建筑施工手冊(cè)施工常用數(shù)據(jù)：常用求面積、體積公式1

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 o平面圖形面積表1-73圖形尺寸符號(hào)面積(A)重心(G)正 方 形a邊長(zhǎng)d對(duì)角線 A- a2a=/A=0.707dd = l414a = l414 燈在對(duì)角線交點(diǎn)上a短邊b長(zhǎng)邊 d對(duì)角線A = a 6 d = J a1 +T2在對(duì)角線交點(diǎn)上A =號(hào)=a6sinCGD = BDCD=DAh高l今周長(zhǎng)a x b、c對(duì)應(yīng)角 A、B、C的邊長(zhǎng)平行四邊形a、b鄰邊h對(duì)邊間的距離A = b b = Q bsinaACBD2sin/?在對(duì)角線交點(diǎn)上CE = ABAF=CDa = CD （上底邊）bAB （下底嗎）h高 a + b ,A

2、 = phKG =r半徑d直徑P圓周長(zhǎng)A = nr1-豐 d4= 0.785 = 0.07958 /2p-nd在圓心上a、b主軸在主軸交點(diǎn)G上a疋18（/r半徑5 弧長(zhǎng)a弧$的對(duì)應(yīng)中心角8氣乎當(dāng)a = 90時(shí)如專.哼,“.6卩r半徑5弧長(zhǎng)a中心角b弦長(zhǎng)h高知吉普當(dāng)a = 180時(shí)GO二話= 0.424圖形尺寸符號(hào)面積(A)重心(G)B環(huán)十rQ Q4r- -Mr-R外半徑r內(nèi)半徑D外直徑d內(nèi)直徑t環(huán)寬Dw平均直徑A =開(R2- r2)=手(D2-d2)在圓心O部 分 圓 環(huán)i釦R外半徑r內(nèi)半徑D外直徑d內(nèi)直徑Rp,圓環(huán)平均半徑/環(huán)寬A =逛_ (R2- r2360 J二遲R .t180 wR3

3、 - r3M-38.2 魚2 ；2血手xa新月 形OOX = L圓心間的 距離d克徑l 孟 a + sina)=FpR= K 180a +P值見F表02平？丄LX102d10104d10106d107d103d109d10P0.400.791.181.561.912.252.552.813.02物線形Ab底邊h-高1 曲線長(zhǎng)SAABC的面積Z ” + i 3333人 2A -b-h亠S3 s等 邊 多 邊 形囤a邊長(zhǎng)K,系數(shù)，指多邊形 的邊數(shù)R外接圓半徑P.系數(shù)，F(xiàn)指正多邊 形的邊數(shù)A嚴(yán)K,a2=RR2正三邊形 K3 = 0.433 P3-1.299 正四邊形K4=1.000 P產(chǎn)2.000

4、正五邊形 K5= 1.720. Ps = 2.375 正六邊毎 K6 = 2598. P6-2.598 正七邊形K7 = 3.634t P產(chǎn)2.736 正八邊形K8 = 4.82&戶8 = 2828 正九邊形 K9=6a82 P9= 2.893 正十邊形K】o = 7694, P10 = 2.939 正十一邊形 KH = 9.364t Pn-2.973 正十二邊形 Ki2=H196, Pu = 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 o表 1-74多面體的體積和表面積尺寸符號(hào)a棱d對(duì)角線S表面積S,側(cè)表面積V = a3S = 6a2S

5、, = 4a2在對(duì)角線交點(diǎn)上長(zhǎng)方體(棱柱)aa、b、h 邊長(zhǎng)O底面對(duì)角線交點(diǎn)V = a *6 AS = 2 b 七 ah + b h)S、= 2h (a + 6)d = J a2 bz + h2GO = 三梭柱GO = ya b、c邊長(zhǎng)h高A底面積O底面中線的交點(diǎn)V = A-AS = (a + b + c) 9h + 2ASi = (a + 6 + c) AV=A-hS=n-/+ASi= n-/GO =f一個(gè)組合三角形的 面積H組合三角形的個(gè)數(shù)0錐底各對(duì)角線交 點(diǎn)V =(A1 + A2+ 7a7a2)S = an + A】+ A2Sj = anA】、A2兩平行底面的面積h底面間的距離a個(gè)組合梯

6、形的面積n組合梯形數(shù)GO =令4x A】 + 2 yA2 + 3A2 A| + /A A 2 + Aj圓柱和空心圓柱(管R外半徑r內(nèi)半徑t柱樂厚度P平均半徑Si內(nèi)外側(cè)面積圓柱：VrR?hS = 2問十2加St = 2nRh 空心直圓柱t V -nh (R2- r2)=ZRPthS = 2k (/? + r) h 2kx (Rj2)S,=2 (R +J h斜截直圜柱h最小高度h2最大高度r底面半徑Si = nr (Ai + hi)GO =GK=i2hT2tgQr底面半徑 h高I母線長(zhǎng)v=fS| = jrr Jr2 + h2 = Krll = 7 r2+ h2S = S i + nr2R、r底面半

