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
The density of a certain type of marble stone is 2.3000 grams per cubic centimeter. If a sample of this type of stone is in the shape of a sphere with a diameter of 33.000 centimeters, what is the mass of this sample, in grams, to the nearest whole number? (Use 3.14159 for π.)
原库参考解析
表面积・体积与相似立体挑战
The table shows the volume of two similar solids, right circular cylinder A and right circular cylinder B. The radius of right circular cylinder A is 7 units. The surface area of right circular cylinder A is \( kπ \) square units, and the surface area of right circular cylinder B is \( nπ \) square units, where k and n are constants. What is the value of \( n−k \)? (The surface area of a right circular cylinder with radius r and height h is \( 2πr^{2}+2πrh \).)
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 72 inches, a wall thickness of \( \frac{5}{8} \) inches, and a height of 138 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
The height of a right circular cylinder is 36 inches, and the circumference of its base is 360 inches. Which expression represents the total surface area, in square inches, of the cylinder?
原库参考解析
表面积・体积与相似立体挑战
A right circular cylinder has a volume of \( 35,152π \) cubic inches. The height of the cylinder is 2 times its radius. What is the radius, in inches, of the cylinder?
原库参考解析
表面积・体积与相似立体进阶
A right circular cylinder has a volume of \( 31,944π \) cubic inches. The height of the cylinder is 3 times its radius. What is the radius, in inches, of the cylinder?
原库参考解析
表面积・体积与相似立体挑战
The figure shown is a trirectangular tetrahedron, a three-dimensional figure with four triangular faces, three of which are right triangles. A chemistry professor uses trirectangular tetrahedrons to create large-scale models of various carbon-based molecules. The base of one of these tetrahedrons is an isosceles right triangle whose legs each have length 14 inches (in). The height of this tetrahedron is 15 in. Which of the following is closest to the volume of this tetrahedron, in pints? (Note: 1 pint = 0.5 quarts; use 1 quart = 57.75 in3. The volume of a trirectangular tetrahedron is equal to \( \frac{1}{3}Sh \), where S is the area of its base and h is its height.)
原库参考解析
表面积・体积与相似立体进阶
The height of a right circular cylinder is 31 inches, and the circumference of its base is 310 inches. Which expression represents the total surface area, in square inches, of the cylinder?
原库参考解析
表面积・体积与相似立体进阶
A right circular cylinder has a volume of \( 6,591π \) cubic inches. The height of the cylinder is 3 times its radius. What is the radius, in inches, of the cylinder?
原库参考解析
表面积・体积与相似立体进阶
A conservation specialist hung artificial nesting structures each in the shape of a right rectangular prism for a species of native woodpecker. Each structure has a height of 20 inches. The length of each structure's base is x inches, which is 2 inches 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
The height of a right circular cylinder is 21 inches, and the circumference of its base is 210 inches. Which expression represents the total surface area, in square inches, of the cylinder?
原库参考解析
表面积・体积与相似立体挑战
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
