1-3常用求面積體積公式37777

1、1-3常用求面積、體積公式1-3-1平面圖形面積平面圖形面積見表1-73。平面圖形面積表1-73圈形尺寸符號(hào)面積重心（G正形因a邊枝d對(duì)角線1a=yA = 0-7Q7dd = L414a = 1.414/A.在對(duì)博線交點(diǎn)上頰邊b長(zhǎng)邊d對(duì).角線A = a *d a2 + b1在對(duì)博線交點(diǎn)上三角形 平行四邊形BA I Dc對(duì)應(yīng)痢A、E、鄰邊 對(duì)邊何的距離GD= -BDCD - DAA = ft Ji a * ft sm s 5ih0在對(duì)甬線交點(diǎn)上CE = ABAF = CDCD （上底邊）i = AB 下底邊）A高a +2酉HG =專IE：r半栓d直栓衛(wèi)一圓間長(zhǎng)A = nr2=Yicd24= 0.

2、785=0.07958p- nd在0心上a b-1主軸在主軸交點(diǎn)G上r半隹5弧長(zhǎng)a孤s的了才應(yīng)中(?角*1住1QJT, = 180rGO =寺乎當(dāng)曲=9時(shí)GO 二手3 jcr一半徑 孤畏= 2.893正十邊形 Ki(j = 7.694 P10 = 2.939正邊形 Kn = 9.364, F|j =2.973 正十二邊形KL2=1U196,珂計(jì)3.000!在內(nèi)接 圓心或外 接圓心處1-3-2 多面體的體積和表面積多面體的體積和表面積見表 1-74。表 1-74多面體的體積和表面積重 心(G)a棱d-一對(duì)角線S表面積Si値表面積在對(duì)角線交點(diǎn)上K方侔棱柱)心、占、h邊長(zhǎng)O底面對(duì)熾線左點(diǎn)V=abhS

3、 =2 a * b + a * h + t h)St = 2h (d + i)d=三棱柱H、b c一邊長(zhǎng)h高A底面積O底面中線的交點(diǎn)S= (a + b 十亡)f +2AS=n*/ +ASi = /GO峠f個(gè)組合三角形的面積n-一組合三箱形的個(gè)數(shù) O錐底備對(duì)角線交點(diǎn)AA】、A3-兩平行底面V = y/i (Aj + A2 + VAt/G)S = dn + A1 + ASt = an的面積A-底面間的距離a個(gè)組合梯形的面-一組合梯形數(shù)云扣+ 2 /創(chuàng)址+3加 Ai + /AjAj + Az圓柱和空心K柱(管)R外半徑r內(nèi)半徑柱壁厚度P一平均半徑51內(nèi)外儼面積圓柱；V=腫hSt = 2Rh空心直圓柱

4、* =rrh (R2-)= 2nRPthS = 2k (R + r) A + 2(rx /)Si =2ir (J? + r) h心（G）無直圖柱ft I最小離度A 2Jft大高度r底面半徑4 （A | + A;）休積（V） 底面積（A）表茴積CS）側(cè)表面積（S1）r半徑d直徑在球心上球扁晤球鍥)r球半徑圧弓形底圓克徑h弓形高V=yjrr2h 2.0944r2hS+刃= 1*5 +T）V =nr2= = 0.5236/3S = 4rr2 wd2h球缺的高r球缺半鋰d平切圓直鏡 s*曲面面租S球缺表羽積y=廠-號(hào)）$曲=2燦=令+刊S = jrfi （4r h）d2 = 4h （2r-fi）尺寸符

5、號(hào)R圓環(huán)體平均半桎 D圓環(huán)悴平均宜徑 dIE環(huán)體哉面直植 r圓環(huán)體截面半徑體積(V) 底面積(A)表面積(S)側(cè)表面積 h.V =2?t2Kt3峠畑zS =4v2Rr=n2 Dd = 39.47Rr 心(G)在環(huán)中応上交叉圓柱休R一球半徑口、r；底面半徑h麗奇球心0至帶底 圓心Q 的距 離V= (3十+ 3沁+屏)St = 2RhS = 2it/?A + ?r ( ri + r；)0中間斷面直鋰 d底直徑t一桶高4 b、c半軸r圓柱半徑 g I同柱長(zhǎng)莊、b卜底邊長(zhǎng)対、bl上底邊快h上、下底邊膽離(高對(duì)于拋樹線形桶板；X ( 2DJ+ a/ + y3)對(duì)于圓形糯板：v = -j (2DJt rf

