常用面积体积公式

1-3常用求面积.体积公式图形尺寸符号 积（Att心 /A 0.707d = 1.414a = 1.414ZA在对角统交点上长方形4短边b 长边d 对角线在对角线交点上三角形h高I寺周长q、b、c对应角 A、B、I = “ ；C的边长GDyBDCD=DA平行四边形a、b邻边h对边间的距离A 二 b h = q 6sinaACBD2在对角线交点上CE = ABAF=CDa = CD （上底边）6 = AB （下底咎）h高KG =r半径 d直径p圆周长A = nr2 -三点24=0.785J2 = 0.07958/2p - nd在圆心上椭圆形a、b主轴在主轴交点G上A = Trs = r2a疋$一18（/r半径5 弧长a弧$的对应中心角r半径 s弧长 a中心角 b弦长 h高s = r，a * jig = 0175r a心吉普当a = 180时GO = = 0.4244r图形尺寸符号面积（A）重心（G）QS环31R外半径r内半径D外直径d内直径环宽 Dr平均直径A = n (- r2)弋(D2-d2)在圆心O部 分 圆 环釦R外半径r内半径D外应径d内直径一圆环平均半径i环宽A = (R2- r2”360 r 丿二述R .t180 wr3GO382館JsinZ x-_ a T新形OO = L圆心间的 距离d直径A =嵩a + sina)= r2-PRr-盂a+sinap值见F表02 -2PLd102dW3d104d105106d10Id1081010P0.400791.181.561.912.252.552.813.02物线形Ab底边h高1曲线长SA ABC的面积Z=762 + 1.3333A2A =bh诗s等 边 多 边 形a边长&系数.指多边形 的边数R外接圆半径Pi系数，（指正多边 形的边数A严Ki宀P,r2正三边形K = 0433P3-1.299正四边形 K4= 1.000. P4 = 2.000正五边形 K5= 1.720 P5 = 2.375正六边形 K6=2.598 P6 = 2598正七边形K73.634t P产2736正八边形 Ks = 4.828 Ps-2.828正九边形K9=6a82 P9=2893 正十边形 K)0 = 7.694, P10 = 2.939正十一边形 Kn = 9.364. Pn = 2.973 正十二边形 Kn=U.196 Pn = 3.000在内接 圆心或外 接圆心处1_32多面体的体积和表面积多面体的体积和表面积表1 - 74心(G)a棱d对角线S表面积S,侧表面积V = a3S = 6a2S, = 4a2在对角线交点上长方体(棱柱)a、b、h边长O底面对角线交点V = a *6 AS = 2 (qb七ah + bh)S = 2h (a + b)d = /a2+ Z2 +A2三梭柱a b、c边长h高A底面积O底面中线的交点V = A-AS = (a + b + c) *h + 2ASi = (a + 6 - c) *h”人S=n-/+ASi= n-/GO =f一个组合三角形的 面积n组合三角形的个数0底各对角线交 点V = yA (At +A：4 /A；A2)S = an + A】+ A2S| = anA】、A2两平行底面的面积A底面间的距离a个组合梯形的面积n组合梯形数GO = -4x A】+ 2 y)A2 + 3A2A1 + /A A 2 + A 2圆柱和空心圆柱(管R外半径r内半径t柱樂厚度P平均半径St内外侧面积圆柱：V=nR2-hS = 2irRh+2R2St = 2nRh 空心直圆柱t V 二汕(R2-r2)=ZRPthS = 2霖(R + r) h 2kx (R2-r2)S严2打(R + 刀hGO =今斜截直圆柱h最小高度h2最大高度r底面半径S =时(hl + 人2)+ 时2 丄) cosa /Si = nr (Ai + hi)GO =球扁形(球楔)r底面半径h高I母线长I - J r2+ h2S = S i + nr2R、r底面半径h高I母线V=y-(R2+r2+Kr)SlkI (R+r) 心丿(尺二厂)2*2 S=S】 +次(炉+ /)GO七4v R2 2Rr 3r2R2+R 十2r半径 d直径r球半径d弓形底圆直径h弓形高h球缺的高r 球缺半径d平切圆直径S曲曲面面积S球缺表面积S 二 4nr J 打V=yirr2A=2.0944r2hS = (4h 十 d)= 1.57r (4h 十 d)v=2( r-f) Sa -2vrh - ir (令十人2 |S nh (4r - h )X = 4h (2r-A)在球心上R圆环体平均半径 D圆环体平均直径 d圆环体截面直径 r圆环体戳面半径V =2n2R-r2=ir2ZW = 39.478Rr在环中心上GO = Ai + -yR球半径厂1、厂2底面半径h腰高心一球心O至带底 圆心0、的距 离v=y (3卄+ 3述+人2)S = 2 托 RhS = 2nRh * k (F)+ r|)D中间断面直径d底宜径I桶离在轴交点上对于抛物线形桶板:V=15X ( 2D2+Dd + yd2|对于圆形桶板：V =(2D?