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
To study fluctuations in leaf water potential, samples of wood were taken from 24 trees and cut in the shape of a cube. The length of the edge of one of these cubes is 2.00 centimeters. lf this cube has a mass of 4.48 grams, what is the density of this cube, in grams per cubic centimeter?
原库参考解析
表面积・体积与相似立体进阶
The length of an edge of cube A is one-fourth the length of an edge of cube B. What is the ratio of the surface area of cube A to the surface area of cube B ?
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 82 inches. A solid sphere with a radius of 41 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 28 centimeters. If the height of the cone is 14 centimeters and the volume of the cone is kπ cubic centimeters, what is the value of k?
原库参考解析
表面积・体积与相似立体进阶
A chemist combines water and isopropanol to make a mixture with a volume of 38 milliliters (mL). The volume of isopropanol in the mixture is 10 mL. What is the volume of water, in mL, in the mixture? (Assume that the volume of the mixture is the sum of the volumes of water and isopropanol before they were mixed.)
原库参考解析
表面积・体积与相似立体挑战
A cylinder has a diameter of 6 inches and a height of 21 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 chemist combines water and isopropanol to make a mixture with a volume of 41 milliliters (mL). The volume of isopropanol in the mixture is 10 mL. What is the volume of water, in mL, in the mixture? (Assume that the volume of the mixture is the sum of the volumes of water and isopropanol before they were mixed.)
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 34,496π cubic centimeters and the area of its base is 3,136π 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
A cube has a volume of 79,507 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 cylinder has a radius of 18 centimeters and a volume of 1,296π cubic centimeters. What is the height, in centimeters, of the cylinder?
原库参考解析
表面积・体积与相似立体挑战
A raised garden is in the shape of a right rectangular prism. Its base has a width of 3 feet and a length of 27 feet, and it will be filled 24 inches high with topsoil. The total cost of the topsoil needed to fill the garden to this height is 288 dollars. What is the unit cost, in dollars per cubic yard, for the topsoil? (1 yard = 3 feet; 1 foot = 12 inches)
原库参考解析
表面积・体积与相似立体挑战
A right square pyramid has a total surface area of 56,784 square inches, and the combined surface area of the four lateral faces of this pyramid is 28,560 square inches. What is the height, in inches, of this pyramid?
原库参考解析
