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
A right rectangular prism has a length of 12 centimeters (\( \ \ cm \)), a width of 11, and a height of 16\( \ \ cm \). What is the surface area, in \( \ \ cm^2 \) , of the right rectangular prism?
原库参考解析
表面积・体积与相似立体挑战
A conservation specialist hung artificial nesting structures each in the shape of a right rectangular prism for a species of native owl. Each structure has a height of 11 inches. The length of each structure's base is x inches, which is 1 inch more than the width of the structure's base. Which function V gives the volume of each structure, in cubic inches, in terms of the length of the structure's base?
原库参考解析
表面积・体积与相似立体挑战
A triangular prism has a height of 9 centimeters (cm ) and a volume of 234 \( cm^{3} \) . What is the area, in \( cm^2 \), of the base of the prism? (The volume of a triangular prism is equal to Bh, where B is the area of the base and h is the height of the prism.)
原库参考解析
表面积・体积与相似立体进阶
A hemisphere is half of a sphere. If a hemisphere has a radius of 56 inches, which of the following is closest to the volume, in cubic inches, of this hemisphere?
原库参考解析
表面积・体积与相似立体进阶
A hemisphere is half of a sphere. If a hemisphere has a radius of 57 inches, which of the following is closest to the volume, in cubic inches, of this hemisphere?
原库参考解析
表面积・体积与相似立体进阶
A right square prism has a height of 16 units. The volume of the prism is 64 cubic units. What is the length. in units. of an edge of the base?
原库参考解析
表面积・体积与相似立体挑战
Cylinder A and Cylinder B are similar right cylinders. The radius of Cylinder A is 9, and its surface area is dπ. The surface area of Cylinder B is tπ . What is the value of t − d ?
(Note: surface area of a cylinder is \( 2 \)πrh\( +2 \)π\( r^{2} \) )
原库参考解析
表面积・体积与相似立体进阶
A right circular cylinder has a height of 4 meters (m) and a base with a radius of 18 m. What is the volume, in m3, of the cylinder?
原库参考解析
表面积・体积与相似立体挑战
The volume of a right rectangular prism with a square base is 2448 cubic centimeters. If the area of the square base is 144 square centimeters, what is the area, in square centimeters, of one of the four lateral faces of the prism?
原库参考解析
表面积・体积与相似立体进阶
A right circular cylinder has a height of 4 meters (m) and a base with a radius of 18 m. What is the volume, in m3,of the cylinder?
原库参考解析
表面积・体积与相似立体挑战
The volume of a right rectangular prism with a square base is 2,448 cubic centimeters. If the area of the square base is 144 square centimeters, what is the area, in square centimeters, of one of the four lateral faces of the prism?
原库参考解析
表面积・体积与相似立体进阶
The right square pyramid shown has a height of 9 centimeters (cm). The length, in cm, of one edge of the pyramid's base is \( \frac{13}{3} \) times the height of the pyramid. What is the volume, in cm3, of the pyramid?
原库参考解析
