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 surface area of 75,264 square inches. which includes a base area of 36,864 square inches. What is the slant height, in inches. of this pyramid?
原库参考解析
表面积・体积与相似立体挑战
As the size of a firework's shell increases, the height above the ground when the firework bursts increases.For a firework to burst 1,075 feet above the ground, a firework's shell in the shape of a sphere with a volume of 514.23 cubic inches could be used. To the nearest whole number, what is the volume of this firework's shell in cubic centimeters? ( 1 inch = 2.54 centimeters)
原库参考解析
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 52 square centimeters (cm2), and the surface area of right rectangular prism Y is 3,328 cm2. The volume of right rectangular prism X is 10.5 cubic centimeters (cm3). What is the volume, in cm3 , of right rectangular prism Y?
原库参考解析
表面积・体积与相似立体进阶
The figure shown is a right circular cylinder with a radius of r and height of h. A second right circular cylinder (not shown) has a volume that is 576 times as large as the volume of the cylinder shown. Which of the following could represent the radius R, in terms of r, and the height H, in terms of h, of the second cylinder?
原库参考解析
表面积・体积与相似立体挑战
The length of each edge of a box is 79 centimeters. Each side of the box is in the shape of a square. The box does not have a lid. What is the exterior surface area, in square meters, of this box without a lid? (1 meter = 100 centimeters)
原库参考解析
表面积・体积与相似立体进阶
A right rectangular prism has a length of 26 centimeters (cm), a width of 16 cm, and a height of 14 cm. What is the surface area, in cm², of the right rectangular prism?
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 12 centimeters and a volume of 2,448π 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 hemisphere is half of a sphere. If a hemisphere has a radius of 59 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・体积与相似立体挑战
As the size of a firework's shell increases, the height above the ground when the firework bursts increases. For a firework to burst l,075 feet above the ground, a firework's shell in the shape of a sphere with a volume of 542.68 cubic inches could be used. To the nearest whole number, what is the volume of this firework's shell in cubic centimeters? (1 inch = 2.54 centimeters)
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 16t square centimeters (cm²). The length of the base of the rectangular prism is \( \frac{8}{3} \) cm, and the height of the rectangular prism is 9 cm. Which expression represents the surface area, in cm², of the right rectangular prism?
原库参考解析
表面积・体积与相似立体挑战
The circumference of the base of a right circular cylinder is 6π meters, and the height of the cylinder is 19 meters. What is the volume, in cubic meters, of the cylinder?
原库参考解析
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 25 trees and cut in the shape of a cube. The length of the edge of one of these cubes is 3.000 centimeters. This cube has a density of 0.210 grams per cubic centimeter. What is the mass of this cube, in grams?
原库参考解析
