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
For the right circular cone shown, \( OP \) is a radius of the base and V is the vertex. The cone's lateral surface area, in square meters, is \( rLπ \) , where r is the length, in meters, of \( OP \) and L is the length, in meters, of \( VP \). The cone has a base area of \( 36π \) square meters and a lateral surface area of \( 143π \) square meters. What is the height of the cone, to the nearest meter?
原库参考解析
表面积・体积与相似立体挑战
A right rectangular prism has a base area of 32t square centimeters (cm²). The length of the base of the rectangular prism is \( \frac{16}{3} \) cm, and the height of the rectangular prism is 3 cm. Which expression represents the surface area, in (cm²), of the right rectangular prism?
参考答案
表面积・体积与相似立体挑战
For the right circular cone shown, \( OP \) is a radius of the base and V is the vertex. The cone's lateral surface area, in square meters, is rLπ, where r is the length, in meters, of \( OP \) and L is the length, in meters, of \( VP \). The cone has a base area of 81π square meters and a lateral surface area of 128π square meters. What is the height of the cone, to the nearest meter?
参考答案
表面积・体积与相似立体挑战
Cylinder A is similar to cylinder B. The volume of cylinder A is 5.13 cubic units and the volume of cylinder B is 1.52 cubic units. The surface area of cylinder A is 22.41 square units. What is the surface area, in square units, of cylinder B?
参考答案
表面积・体积与相似立体进阶
The height of a right circular cylinder is 47 inches, and the circumference of its base is 376 inches. Which expression represents the total surface area, in square inches, of the cylinder?
参考答案
表面积・体积与相似立体挑战
A right square prism has a height of 5 units. The volume of the prism is 405 cubic units. What is the length, in units, of an edge of the base?
参考答案
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 \( 2πr^{2}+2πrh \).)
参考答案
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?
参考答案
表面积・体积与相似立体挑战
Right rectangular prism X is similar to right rectangular prism Y. The surface area of right rectangular prism X is 50 square centimeters (cm²), and the surface area of right rectangular prism Y is 1,250 cm². The volume of right rectangular prism X is 14.4 cubic centimeters (cm³). What is the volume, in cm³, of right rectangular prism Y?
参考答案
表面积・体积与相似立体进阶
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?
参考答案
表面积・体积与相似立体挑战
Right circular cylinder A is similar to right circular cylinder B, where the ratio of the diameter of cylinder A to the diameter of cylinder B is 2 to 3. The height of cylinder A is equal to its diameter. If the surface area of cylinder A is 600π square inches, what is the volume, in cubic inches, of cylinder B?
参考答案
表面积・体积与相似立体进阶
The height of a right circular cylinder is 59 inches, and the circumference of its base is 590 inches. Which expression represents the total surface area, in square inches, of the cylinder?
参考答案
