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 triangular pyramid has exactly six edges. Each of these edges is 74 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・体积与相似立体进阶
The edge length, in inches, of cube Y is \( \frac{3}{64} \) the edge length, in inches, of cube X . The surface area, in square inches, of cube Y is n times the surface area, in square inches, of cube X . What is the value of n ?
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 83 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
A right square pyramid has a total surface area of 28,160 square inches, and the combined surface area of the four lateral faces of this pyramid is 16,060 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 right square pyramid has a height of 11 centimeters (cm). A second right square pyramid has a height of 22 centimeters. The area of the bases of each of the two pyramids is 100 cm². Which of the following is closest to the difference in the surface area of the second pyramid and the surface area of the first pyramid, in cm²?
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 48 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 circular cone has a height of 13 centimeters (cm) and a base with a radius of 6 cm. What is the volume, in cm³, of this cone?
原库参考解析
表面积・体积与相似立体进阶
A sphere has a radius of \( \frac{8}{3} \) feet. What is the volume, in cubic feet, of the sphere?
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 3 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・体积与相似立体进阶
A hemisphere is half of a sphere. If a hemisphere has a radius of 86 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・体积与相似立体进阶
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. If 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
A triangular prism has a height of 5 centimeters (cm) and a volume of 115 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
