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1、用求面积、体积公式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形尺寸符号面积(A)重 心(G)正形0a 边长d对角线A = /a =a/A = 0.707drf = 1.414a = 1.414-ZA在对角线交点上a短边b长边 d对角线在对角线交点上h 高Z+周长Ja b、c对应角 A、B、C的边长A bh 1 尸A = = rabsnC匕. a + b + cGD = *BDCDDA平行四边形Q、b邻边h对边间的距离A b h = a 6 sin aAC-BD . Zsm# 乙在对角线交点上CE = ABAF = CDa = CD （上底边） b = AB （下底边） h高
2、hHG =KG =h a + 2ba + bh 2a + b - * 3 a + br半径d直径P圆周长d 1A = nr - itd4= O.785d2 = 0.O79582在圆心上A n ,A = ra b4在主轴交点G上r半径5弧长a弧s的对应中心角r1802 rb3 s当u = 90时GO = .6r3 兀r半径s弧长a中心角b弦长ha 121 a?r .A = Tr I 面 Fna）1 l-r= Er （5-6） + bh JJjrs - r* a ,gQ = 0.0175ra九=1m 1 b2 GO=12，T当a = 180时GO =变=0.4244 厂图形尺寸符号面积(A)重 心
3、(G)圆环十r如-Mr-R外半径 r内半径D外直径 d内直径 r环宽Dp：平均直径A =兀(R2 r2)=奇(D2-d2)在圆心O部 分 圆 环争。R外半径r 内半径D外直径d 内直径Ra圆环平均半径t环宽A嗤()=F? 、. f180 PJR3 - r3 GO 38.2 土 2 a sin Xa T新 月 形OOi = L圆心间的 距离d直径A r 兀一T?7a + sin a 1loUI=r2*PK 18Qa + sinaP值见下表Ld_102d103d104d105d106d10Td108d109d10P0.400.791.181.561.912.252.552.813.02抛 物 线
4、形4-4-b底边h高1曲线长SA ABC的面积1= V 62+ 1.3333A2A = h3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，t指正多边 形的边数Af = Kf a? = pr2正三边形 K3 = 0.433, P3 = 1.299 正四边形 K4= 1.000, P4 = 2.000 正五边形 Ks = 1.720 0 = 2.375 正六边形 K6 = 2.598, P6 = 2.598 正七边形 K7-3.634, P7 = 2.736 正八边形 珞=4.828, P8 = 2.828 正九边形 K9 = 6.182 P9 = 2.893 正十边形 K
5、io = 7.694, P10 = 2.939 正十一边形 Ku = 9.364, Pn-2.973 正十二边形 K 12=11-196, Pl2 = 3.000在内接 圆心或外 接圆心处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1-74心(G)a棱d对角线S表面积S1侧表面积V = a3S = 6a2S1 = 4a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 b * hS = 2 (a * b + a h + b h)Si = 2h (a + b) d= -J a2b2 + h2GO = - 乙GO = a b、c边长h高A底面积O
6、底面中线的交点V = A*hS = (a + b + c) h + 2ASi = (a + b + c) * hf一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VA-hS = n /+ ASi= ,/GO = *V = h (At + A2+A1A2)S = an + Ai + A?Si = anA】、A2两平行底面 的面积h底面间的距离a一个组合梯形的面 积组合梯形数GO = 4、了 Ai + 2 JAA2 + 3A2 Ai + /A1A2 +A?R外半径r内半径t柱壁厚度P平均半径 S内外侧面积GO-3-圆柱：V = irR2t hS 2jvRh + 2ttR2Si = 2 冗R
7、h空心直圆柱:V =祐(K2- r2)=2nRPthS = 2n (R + r)九+ 2ttx (R2-r2)Si = 2tt (R + r) hhx最小高度h2最大高度r底面半径S = Jtr (Ai + 为)+ nr2Si = nr (hi + hy)八八 Al + 2GO = . L4+4 (妇 + h2)1 r2GK = *77tgr底面半径h高I母线长V = -jrr2hS = jrr Vr2 + h2 = trrl 1=7 r2 + h2S = S + Jrr2GO = 4R2 + 2Rr + 3r2R2 + Rr+ r2R、r一底面半径h高I母线V- ,(K2 + r2 + Rr