6、1)V =亍赧JTS= 22 br /a + b2V =石：2 也 + ai)6+(2*| +在軸交點(diǎn)上在軸兗直上在二軸線兗點(diǎn)上1-3-3物料堆體積計(jì)算物料堆體積計(jì)算見表1-75。1-3-4殼體表面積、側(cè)面積計(jì)算1-3-4-1圓球形薄殼（圖1-1）圖1-1圓球形薄殼計(jì)算圖球面方程式：X2 +（對(duì)坐標(biāo)系XYZ,原點(diǎn)在0）式中R半徑；x、y. z在球殼面上任一點(diǎn)對(duì)原點(diǎn)o的坐標(biāo)。殳 一0 F弦lx (AC); 己擴(kuò)匸 f 4 u X T、厶 jT-izS / j26弦長(zhǎng)(BC)iF、GABt BC的中點(diǎn)；f弓形AKC的高（KOJ）；一弓形AEB的高（EF）；弓形BDC的高（DG）;S,弧AEB的長(zhǎng);

7、Sy-一弧另DC的獲；Ak弓形AEB的面積（側(cè)面積”Ay弓形BDC的面積；2丸對(duì)應(yīng)弧的圓心角（弧度）；20$對(duì)應(yīng)弧贏的圓心角（弧度）；CT新坐標(biāo)系工平的原點(diǎn)（XOY平面平移JR（專后與Z軸的交點(diǎn)）。SJ.aX = arcsin 占產(chǎn)Z咅Ay R - a - J弧AEE與EDC之曲線方程式分別為:/+ z2= (R2- 62)(AEB)y2 + z2 = R2-a2 (HDC)1*弧長(zhǎng)按下式計(jì)算:SK = 2 Jr2 - b2 * arcsin 富 二 J疋-h2Sy = 2 JR2 - a2 arcsin .2.側(cè)面積按下式計(jì)算：Ax = (K2 b2) arcsin a J R2 - a2

8、b2用-b2 (R2 a2) arcsin 7- - zr - b JR1 - a2 b2Jr2 - a23*殼表面積按下式計(jì)算：A = Sx-Sy其一次近似值為：LA = 4oR arcain = _ = 4 aR* r其二次近似值為:A 二 4 aKarcsiti +a3b1-3-4-2橢圓拋物面扁殼（圖1-2）圖1-2橢圓拋物面扁殼計(jì)算圖殼面方程式：Z吟疋+令嚴(yán)1 0在殼面上任一點(diǎn)對(duì)原點(diǎn)0的坐標(biāo)；X. Y、Z加一對(duì)應(yīng)弧ADB的弦長(zhǎng)匚2b對(duì)應(yīng)弧BEC的弦長(zhǎng);Ak弓形ADB的高；hv弓形BEC的高冷假設(shè)：Sx弧忑的長(zhǎng):；W 、Sy弧BEC的長(zhǎng)；弓形ADB的面積;Ay弓形EEC的面積。1.弧長(zhǎng)

9、按下式計(jì)算式中或者:SI = 2ax系數(shù)KSy2&x系數(shù)心= ci + am Jn4- mi a+m2 o JW1 = 2A；式中系數(shù)K外Kh可分別根據(jù)尹、的值，查表1-兀得到。3 2b2.殼表面積按下式計(jì)算A = SH * Sy3.側(cè)面積按下式計(jì)算1-3-4-3橢圓拋物面扁殼系數(shù)計(jì)算見圖1-2，殼表面積（A）計(jì)算公式:A=SX Sy= 2ax系數(shù) KaX2bx系數(shù) Kb式中 Ka、Kb橢圓拋物面扁殼系數(shù)，可按表 1-76查得。橢圓拋物面扁殼系數(shù)表表1-76系數(shù)或瓦2a 2b系數(shù) K.或$Aa歳2a2b系數(shù) K.或心Aa或H2a *2&垂數(shù) K,或2a2bM數(shù) 氐或心0.0501.00660.