十屮)a、b、c半轴V = abcnS = 272-Va2 + b2在轴交点上交叉圆柱体r一圆柱半径 几、I圆柱长在二轴线交点上V =(2a + at)b十(2ai + a)ZiJa、b下底边长21 61上底边长h上、下底边距离（高）1-3-3物料堆体积计算物料堆体积计算表1-75图形计算公式I芝 4-號）XH物料自然堆积角JX tgaV =書(3b - a)1- a 丄Xor7Vo （延米体积）=盍+ bH - %tga1-3-4壳体表面积、侧面积计算1-3-4-1圆球形薄壳（图1-1）图1-1圆球形薄壳计算图球面方程式：X2+W + Z2 = R2 （对坐标系XYZ,原点在O）式中 R半径；x、y、z在球壳面上任一点对原点o的坐标。 假设 C弦长（AC）;2a弦长（AB）;2b弦长（BC）;F、GAB, EC的中点；f弓形AKC的高（K03;hx弓形AEE的高（EF）;Ay弓形BDC的高（DG）;Sx弧AEB的长；Sy弧BDC的长；Ax弓形AEB的面积（侧面积）；Ay弓形BDC的面积；2技对应弧益的圆心角（弧度）; 2肩对应弧蹴的圆心角（弧度）;of新坐标系發的原点（XOY平面平移血二石后与Z轴的交点）。duX V n n 1 I s sQ-R 6-Rx = arcsin *hx= y r1 -72 -丿应二尹以Ay = /R2 - a2 - J W _ a1 b?弧AEB-BDC之曲线方程式分别为:x2 + z2 = (R2-b2)(AEB)y2 + z2 = R2-a2 (BDC)1. 弧长按下式计算:S = 2 J R2 _ 护 arcsin / 幺 三JR2 - b2Sy = 2 v jR2 - a2 arcsin =：Jr2 - a2Ax = (R2 - 62) arcsin =z - a J R2 - a2 - b22侧面积按下式计算:JRZ _ hAy = (R2 - a2) arcsin (上二二-b JR2 - a2 - b2JR? - a13.壳表面积按下式计算:A = S/Sy其一次近似值为:A = 4aRarcsin =令=4aRy其二次近似值为:A =4 aRarcsin +a3b,“( asin人tg丸=4aRM1+1-3-4-2椭圆抛物面扁壳（图1-2）图1-2椭圆抛物面扁壳计算图壳面方程式：Z吟X2 +卽2X、Y、Z在壳面上任一点对原点O的坐标；2a对应弧孟的弦长；2b对应弧BEC的弦长；h x弓形ADB的高；hy弓形BEC的高。假设：Sx弧忑的长；Sy弧EEC的长；Ax弓形ADB的面积；Ay弓形BEC的面积。1. 弧长按下式计算Sx =ci + am iC2 + bmr式中Q=儿2 + 4疋c2= J b2 + 4hyb或者：Sx = 2x系数心Sy = 26X系数心式中 系数心、心一可分别根据筝、的值，査表1-76得到。 la 2b2. 壳表面积按下式计算A = Sx Sy3. 侧面积按下式计算A* =討 HAy =铁 hy1-3-4-3椭圆抛物面扁壳系数计算见图1-2,壳表面积（A）计算公式：A=S, * Sy=2aX 系数 K.tX2bX 系数仏式中K；,、仏一一椭圆抛物面扁壳系数，可按表1-76查得。椭圆拋物面扁壳系数表表1-76L或虹2a2b系数K.或 Kb系数K或 Kb务或發系数K或 Kb系数K或 KbAa 或 Ax2a2b系数 K或 Kb0.0501.00660.0801.01680.1101.03140.1401.05000.1701.07240.0511.0069 j0.0811.01720.1111.03200.1411.05070.1711.07330.0521.00720.0821.01770.1121.03250.1421.05140.1721.07410.0531.00740.0831.01810.1131.03310.1431.05210.1731.07490.0541.00770.0841.01850.1141.03370.1441.05280.1741.07570.0551.00800.0851.01890.1151.03420.1451.05350.1751.07650.0561.00830.0861.01940.1161.03480.1461.05420.1761.07730.0571.00860.0871.01980.1171.03540.1471.05500.1771.07820.0581.00890.0881.02030.1181.03600.1481.05570.1781.07900.0591.00920.0891.02070.1191.03660.1491.05640.1791.07980.0601.00950.0901.02120.1201.03720.1501.05710.1801.08070.0611.00980.0911.02170.1211.03780.1511.05780.1811.08150.0621.01020.0921.02210-1221.03840.1521.05860.1821.08240.0631.01