7、徑h高I母線乎(R2+/+&) Sb 削(R + r) 2 /(R二r)2 *2 S=Si +狄(2+ r2)GO七4v R2 2Rr 4 3r2R2+R 十2r半徑d直徑=琴 =05236d，S = 4rr2= nd2在球心上球氏形(球楔)r球半徑d弓形底圓直徑h弓形高V = r2h = 2.0944r2hS =y (4A + d)= 1.57r (4A 十 d)h球缺的高r球缺半徑d平切圓直徑S曲面面積S球缺表面積v=2(r-f)Sa -2rh - 令十&2 I8 Kh (4r - h )dh (2r-A)R圓環(huán)體平均半徑 D圓環(huán)體平均直徑 d圓環(huán)體截面直徑 r圓環(huán)體戡面半徑V =2n2R

8、-r2=jc2EW = 39.478Rf在環(huán)中心上R球半徑廠1、r2底面半徑h稷高心一球心O至帯底 圓心0的距 離V=Y （3卄 + 3蟲+人2）S = 2RhS = 2nRh * k （F）r|）中間斷面直徑底宜徑I桶高在軸交點(diǎn)上對(duì)于拋物線形桶板：v = 15X 2Da + Drf + yd2）對(duì)于圓形桶板：V =（2。2十屮）a、b、c半軸V = abcnS = 272-6- J卄金在軸交點(diǎn)上交叉圓柱體r圓柱半徑 仃、I圓柱長(zhǎng)在二軸線交點(diǎn)上V =g（2a + ai）b十（2s + a ）6% b下底邊長(zhǎng) 1、b上底邊長(zhǎng)h一上、下底邊距離（高）1-33物料堆體積計(jì)算物料堆體積訃算見表V5o物

9、料堆體積計(jì)算表-75圖形計(jì)算公式1芝 4-號(hào)）XH物料自然堆積角/SBtgaV=書（36 -a）M-4 1Xq7Vo （延米體積）二盍H牛ga1-3-4殼體表面積、側(cè)面積計(jì)算1-3-4-1圓球形薄殼（圖N）圖“圓球形薄殼計(jì)算圖球面方程式：X2+y2 + z2 = R2 （對(duì)坐標(biāo)系XYZ,原點(diǎn)在O）式中 R半徑；X、丫、Z在球殼面上任一點(diǎn)對(duì)原點(diǎn)O的坐標(biāo)。 假設(shè) C弦長(zhǎng)（AC）;2a弦長(zhǎng)（AB）;26弦長(zhǎng)（BC）;F、GAB, BC的中點(diǎn)；f弓形AKC的髙（K0）；H弓形AEB的高（EF）；hy弓形BDC的高（DG）;Sx弧AEB的長(zhǎng)；Sy弧BDC的長(zhǎng)；Ax弓形AEB的面積（側(cè)面積）；Ay弓形BD

10、C的面積；2人對(duì)應(yīng)弧缶的圓心角（弧度）;2軻對(duì)應(yīng)弧盒的圓心角（弧度）;(X新坐標(biāo)系工yz的原點(diǎn)（XOY平面平移JR? -（手后與z軸的交點(diǎn)）。X V n n 1 1 s s=arcsin 管tgx =a7rAx= -b2 -h 產(chǎn)丿臣二2 _ J R2 _ 口? _ 弧AEE與BDC之曲線方程式分別為：a：2+z2 = （R2-b2）（AEB）、 y2z2 = R2-a2 （BDC）1. 弧長(zhǎng)按下式計(jì)算：Sx = 2、/r2 _ b2Sy = 2R2 二a $ arcsin =7 R2 a2側(cè)面積按下式計(jì)算:Ax = (R2 - b2) arcsin 7= _ a JR2 - a2 - 62

11、JR1 - b2Ay = (R2 - a2) arcsin t=- - b JR2 - a2- b2 用- a23殼表面積按下式計(jì)算：A = Sx* Sy其一次近似值為：A =4aRarcsin= # = 4aR0y 其二次近似值為：人赳叫皿+川壽卜偵小+嚅鬥1-3-4-2橢圓拋物面扁殼（圖12）圖1-2橢圓拋物面扁殼訃算圖殼面方程式：Z吟X2 +樹2X、Y、Z在殼面上任一點(diǎn)對(duì)原點(diǎn)O的坐標(biāo)；2a對(duì)應(yīng)弧孟的弦長(zhǎng)；2b對(duì)應(yīng)弧BEC的弦長(zhǎng)；h x弓形ADB的高；hy弓形BEC的高。假設(shè)：Sx弧忑的長(zhǎng)；Sy弧EEC的長(zhǎng)；Ax弓形ADB的面積；Ay弓形BEC的面積。1. 弧長(zhǎng)按下式計(jì)算式中C1= J a

