness of \( \frac{5}{8} \)inches, and a height of 135 inches. Which of the following is closest to the volume, in cubic inches, of the wall of this steel pipe?
原库参考解析
表面积・体积与相似立体挑战
A right square pyramid has a height of 12 inches and a volume of 196 cubic inches. What is the perimeter, in inches, of the base of this pyramid?
原库参考解析
表面积・体积与相似立体进阶
A right rectangular prism has a height of 24 inches. The length of the prism's base is x inches, which is 10 inches more than the width of the prism's base. Which function V gives the volume of the prism, in cubic inches, in terms of the length of the prism's base?
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 \( 4,800π \) cubic centimeters and the area of its base is \( 1,600π \) square centimeters. What is the slant height, in centimeters, of this cone?
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
表面积・体积与相似立体挑战
The volume of right circular cylinder A is 22 cubic centimeters. What is the volume, in cubic centimeters, of a right circular cylinder with twice the radius and half the height of cylinder A?
原库参考解析
表面积・体积与相似立体挑战
The dimensions of a right rectangular prism are 4 inches by 5 inches by 6 inches. What is the surface area, in square inches, of the prism?
原库参考解析
表面积・体积与相似立体挑战
A cube has a surface area of 54 square meters. What is the volume, in cubic meters, of the cube?
原库参考解析