050.0931.02260.1231.03900.1531.05930.1831.08320.0641.01080.0941.02310.1241.03960.1541.06010.1841.08410.0651.01120.0951.02360.1251.04020.1551.06080.1851.08490.0661.01150.0961.02410.1261.04080.1561.06160.1861.08580.0671.01180.0971.02460.1271.04150.1571.06230.1871.08670.0681.01220.0981.02510.1281.04210.1581.06310.1881.08750.0691.01260.0991.02560.1291.04280.1591.06380.1891.08840.0701.01290.1001.02610.1301.04340.1601.06460.1901.08930.0711.01330.1011.02660.1311.04400.1611.06540.1911.09020.0721.01370.1021.02710.1321,04470.1621.06610.1921.09100.0731.01400.1031.02760.1331.04530.1631.06690.1931.09190.0741.01440.1041.02810.1341.04600.1641.06770.1941.09280.0751.01480.1051.02870.1351.04670.1651.06850.1951.09370.0761.01520.1061.02920.1361.04730.1661.06930.1961.09460.0771.01560.1071.02970.1371.04800.1671.07000.1971.09550.0781.01600.1081.03030.1381.04870.1681.07080.1981.09640.0791.01640.1091.03080.1391.04940.1691.07160.1991.0973查表说明例已知2a = 240nb 2b = 16. Om, hx=3. Om, hy=2. 8m,试求椭圆抛物面扁壳表面积A。先求出 hx/2a=3. 0/24. 0=0. 125h、./2b = 2 8/16.0 = 0. 175分别查表得系数K；,为1.0402和系数心为1.0765,则扁壳表面积A=24. OX 1. 0402X16. 0 XI. 0765=429. 99m21 -3-4-4圆抛物面扁壳(图1-3)图1-3圆抛物面扁壳计算图式中X、Y、Z在壳面上任一点对原点0的坐标;R半径；X假设2a对应弧AGE的弦长； 2b对应弧赢的弦长；Sx弧AGE的长；S弧BDC的长； hx弓形AGB的高；hy弓形BDC的高； 入弓形AGB的面积；Ay弓形BDC的面积； f壳顶到底面距离； cAC的长。则：C =2九2 +护壳面方程式：Z =(X2 + Y2)8R2R_ 一 2R1.弧长按下式计算Sx =貰 vZK2 + a2 + R 叫責 + * Jr2 + aSy = |- JR? + 0 + 尺】n怯 + *+32.壳表面积按下式计算A = Sx * Sy3.侧面积按下式计算A2启 4A2b34“Ay _ 3R _ 3 bhy1-3-4-5单、双曲拱展开面积1. 单曲拱展开面积=单曲拱系数X水平投影面积。2. 双曲拱展开面积=双曲拱系数（大曲拱系数X小曲拱系数）X水平投影面积。单、双曲拱展开面积系数见表1-77。单双曲拱展开面积计算图见图1-4。图1-4单、双曲拱展开面积计算图L-拱跨；F-拱高单、双曲拱展开面积系数表表1-77单曲拱F/L1/21/31/41/51/61/71/81/91/10f/l系数单曲拱系数1.501.251.151.101.071.051.041.031,02双曲拱系数1/21.502.251.8751.7251.6501.6051.5751.5691.5451.5301/31.251.8751.5631.4381.3751.3381.3131.3001.2881.2751/41.151.7251.4331.3231.2651.2311.2081.1961.1851.1731/51.101.6501.37512651.2101.1771.1551.1441.1331.1221/61.071.6051.3331.2311.1771.1451-1241.1131.1021.0911/71.051.5751.3131.2031.1551.1241.1031.0921.0821.0711/81.041.5601.3001.1961.1441.1131.0921.0821.0711.0611/91.031.5451.2881J851.1331.1021.0821.0711.0611.0511/101.021.5301.2751.1731.12211.0711.0611.0511.040