8、)Sj = nl (R + r)l J (/? - r )2 + A2S-Si + n- (K2+ r2)r半径d直径V = r ?rr23=晔-=0.5236/3 o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=y?rr2/i = 2.0944r2/iS =竽(4A + n=Go22 r +21 r对于抛物线形桶板:D中间断面直径 d底直径 I桶高Trz-15 -Vx ( 2D2+ Dd + -d2)对于圆形桶板：V =(2D2 + d2)JL Xr在轴交点上a、b、c半轴V=yatoS = 2a/2 b , a/ a2 + 62在轴交点上交叉圆柱体r-圆柱半径Zi、I圆柱长在二轴
9、线交点上下底边长V - (2a + ai)b上底边长+ (2ai + a)6iJ一上、下底边距= ab + (a + aQ (b +离（高）0bi) + Qi 如3物料堆体积计算物料堆体积计算见表1-75。4壳体表面积、侧面积计算球形薄壳（图1-1）图1-1圆球形薄壳计算图球面方程式：X2 + Y2 + Z2 = R2 （X4*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）;26弦长（BC）;F、GAB,的中点；f弓形AKC的高（KO）;hx弓形AEB的高（EF）；hy弓形BDC的高（DG）;尸fSx弧AEB的长；Sy弧BD
10、C的长；Ax弓形AEB的面积（侧面积）；Ay弓形BDC的面积；2化对应弧;S的圆心角（弧度）；2夕y对应弧敲:的圆心角（弧度）；0新坐标系工四的原点（X0Y平面平移Jr2 一 （言）2后与z轴的交点）。则R = g+#OJ N. j Qsinp = 7Tx = arcsiny = arcsintcr = h尸扁序寸译、=JR2 a2 JR2 - / 一 力2弧蓝与长之曲线方程式分别为：x2+ z2=(R2b2)(AEB)y2+ z2 = R2-a2 (BDC)1. 弧长按下式计算：Sx = 2 VR2 - b2 , arcsin . a Sv = 2 Jr2 _ / . arcsin ,y2
11、.侧面积按下式计算：Ax = (R2 - b2) arcsin- a JR2 - a2 - 62Av = (R2_/) . arcsin、 二二-b JR2 - a1 - b2 y岳-a13.壳表面积按下式计算：A = SxS其一次近似值为:A = 4aR arcsin 二合=4aR其二次近似值为:A = 4 aR arcsin4-2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 = 2 +知 2X、y、Z在壳面上任一点对原点O的坐标；2a对应弧ADB的弦长；2b对应弧BEC的弦长； 五x弓形ADB的高； hy弓形BEC的高。假设：Sx 弧ADB的长；Sy弧BEC的长；A
12、*弓形ADB的面积；Ay弓形BEC的面积。1.弧长按下式计算Sx = ci + amnm ia(Ic iSv =+ bm21n + 5y m2 bl式中门=+C 2= J I）。+ 4/i*m盐或者：Sx = 2aX系数兀Sy = 2bx 系数 Kb式中 系数Ka、Kb可分别根据多、号的值，查表1-76得到2. 壳表面积按下式计算3. 侧面积按下式计算Ax 3 Q h*Ay = 6 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:4=乩 =2系数KaX2bx系数Kb式中Ka、椭圆抛物面扁壳系数，可按表1-76查得。椭圆抛物面扁壳系数表表1-76h V 2a2b系数
13、皿或Kb2a2b系数 兀或珞2a2b系数Ka 或 KbAx 2a2b系数 &或Kb心或上 2a i 2b系数Ka或K0.0501.00660.0801.01680.1101.03140.1401.05000.1701.07240.0511.00690.0811.01720.1111.03200.1411.05070.1711.07330.0521.00720.0821.01770.1121.03250.1421.05140.1721.07410.0531.00740.0831.01810.1131.03310.1431.05210.1731.07490.0541.00770.0841.0185
14、0.1141.03370.1441.05280.1741.07570.0551.00800.0851.01890.1151.03420.1451.05350.1751.07650.0561.00830.0861.01940.1161.03480.1461.05420.1761.07730.0571.00860.0871.01980.1171.03540.1471.05500.1771.07820.0581.00890.0881.02030.1181.03600.1481.05570.1781.07900.0591.00920.0891.02070.1191.03660.1491.05640.1
