常用求面积体积公式五

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图形尺寸符号面积(A)重 心(G)正A = a2方1/4:一对角线。=0.7078在对角线交点上形drf = l,414a = 1.414ZA长 方 形国。a短边b长边d对角线A = q 6d = J a2 + b2在对角线交点上角形ih高/£周长4、6、C对应角A、B、 C的边长A = 祟 = a6sinC.a + a，c-2GO = 43DJCD=DA平行四边形MlA b D十、b邻边h对边间的距陶A = b h = a OsinaAC BD . o 1- 2 一smp在对角线交点上梯

2、形V JMC BCE= AB AF=CD a=CD （上底边） 6 = AB （下底边） .h高a a + 6 lA - 2 1HG = A.a±24 3 a bKG=勺汽T圆形$3r半径d直径P圆周长A = nr2 =；仅产4= 0.785, = 0.07958/p-nd在圆心上楠 圆形、b主轴AkiA = -ra-6 4在主轴交点G上扁形气 0r半径s孤长a弧s的对应中心角A = »"急"24-180r60 =春卫3 s当a = 90°时GO - g0673 n弓形r半径s弧长a中心角b弦长h高A = /户（费一加J= ；tr （s-6）十

3、 6九 Ds = r*ae77Z =0.0175r loUh = r- J/-/8 =盍，号当a =180时GO = g = 0.424”3图形尺寸符号面积(A)重 心(G)圆环0i十rQ Q>-Mr-R外半径 r内半径D外直径 d内直径C环宽Dpj平均直径A = K （r2-产）=-f（D2-a2）在圆心O部分P力卜期F径 性径 茎径A =里工（r2_小a 360 '）8-38.2sin x-a Tk7r=ir内斗D外工四环d内豆俭R力一圆环平均半径/环宽=-t180 p;新 月 形OOi = L圆心间的距离d直径A = r2（，r-180a + sino）=r2*P昼=K _

4、 if6a +p值见下表OiG-巴尸LI 2d103d1044105d106d10Id108d109d10P0.40|0.791.181.561.912.252.552.813.02抛 物 线 形小 4。4-b底边h1曲线长SAABC的面积2=，户+ 1,3333标A =%b、hW-s等 边 多 边怂a边长K.系数，i指多边形 的边数R外接圆半径P.系数,£指正多边 形的边数Aj=K"2=p,R2正三边形 K3 = 0.433, P3= 1.299 正四边形 K,= 1.000, P4 = 2.000 正五边形Ks = L720, P$ = 2,375 正六边形 K6 =

5、2.598, P6 = 2.598 正七边形K7 = 3.634, P产2.736 正八边形 K& = 4.828, Ps = 2.828在内接 圆心或外 接圆心处形正九边形 K9=6.182,29=2.893正十边形 Ki0 = 7.694, Pio = 2.939正十一边形 Kn = 9.364, Pn =2.973正十二边形 Kt2Mli 196, Pi2=3.0001-3-2多面体的体积和表面积多面体的体积和表面积见表l-74o多面体的体积和表面积表1-74图形尺寸符号体积（V） 底面积（A） 表面积（S）侧表面积（Si）重心（G）立 方体Q梭d对角线S表面积S1到表面积V=a

6、3S = 6a?Si =4a2在对角线交点上I长方体(棱柱)2 a1«、8、h边长O底面对角线交点V= a6入S = 2 (a，b + bh)Si = 2h (a + b)d = Va2 b2 h2GO = y柱<1Ja、b、c边长h高A底面积O底面中线的交点Y= QhS = (a + 6 + c) A + 2ASi= (a + 6 + c)，hGO = y械锥/一个组合三角形的 面积n组合三角形的个数O一锥底各对角线交 点V=A-AS=/+AS】=fGO = 4梭台g 4A）、Aj两平行底面的面积A底面间的距离a一个组合梯形的面积n组合梯形数V = A (Aj + A2+ J

