1-3常用求面积、体积公式

1、1-3 常用求面积、体积公式1-3-1平面图形面积平面图形面积见表1-73。平面图形面积表1-73圈形尺寸符号面积(A)重心(tn正肓形因a边抚d对角线1'A = fl2a=TA=0.707dci = L414ci = 1.414 ZA.在对划线交点上A = a *6d = x/ b2在对角线交点上h高l寺周长a、b、c对应角A、B、C的边长A =号=*absinCGD = *BDCD=DA平行四边形/4. b邻边h对边间的距离A = h = a bsinaACBD2在对角线交点上ECE = ABAF=CDa = CD （上底边）6 = AB （下底边）h高a十26r半径d直径P圆周长

2、A = nr2 - 4=0.785/2 = 0 07958p2p- nd在圆心上“、b主轴在主轴交点G上F半径S弧长a弧s的对应中心角GO甘呼当a = 90时刍.各0.6,a短边b长边d对角线r-a = 0.0175r-a°°12 A当a = 180*时GO = TC = 0.4244r3kr一半径5弧长a中心角b弦长h高图形尺寸符号面积(A)重心(G)圆环十r> -Mr-R外半径r内半径D外直径d内直径t环宽Dw平均直径A =n (R2-/2)- (D2-d2)在圆心O部 分 圆 坏R外半径r内半径D外直径d内直径Rpj-圆环平均半径i环宽”360 'r &

3、gt;ax门p3 - r3GO 一 38.2® ；2sin号XaT新月形OO = L圆心间的距离d直径A击 a + sinj= r2.pR= re -+ sinaP值见F表O2 = S#)ZL2d10104d10106d10Id108d109d10P0.400.791.181.561.912.252.552.813.02物线形b底边h-高1曲线长SABC的面积Z=丿Q+ 1.3333人 2A亠S3 d等 边 多 边 形a边长Kt系数，i指多边形的边数R外接圆半径P,系数，r指正多边形的边数AKra2=PrR2正三边形 K3 = 0.433, Pa-1.299正四边形K4= 1-000

4、.卩产2000正五边形K5= 1.720> P产2.375正六边形K6 = 2.59& P6 = 2598正七边形K7 = 3.634t卩产2736 正八边形Kg = 4.82& Pg = 2828 正九边形 心=6182, P9 = 2893 正十边形Kio = 7694. Pio = 2939 正十一边形 Kn = 9.364, Pn = 2.973 正十二边形 Kn= 11.196, Pu3.000在内接 圆心或外 接圆心处1-3-2 多面体的体积和表面积多面体的体积和表面积见表 1-74。多面体的体积和表面积 表 1-74a棱d对角线S表面积Si侧表面积V = a

5、3S = 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 9 b* hS = 2 (ab十ah + bh)Si = 2h (a + b)J-丿/宀以三棱柱GO = yax b、c边长h高A底面积O底面中线的交点V= A-AS = (a + 6+c) h + 2ASi = (a + b + c) AS= «/+ASi = n/f个组合三角形的面积n组合三角形的个数O锥底各对角线交点台V = h (At + A2+ /a2)S = an + A】+ A2Si = anA】、A2两平行底面的面积A底面间的距离a一个组合梯形的血 积H组合梯形数GO = 4

6、* Ai + 2 J瓦A2TA2A i + y/ AjAj + A 2圆柱:圆柱和空心圆柱(管)R外半径r内半径/柱壁厚度P平均半径S】一内外儷面积S = 2ttRh +2肿Si = 2 乐 Rh空心直圆柱：V -nh (R2-r2)=InRPthS-2n (R r) h + 2kx (应-厂2)Sl2兀(R+r) hGO = y斜截直圆柱h i最小高度A 2最大高度r底面半径S = nr (人+人2)+ 时$Si = nr (hi + 人2)r底面半径h高I母线长SI = nr a/f2 + h 2 = nrll = y r2+ h2S = S i + nr2R、r底面半径h高I母线V=ye

7、(R2 + r2 + Rr)Sx=zrZ (R+r)I = -/(R - r)2 h2S = Si + ” (K2+f2)r半径 d直径V =寺2=誓= 0.5236/S = 4/rr2 = Kd2在球心上球扁形(球楔)r球半径d弓形底圆直径h弓形高V=y7rr2fc=2.0944r2hS =十/)= 1.57r (4A + J)h球缺的高r球缺半径d平切圆直径S筒曲面面积S球缺表面积g剧L专) 冷曲=2nrh =凭(+ h2jS nh (4r - h )dh (2r-h)体积（V） 底面积（A）表面积（S）侧表面积（SJ在环中心上R一圆环体平均半径 D 圆环体平均直径 d圆环体截面直径 r圆

8、环体截面半径V =2异&“=n2Dd24S = 4k2 Rr= 2EW = 39.478BrR球半径 厂1、厂2底面半径h讓高hx球心0至带底 圆心6的距 离V=y (3r? + 3ri+A2)Si = 2鈕S = 2nRh + x ( r| + r|)D中间断面直径 d底直径桶高对于抛物线形桶板：x（2D2+LW + yrf2）对于圆形桶板：在釉交点上鸟削（21）2 十/2）a、b、c半轴V = abcnS = 2>/2 - b J+ b2在轴交点上交叉圆柱体r圆柱半径h. I圆柱长在二轴线交点上s b下底边长21S b上底边长h上.下底边距离（高）V =(2a+ ax)d*十