10、0801.0168OJW1.03140.1401.05000.1701.07240.0511.00690.081L.01720.1111.03200.1411.05070.1711.07330.0521.00720.0821.01770J121,03250,1421.05140.1721.07410.0S31.00740.083L.018L0.1131.03310.1431.05210.1731.07490.0541.00770.0S41.01850J141.03370.144L05280.1741.07570.0551.00800.0851.01890.1151.03420.1451.053

11、50.1751.07650.0561.00830.0861.01940.1161.03480.146L05420.1761.07730.0571.00860.0871.01980.1171.03540.1471.05500.1771.07820+0581.00890.0&8U02030.1181.03600.1481.055703781,07900.0591.00920.089L.02070.1191.03660L1491.05640.1791.07980+C601.00950,0901.02120.1201.03720,1501,05710.1801.08070,0611.QQ9Bth 09

12、11.02170.1211.03780.1511.057S0.1811.08150.0621.01020.0921.02210.122L03B40,1521,05860.1821,08240.0631.01050.093L.02260.1231.03900.1531.05930.1831.08320.0641.010S0.0941.02310.1241.03960.154L06010.1841.08410.0651.G1120.0951.02360.1251.04020.1551.06080.1851.08490.0661.01150.0961.02410.1261440801561.0616

13、0,1861.08580.0671.011&0.097L.02460.1271.04150.1571.06230.1871.08670.0681,01220.0981.02510.1281.0421OhISSL06310.1881.08750.069L01260.091.02560J291.0428UJ591.063B0.1891.08840.0701.01290.1001.02610.1301.04340.1601.06460.1901.W30.0711.0133oaoiL02660.1311.04400161L065401911.09020.0721.01370.1021.02710.13

14、21.04470.1621.06610.1921.09100.073J.O14Ooao31.02760,1331.04530.1631.06690.1931.09190.0741.01440.1041.02810.1341.04600.1641.06770.1941.09280.0751.01480.1051.02&70.1351.04670.1651.068$0.195IM370.0761.01520.106L.02920.1561.04730.1661.06930.1961.09460.0771.015&0.1071,0270,1371.O4S00.1671.07000.1971.0955

15、0.0781.01600.1081.Q3030.1381.04870.1681,07080.1981.09640.0791.01640.1091.03080.1391.04940.169L07160.1991.0973查表說明例已知 2a= 24.0m, 2b= 16.0m, hx = 3.0m, hy= 2.8m，試求橢圓拋物面扁殼表面積A。先求出 hx/2a= 3.0/24.0= 0.125hy/2b = 2.8/16.0= 0.175分別查表得系數(shù)Ka為1.0402和系數(shù)Kb為1.0765,則扁殼表面積A = 24.0X1.0402X16.0X 1.0765= 429.99m21-3-4

16、-4圓拋物面扁殼(圖1-3)圖1-3圓拋物面扁殼計(jì)算圖殼面方程式：Z二占(X2+Y2)式中 X、Y、一在殼面上任一點(diǎn)對(duì)原點(diǎn)O的坐標(biāo);R 一半徑；假設(shè)2a一對(duì)應(yīng)弧忌的弦長(zhǎng)；2b一對(duì)應(yīng)弧贏的弦長(zhǎng)；S衣一弧AGB的長(zhǎng)；、弧BDC的長(zhǎng)；弓形AGE的高*嘰弓形BDC的冋;AS Tfx a /ZD i!j4t fSr in ./p /iL/iJ trjwti 怩；弓形BDC的面積；f一殼頂?shù)降酌婢嚯x；AC的長(zhǎng)。M：c=2 J a1 + b1= 8R,一 Z 2R k 知-2RI-弧長(zhǎng)按下式計(jì)算S” =黃VJ?2 + a2 + K 山冷 +寺% =董 JR2 + b2 + R X詰 + 影 JR + b22

17、.殼表面積按下式計(jì)算A = Sx * Sy3-側(cè)面積按下式計(jì)算A2a34 ,A-3R = 3 朋耳a2臚4 ,Ar 3R亍叭1-3-4-5單、雙曲拱展開面積1 單曲拱展開面積=單曲拱系數(shù)x水平投影面積。2.雙曲拱展開面積=雙曲拱系數(shù)（大曲拱系數(shù)X小曲拱系數(shù)）X水平投影面積單、雙曲拱展開面積系數(shù)見表1-77。單雙曲拱展開面積計(jì)算圖見圖1-4圖1-4單、雙曲拱展開面積計(jì)算圖L-拱跨；F-拱高單、雙曲拱展開面積系數(shù)表表1-77單-曲拱F/L1/21/31/41/51/61/71/81/91/10/系數(shù)單曲拱系數(shù)1.501.251,15L101.071.051.1J41.031.02雙曲拱鬲數(shù)1/21.5D2.251.8751.725L.6501.6051.5751.5691.5451.5301/3U25KS751.5631.43B1.3751.3381.3131.3001.2&81.2751/41.151.725L4331.3231.2651,2311,2081A961.1851.1731/51.101,6