12、? + 4就W1 = 2t；c2= J b2 + 4hyb或者:Sx = 2 X 系數(shù) KaSy = 26X 系數(shù) Kb式中系數(shù)Ka、Kb可分別根據(jù)辭、空的值，査表1-76得到。2a 2b2殼表面積按下式計(jì)算A = Sx Sy3.側(cè)面積按下式計(jì)算A-= 3a9 孔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 =2d X 系數(shù) Ka x 2b X 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 Kb系數(shù)K,或 Kb系數(shù)K或 Kb曲系數(shù)K或 Kb餉舒系數(shù)K或心0.0501.00660.0801.016

13、80.1101.03140.1401.05000.1701.07240.0511.00690.0811.01720.1111.03200.1411.05070.1711.07330.0521.00720.0821.01770.1121.03250.1421.05140.1721.07410.0531.00740.0831.01810.1131.03310.1431.05210.1731.07490.0541.00770.0841.01850.1141.03370.1441.05280.1741.07570.0551.00800.0851.01890.1151-03420.1451.05350.

14、1751.07650.0561.00830.0861.01940.1161.03480.1461.05420.1761.07730.0571.00860.0871.01980.1171.03540.1471.05500.1771.07820.0581.00890.0881.02030.1181.03600.1481.05570.1781.07900.0591.00920.0891.02070.1191.03660.1491.05640.1791.07980.0601.00950.0901.02120.1201.03720.1501.05710.1801.08070.0611.00980.091

15、1.02170.1211.03780.1511.05780.1811.08150.0621.01020.0921.02210.1221.03840.1521.05860.1821.08240.0631.01050.0931.02260.1231.03900.1531.05930.1831.08320.0641.01080.0941.02310.1241.03960.1541.06010.1841.08410.0651.01120.0951.02360.1251.04020.1551.06080.1851.08490.0661.01150.0961.02410.1261.04080.1561.0

16、6160.1861.08580.0671.01180.0971.02460.1271.04150.1571.06230.1871.08670.0681.01220.0981.02510.1281.04210.1581.06310.1881.08750.0691.01260.0991.02560.1291.04280.1591.06380.1891.08840.0701.01290.1001.02610.1301.04340.1601.06460.1901.08930.0711.01330.1011.02660.1311.04400.1611.06540.1911,09020.0721.0137

17、0.1021.02710.1321.04470.1621.06610.1921.09100.0731.01400.1031.02760.1331.04530.1631.06690.1931.09190.0741.01440.1041.02810.1341.04600.1641.06770.1941.09280.0751.01480.1051.02870.1351.04670.1651.06850.1951.09370.0761.01520.1061.02920.1361.04730.1661.06930.1961.09460.0771.01560.1071.02970.1371.04800.1

18、671.07000.1971.09550.0781.01600.1081.03030.1381.04870.1681.0708 0.1981.09640.0791.01640.1091.03080.1391.04940.1691.07160.1991.0973查表說明例已知2d = 24.om, 2b=i6.om, hx=?.om, hy=2.8m,試求橢圓拋物面扁殼表面 積A。先求出hx/23 = ?O/24O = O125hy/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Xi.0765 = 429-99m21-3-4-4圓

19、拋物面扁殼(圖卜3)圖圓拋物面扁殼訃算圖殼面方程式：Zz=2R (X+Y?)式中 X、Y、Z在殼面上任一點(diǎn)對(duì)原點(diǎn)0的坐標(biāo);R半徑；假設(shè)2a對(duì)應(yīng)弧憶懇的弦長(zhǎng)；2b對(duì)應(yīng)弧嚴(yán):的弦長(zhǎng)；、Sx弧AGE的長(zhǎng)；Sy弧BDC的長(zhǎng)； h x 弓形AGB的高； h y 弓形BDC的高；Ax 弓形AGB的面積； Ay 弓形BDC的面積； f殼頂?shù)降酌婢嚯x； CAC的長(zhǎng)。則:c =2 J a1 + b2b一 2R1.弧長(zhǎng)按下式計(jì)算Sx =責(zé)、/爐+屏+ RIn責(zé)+卡加+訝 Sy =食 1r2 + 0 + R ln| + 寺丿爐 + 兩A = Sx Sy2. 殼表面積按下式計(jì)算3. 側(cè)面積按下式計(jì)算=Jbhy1-3-4-5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