9、(a + s)(6 +1-3-3物料堆体积计算物料堆体积计算见表1-75。1-3-4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 + YZR2 （对坐标系XYZ,原点在0）式中R半径；x、y. z在球壳面上任一点对原点o的坐标。殳 £一0 F弦lx (AC); 己扩匸 f 4 u X T、厶 jT-izS / j26弦长(BC)iF、GABt BC的中点；f弓形AKC的高（KOJ）；一弓形AEB的高（EF）；弓形BDC的高（DG）;S,弧AEB的长;Sy-一弧另DC的获；Ak弓形AEB的面积（侧面积”后与Z轴的交点）。Ay弓形BD

10、C的面积；2仪一对应弧AEB的圆心角（弧度）；2帕一对应弧赢?的圆心角（弧度）；O'一新坐标系XVZ的原点（XOY平面平移X2a-RARn n1 s s a =arcsin 三K=arcsinhy= 匸了- y/R2-a2-b2弧AEB与BDC之曲线方程式分别为：x2 + z2=(R2-62)(AEB)y2- z2 = R2-a2 (BDC)1弧长按下式计算：Sx = 2 -JR2 - b2 arcsin r. frSy = 2 -JR2 - a2 arcsin y丿R2十22. 侧面积按下式计算：Ax = (R? - 62) arcsin a 7R2 - a2 b2金- b2Ay =

11、 (R2 - a2) arcsin ,- - b 71?2 - a2 -VR2 - a23. 壳表面积按下式计算：A = Sx-Sy其一次近似值为:LA = 4oR arcsin = _ = 4 aR 忙其二次近似值为:A叫品品"灵亦司十阴1-3-4-2椭圆抛物面扁壳（图1-2）|z图1-2椭圆抛物面扁壳计算图壳面方程式:X. Y、Z在壳面上任一点对原点0的坐标；2a对应弧ADE的弦长；2b对应弧BEC的弦长；h装弓形ADB的高；hy弓形BEC的髙。 假设：Sx弧忑的长;Sy弧EEC的长；弓形ADB的面积;Ay弓形EEC的面积。1. 弧长按下式计算式中或者:SI = 2ax系数KSy

12、2&x系数心= ci + am Jn1- 一 mi a+m2 o J式中系数K外Kh可分别根据尹、的值，查表1-兀得到。3 2b2. 壳表面积按下式计算A = SH * Sy3. 侧面积按下式计算1-3-4-3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= SX Sy = 2ax系数 KaX 2bx系数 Kb式中 Ka、Kb椭圆抛物面扁壳系数，可按表 1-76查得。系数或瓦Aa歳乩2a 2b系数 K.或$Aa歳2a2b系数 K.或心Aa或H2a *2&垂数 K,或 KbAa或虹2a2bM数 氐或心0.0501.00660.0801.0168OJW1.03140.

13、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.018L0.1131.03310.1431.05210.1731.07490.0541.00770.0S41.01850J141.03370.144L05280.1741.07570.0551.00800.0851.01890.1151.03420.1451.05350.1751.07650.0561.0

14、0830.0861.01940.1161.03480.146L05420.1761.07730.0571.00860.0871.01980.1171.03540.1471.05500.1771.07820+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 0911.02170.1211.03

15、780.1511.057S0.1811.08150.0621.01020.0921.02210.122L03B40,1521,05860.1821,08240.0631.01050.093L.02260.1231.03900.1531.05930.1831.08320.0641.010S0.0941.02310.1241.03960.154L06010.1841.08410.0651.G1120.0951.02360.1251.04020.1551.06080.1851.08490.0661.01150.0961.02410.1261440801561.06160,1861.08580.067

16、1.011&0.097L.02460.1271.04150.1571.06230.1871.08670.0681,01220.0981.02510.1281.0421OhISSL06310.1881.08750.069L01260.09»1.02560J291.0428UJ591.063B0.1891.08840.0701.01290.1001.02610.1301.04340.1601.06460.1901.W30.0711.0133oaoiL02660.1311.04400161L065401911.09020.0721.01370.1021.02710.1321.044

17、70.1621.06610.1921.09100.073J.0140oao31.02760,1331.04530.1631.06690.1931.09190.0741.01440.1041.02810.1341.04600.1641.06770.1941.09280.0751.01480.1051.02&70.1351.04670.1651.068$0.195IM370.0761.01520.106L.02920.1561.04730.1661.06930.1961.09460.077I.015&0.1071,0270,1371.04SO0.1671.07000.1971.09

18、550.0781.01600.1081.Q3030.1381.04870.1681,07080.1981.09640.0791.01640.1091.03080.1391.04940.169L07160.1991.0973查表说明例已知 2a= 24.0m, 2b= 16.0m, hx = 3.0m, 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= 0.125hy/2b = 2.8/16.0= 0.175分别查表得系数Ka为1.0402和系数Kb为1.0765,则扁壳表面积A = 24.0X1.0402X16.0 X 1.0765= 429.99m21

19、-3-4-4圆抛物面扁壳（图1-3）图1-3圆抛物面扁壳计算图壳面方程式：z = （x2+ y2）式中 X、丫、Z一在壳面上任一点对原点O的坐标;R 一半径；假设2a对应弧的弦长；26 一对应弧蹴的弦长；邑一一弧AGB的长；Sy弧BDC的长；曇一弓形AGB的高，弓形BDC的高；去一弓形AGB的面积；A厂弓形BDC的面积；fC一壳顶到底面距离；AC的长。则：cljab2f =c28R民X二a22RZly =b22RI-弧长按下式计算S” =責VJ?2 + a2 + K 山冷 +寺% =董 JR2 + b2 + R X诘 + 影 JR + b22.壳表面积按下式计算A = Sx * Sy3-侧面积按下式计算A2a34 ,A-3R = 3 朋耳a2胪4 ,Ar 3R亍叭1-3-4-5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