7、 AiAi)S = a。+ A1+ A2Si = anGO = v4x Ai + 2 J A)A2 + 3A2A + J A1A2 + A2圆柱和空心圆柱(管)*111 (J F1G I 1 .LR外半径'r内半径，柱壁厚度P平均半径 S1内外侧面积圆柱：V=;rR2八S = 2kRIi + 2kR2St = 2Rh 空心直圆柱：V = M (R2 户)= 2nRPthS = 2rr (R + r) h + 2k x (R2-2)Si = 2jt (R + r) hGO = y图形尺寸符号体积（V） 底面积（A） 表面积（S）侧表面积（S。重心（G）斜截直圆柱XTf11r五1最小高度入

8、2最大高度r底面半径V-12S =(h + A2) + *弃X ( 1 + ) cosa !Si = nr (hi + &2)GO = 44 (ht + hz)_L 户0GK- 2 % + /12 18a直 圆 傩一旗r底面半径h 高/母线长V = nr2hSi =仃 Vr2 h2 = rrl l - y r2 + h2S = S + Kr2GO44圆台4ItR、r底面半径h高1母线V=号(火2+ r2+ Rr)Si " nl (R + r) 1= J(Rr)2+M S = Si +万(R2r2)GO = 4R2 + 2Rr3r2R2 + & + r2球r半径 d直径

9、V =告括产= = 0.5236J3 0S = 4b2=次舒在球心上球扁形(球楔)r球半径d弓形底圆直径h 弓形高V=yKr2A=2.0944r2hS =苧(4九 + d)= 1.57r (4人+d)GO = T(r-f)球缺(h球缺的高r球缺半径d平切圆直径S岛曲面面积S球缺表面积V =2 r - -3)S住-2nrh =笊(+ ft2)S = nh (4r - h)dh (2r-h)c 3 (2r - h)2 4 3r - h夕尺寸符号体积（V） 底面积（A） 表面积（S）俺表面积（Si）重心（G）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圆环体截面半径V =2MR /4S

10、=4/&= MEU = 39.478Rr在环中心上交叉圆柱体R球半径riA r2底面半径h腰高h球心。至带底圆心Oi的电 离V=R (3rJ + 3r| + A2) bS = 2KRhS = 2nRh + rr ( + r|)GO = hi + -1-n士而断面直怒对于抛物线形桶板：V*15T alWiW 且生X ( 2D2+ Dd + yd2)d-一底直径在轴交点上一一桶高对于圆形桶板：v=gl （2。2 + 笛）a s 6、c半轴r圆柱半径 匕、I圆柱长a、b下底边长打、bi上底边长h上、下底边距离（高）V = abcnS = 2/2-6- Va2b2V = -(2a+ai)6o+

11、 (2ai + a )61=" + ( a + °i)(6 +obi) + 2 J在轴交点上在二轴线交点上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球形薄壳(1-1)图17圆球形薄壳计算图球面方程式：X2+ Y2 + Z2=R2 (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。假设 c弦长(AC);2a弦长(AB);2b弦长(BC);F、GAB,的中点；f弓形AKC的高(K0z);hx弓形AEB的高(EF)；%弓形BDC的高(DG);Sx弧AEB的长；Sy弧BDC的长；Ax弓形AEB的

12、面积(侧面积)；Ay弓形BDC的面积;2 6 x对应弧AEB的圆心角（弧度）;2%对应弧BDC的圆心角（弧度）;O'新坐标系zyz的原点（XOY平面平移JR?一2后与Z轴的交点）。R = 2 + 上R-8f 2 而九=石 siny = ->a>>x = arcsin 记% = arcsin 记弧AEJ3与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:Sx = 2 JR2 - b2 arcsinSy = 2 /r2 - a2 arcsinJR2 - b2b2 .侧面积按下式计算:Ax

13、= (R2 - b2) arcsinAy = (R2 - a2) arcsinJr2 _ b2ba J R2 - a2 - b2- 6 J R2 a2 b23 .壳表面积按下式计算:A = SX9 Sy其一次近似值为:A = 4ajRarcsin=/=4aR6y其二次近似值为:A = 4 aR arcsin g +Ka3b= 4aR“l +asin6x，tg6y6R6y1-3-4-2椭圆抛物面扁壳（1-2)z图1-2椭圆抛物面扁壳计算图壳面方程式：Z = X2 + t?Y2a bX、丫、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；力x弓形ADB的高；hy弓形