15、791.07980.0601.00950.0901.02120.1201.03720.1501.05710.1801.08070.0611.00980.0911.02170.1211.03780.1511.05780.1811.08150.0621.01020.0921.02210.1221.03840.1521.05860.1821.08240.0631.01050.0931.02260.1231.03900.1531.05930.1831.08320.0641.01080.0941.02310.1241.03960.1541.06010.1841.08410.0651.01120.0951
16、.02360.1251.04020.1551.06080.1851.08490.0661.01150.0961.02410.1261.04080.1561.06160.1861.08580.0671.01180.0971.02460.1271.04150.1571.06230.1871.08670.0681.01220.0981.02510.1281.04210.1581.06310.1881.08750.0691.01260.0991.02560.1291.04280.1591.06380.1891.08840.0701.01290.1001.02610.1301.04340.1601.06
17、460.1901.08930.0711.01330.1011.02660.1311.04400.1611.06540.1911.09020.0721.01370.1021.02710.1321.04470.1621.06610.1921.09100.0731.01400.1031.02760.1331.04530.1631.06690.1931.09190.0741.01440.1041.02810.1341.04600.1641.06770.1941.09280.0751.01480.1051.02870.1351.0467 .0.1651.06850.1951.09370.0761.015
18、20.1061.02920.1361.04730.1661.06930.1961.09460.0771.01560.1071.02970.1371.04800.1671.07000.1971.09550.0781.01600.1081.03030.1381.04870.1681.07080.1981.09640.0791.01640.1091.03080.1391.04940.1691.07160.1991.0973b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 h /2a=3.0/24.0 = 0.125xhy
19、/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=24.0X 1.0402 X 16.0X1.0765=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壳面方程式：Z =(X2+ Y2)式中 X、y、Z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；2b对应弧的弦长；Sx一弧AGB的长；Sy弧BDC的长；4弓形AGB的rj;hy一弓形BDC的高；Ax一弓形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 Ja2 + b2c28Rh 二y 2R1.弧长按下式计算
20、Sx =竟 JR2 + a2 + R In（言 + % JR? +质| ISy = * Jr2 + b2 + R Tn g + * Jr? + 叫 2.壳表面积按下式计算3.侧面积按下式计算=Sx SyAxAy2a33R2b33R1-3-4-5单、双曲拱展开面积1. 单曲拱展开面积=单曲拱系数乂水平投影面积。2. 双曲拱展开面积=双曲拱系数（大曲拱系数X小曲拱系数）乂水平投影面积。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
21、/10f/l系数单曲拱系数1.501.251.151.101.071.051.041.031.02双曲拱系数1/21.502.251.8751.7251.6501.6051.5751.5691.5451.5301/31.251.8751.5631.4381.3751.3381.3131.3001.2881.2751/41.151.7251.4331.3231.2651.2311.2081.1961.1851.1731/51.101.6501.3751.2651.2101.1771.1551.1441.1331.1221/61.071.6051.3331.2311.1771.1451.1241.1131.1021.0911/71.051.5751.3131.2031.1551.1241.1031.0921.0821.0711/81.041.5601.3001.1961.1441.1131.0921.0821.0711.0611/91.031.5451.2881.1851.1331.1021.0821.0711.0611.0511/101.021.5301.2751.1731.1221.0911.0711.0611.0511.040
