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
A right square pyramid has a total surface area of 70,560 square inches, and the combined surface area of the four lateral faces of this pyramid is 38,160 square inches. What is the height, in inches, of this pyramid?
原库参考解析
表面积・体积与相似立体挑战
A sphere has a radius of 3 inches and a density of 51 ounces per cubic inch. What is the mass, in ounces, of the sphere?
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 13 inches. What is its volume in cubic inches?
原库参考解析
表面积・体积与相似立体挑战
A right triangular pyramid has exactly six edges. Each of these edges is \( 96 \) centimeters long. If the surface area of the pyramid is \( k\sqrt{3} \) square centimeters, what is the value of \( k \)?
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 cm³. What is the area, in cm², 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
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 86 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?
原库参考解析
表面积・体积与相似立体进阶
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 2 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 2πr²+ 2πrh.)
原库参考解析
表面积・体积与相似立体进阶
Right rectangular prism X is similar to right rectangular prism Y. The surface area of right rectangular prism X is 58 square centimeters (cm²), and the surface area of right rectangular prism Y is 1,450 cm². The volume of right rectangular prism Y is 1,500 cubic centimeters (cm³). What is the sum of the volumes, in cm³, of right rectangular prism X and right rectangular prism Y?
原库参考解析
表面积・体积与相似立体进阶
The density of a certain type of marble stone is 2.6000 grams per cubic centimeter. If a sample of this type of stone is in the shape of a sphere with a diameter of 31.000 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
Cube B has a volume of 125 cubic inches. The length of each edge of cube A is k times the length of each edge of cube B. If cube A has a volume of 3,375 cubic inches, what is the value of k?
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 6 units and a volume of 128 cubic units. What is the length, in units, of one side of the base of the pyramid?
原库参考解析
