# 锥台

${\displaystyle {{n}+{2}}}$
${\displaystyle {{3}\,{n}}}$

 不對稱雙錐體（對偶多面體） （展開圖）

## 性质

### 体积

${\displaystyle {\frac {H-h}{H}}={\sqrt {\frac {S_{u}}{S_{d}}}}}$

${\displaystyle H={\frac {h{\sqrt {S_{d}}}}{{\sqrt {S_{d}}}-{\sqrt {S_{u}}}}}}$

${\displaystyle V={\frac {S_{d}H}{3}}-{\frac {S_{u}(H-h)}{3}}={\frac {(S_{d}{\sqrt {S_{d}}}-S_{u}{\sqrt {S_{u}}})h}{3({\sqrt {S_{d}}}-{\sqrt {S_{u}}})}}={\frac {h}{3}}\left(S_{d}+S_{u}+{\sqrt {S_{d}}}{\sqrt {S_{u}}}\right)}$

${\displaystyle V={\frac {n(a^{2}+b^{2}+ab)h}{12}}\cot {\frac {\pi }{n}}.}$

### 表面积

${\displaystyle S_{c}=\sum _{i=1}^{n}S_{i}}$ ，其中${\displaystyle S_{i},i=1,2\cdots ,n}$ 是第 i 个侧面的面积。

${\displaystyle S_{c}=\sum _{i=1}^{n}S_{i}={\frac {1}{2}}\sum _{i=1}^{n}(a_{i}+b_{i})h_{i}.}$

### 体积公式

${\displaystyle V={\frac {h_{2}B_{2}-h_{1}B_{1}}{3}}}$

B1 指一个底面的面积，B2指另一个底面的面积, and h1h2 指原顶点分别到两底面的面积。 考虑到

${\displaystyle {\frac {B_{1}}{h_{1}^{2}}}={\frac {B_{2}}{h_{2}^{2}}}}$

${\displaystyle V={\frac {h}{3}}(B_{1}+B_{2}+{\sqrt {B_{1}B_{2}}})}$

${\displaystyle V={\frac {\pi h}{3}}(R_{1}^{2}+R_{2}^{2}+R_{1}R_{2})}$

π 等于 3.14159265...，'R1, R2 是两底面的半径

${\displaystyle V={\frac {nh}{12}}(a_{1}^{2}+a_{2}^{2}+a_{1}a_{2})\cot {\frac {180}{n}}}$

a1a2 是底面的边长。

### 表面积公式

{\displaystyle {\begin{aligned}{\text{Lateral Surface Area}}&=\pi (R_{1}+R_{2})s\\&=\pi (R_{1}+R_{2}){\sqrt {(R_{1}-R_{2})^{2}+h^{2}}}\end{aligned}}}
{\displaystyle {\begin{aligned}{\text{Total Surface Area}}&=\pi \left[(R_{1}+R_{2})s+R_{1}^{2}+R_{2}^{2}\right]\\&=\pi \left[(R_{1}+R_{2}){\sqrt {(R_{1}-R_{2})^{2}+h^{2}}}+R_{1}^{2}+R_{2}^{2}\right]\end{aligned}}}

Lateral Surface Area指侧面积，Total Surface Area指总面积，R1 and R2 为底面半径，s 为平截头体的斜高。 一个底面为正n边形的正棱台的表面积是

${\displaystyle A={\frac {n}{4}}\left[(a_{1}^{2}+a_{2}^{2})\cot {\frac {\pi }{n}}+{\sqrt {(a_{1}^{2}-a_{2}^{2})^{2}\sec ^{2}{\frac {\pi }{n}}+4h^{2}(a_{1}+a_{2})^{2}}}\right]}$

a1a2是两底面的边长。

## 参考资料

1. ^ Nahin, Paul. "An Imaginary Tale: The story of the square root of minus one." Princeton University Press. 1998
2. ^ Mathwords.com: Frustum. [17 July 2011]. （原始内容存档于2021-01-26）.