14、BEC的高。假设：Sx弧病的长；Sy弧BEC的长；Ax弓形ADB的面积;Ay弓形BEC的面积。1，弧长按下式计算SxSy =。2 + 6m21n工+孑m2 o式中C2 = y 62 +4/lyb7n2 =西或者:式中Sx = 2a x系数降Sy = 26x 系数 Kbh hy.系数"Kb可分别根据冰前值，查表76得到。2 .壳表面积按下式计算A = Sx Sy3 .侧面积按下式计算b 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=Sx Sv=2a>< 系数 KX2bx 系数长式中 Ka、Kb-椭圆抛物面扁壳系数，可按表1-76查得。椭圆抛

15、物面扁壳系数表表1-76b或4r2G 次 26系数K8 或 Kb髭或Ax2a 取26系数K.或 Kb出系数K.或 Kb聂脸系数Ka 或 Kb2a 取26系数K.或 Kb0.0501.00660.0801.01680.1101.03140.1401.05000.1701.07240.0511.00690.0811.01720.1111.03200.1411.05070.1711.07330.0521.00720.0821.01770.1121.03250.1421.05140.1721.07410.0531.00740.0831.01810.1131.03310.1431.05210.1731.

16、07490.0541.00770.0841.01850.1141.03370.1441.05280.1741.07570.0551.00800.0851.01890.1151.03420.1451.05350.1751.07650.0561.00830.0861.01940.116E03480.1461.05420.1761.07730.0571.00860.0871.01980.1171.03540.1471.05500.1771.07820.0581.00890.0881.02030.1181.03600.1481.05570.1781.07900.0591.00920.0891.0207

17、0.1191.03660.1491.05640.1791.07980.0601.00950.0901.02120.1201.03720.1501.05710.1801.08070.0611.00980.0911.02170.1211.03780,1511.05780.1811.08150.0621.01020.0921.02210.1221.03840.1521.05860.1821.08240.0631.01050.0931.02260.1231.03900.1531.05930.1831.08320.0641.01080.0941.02310.1241.03960.1541.06010.1

18、841.08410.0651.01120.0951.02360.1251.04020.1551.06080.1851.08490.0661.01150.0961.02410.1261.04080.1561.06160.1861.08580.0671.01180.0971.02460.1271.04150.1571.06230.1871.08670.0681.01220.0981.02510.1281.04210.1581.06310.1881.08750.0691.01260.0991.02560.1291.04280.1591.06380.1891.08840.0701.01290.1001

19、.02610.1301.04340.1601.06460.1901.08930.0711.01330.1011.02660.1311.04400.1611.06540.1911.09020.0721.01370.1021.02710.1321.04470.1621.06610.1921.09100.0731.01400.1031.02760.1331.04530.1631.06690.1931.09190.0741.01440.1041.02810.1341.04600.1641.06770.1941.09280.0751.01480.1051.02870.1351.04670.1651.06

20、850.1951.09370.0761.01520.1061.02920.1361.04730.1661.06930.1961.09460.0771.01560.1071.02970.1371.04800.1671.07000.1971.09550.0781.01600.1081.03030.1381.04870.1681.0708- 0.1981.09640.0791.01640.1091.03080.1391.04940.1691.07160.1991.0973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 1

21、卜/2a=3.0/24.0 = 0.125h，2b=2.8/16.0=0.175分别查表得系数Ka为L0402和系数Kb为L0765,则扁壳表面积A=24.0XL0402X16.0X L0765=429.99n1-3-4-4圆抛物面扁壳(图13)图1-3圆抛物面扁壳计算图壳面方程式：Z = 为 (X2+ Y2)式中 X、丫、Z在壳面上任一点对原点0的坐标; R半径；假设2a对应弧病的弦长；2b对应弧嬴:的弦长；/Sx弧AGB的长；Sy弧8DC的长； hx弓形AGB的高； hy弓形BDC的高； A*弓形AGB的面积；Ay弓形BDC的面积；/壳顶到底面距离； cAC的长。则：c=21Q? +户f c2/= 8Rl a2 沃f b2 h2R1 .弧长按下式计算Sx =等 JR2 + a2 + R Inf Jr2 4- a1 K Ki/Sy =旨 Jr2 + b2 + R In倍 + J Jr' + 6?) K' KK/2 .壳表面积按下式计算=Sx Sy3 .侧面积按下式计算A* =Ay 二2a34 工3R = M2b