# 锥台

## 公式

### 体积公式

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

${\frac {B_{1}}{h_{1}^{2}}}={\frac {B_{2}}{h_{2}^{2}}}$ $V={\frac {h}{3}}(B_{1}+B_{2}+{\sqrt {B_{1}B_{2}}})$ $V={\frac {\pi h}{3}}(R_{1}^{2}+R_{2}^{2}+R_{1}R_{2})$ π 等于 3.14159265...，'R1, R2 是两底面的半径

$V={\frac {nh}{12}}(a_{1}^{2}+a_{2}^{2}+a_{1}a_{2})\cot {\frac {180}{n}}$ a1a2 是底面的边长。

### 表面积公式

{\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}} {\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

$A={\frac {n}{4}}\left[(a_{1}^{2}+a_{2}^{2})\cot {\frac {\pi }{n}}+{\sqrt {(a_{1}^{2}-a_{2}^{2})^{2}\sec ^{2}{\frac {\pi }{n}}+4h^{2}(a_{1}+a_{2})^{2}}}\right]$ 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].
{"ft_lang_label":"__label__zh","ft_lang_prob":0.5952731,"math_prob":0.9999989,"size":1196,"snap":"2021-04-2021-17","text_gpt3_token_len":877,"char_repetition_ratio":0.09563758,"word_repetition_ratio":0.0,"special_character_ratio":0.29682276,"punctuation_ratio":0.13679245,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99997437,"pos_list":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"im_url_duplicate_count":[null,8,null,8,null,null,null,8,null,8,null,8,null,8,null,9,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-04-13T07:42:59Z\",\"WARC-Record-ID\":\"<urn:uuid:de6f9a6e-b3c9-4697-99cd-a1eb23c7c0cd>\",\"Content-Length\":\"111499\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:ca8c3f88-faeb-4cec-9008-36916c7f270c>\",\"WARC-Concurrent-To\":\"<urn:uuid:53d323fa-2c0f-41ac-a6ef-a8ff23aecab5>\",\"WARC-IP-Address\":\"208.80.154.224\",\"WARC-Target-URI\":\"https://zh.wikipedia.org/wiki/%E9%94%A5%E5%8F%B0\",\"WARC-Payload-Digest\":\"sha1:4HPNEQUNSJ6YZOHGG3NL2TZUOOJBX62Y\",\"WARC-Block-Digest\":\"sha1:67EG3DGUVRB4JESB47UKQWW7CWLQ67F6\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-17/CC-MAIN-2021-17_segments_1618038072175.30_warc_CC-MAIN-20210413062409-20210413092409-00496.warc.gz\"}"}