Name	Mathematical constant	Value	Notation
One	$$1$$	1	M_ONE
Two	$$2$$	2	M_TWO
One half	$$1/2$$	0.5	M_ONE_HALF
Ludolph's number	$$π$$	3.1415926535898	M_PI
Tau	$$τ=2\cdot π$$	6.2831853071796	M_TAU
Euler's number	$$e=\sum_{n=0}^{\infty}\cfrac{1}{n!}=1+\cfrac{1}{1}+\cfrac{1}{1\cdot 2}+\cfrac{1}{1\cdot 2\cdot 3}+\cdots$$	2.718281828459	M_E
Euler's constant	$$γ=\lim_{n\rightarrow\infty}\left(-\log{n}+\sum_{k=1}^{n}\right)\cfrac{1}{k}$$	0.57721566490153	M_EULER
Apéry's constant	$$ζ(3)=\sum_{n=1}^{\infty}\cfrac{1}{n^3}=1+\cfrac{1}{2^3}+\cfrac{1}{3^3}+\cfrac{1}{4^3}+\cfrac{1}{5^3}+\cdots$$	1.2020569031596	M_APERY
Catalan's constant	$$G=\sum_{n=0}^{\infty}\cfrac{\left(-1\right)^n}{\left(2n+1\right)^2}=\cfrac{1}{1^2}-\cfrac{1}{3^2}+\cfrac{1}{5^2}-\cfrac{1}{7^2}+\cfrac{1}{9^2}-\cdots$$	0.91596559417722	M_CATALAN
Feigenbaum constant α	$$α$$	2.5029078750959	M_FEIGENBAUM_ALPHA
Feigenbaum constant δ	$$δ$$	4.669201609103	M_FEIGENBAUM_DELTA
Lemniscate constant	$$ϖ=2\int_{0}^{1}\cfrac{\text{d}t}{\sqrt{1-t^4}}$$	2.6220575542921	M_LEMNISCATE
Glaisher–Kinkelin constant	$$A$$	1.2824271291006	M_GLAISHER
Khinchin's constant	$$K_0=\lim_{n\rightarrow\infty}\left(a_1a_2\ldots a_n\right)^{1/n}$$	2.6854520010653	M_KHINCHIN
Golden Ratio	$$φ=\cfrac{1+\sqrt{5}}{2}$$	1.6180339887499	M_GOLDEN_RATIO
Silver Ratio	$$δ_S=\sqrt{2}+1$$	2.4142135623731	M_SILVER_RATIO
Supergolden Ratio	$$ψ=\cfrac{1+\sqrt[3]{\cfrac{29+3\cdot\sqrt{93}}{2}}+\sqrt[3]{\cfrac{29-3\cdot\sqrt{93}}{2}}}{3}$$	1.4655712318768	M_SUPERGOLDEN_RATIO
Connective constant	$$μ=\sqrt{2+\sqrt{2}}$$	1.8477590650226	M_CONNECTIVE
Kepler–Bouwkamp constant	$$K'=\prod_{n=3}^{\infty}\cos\left(\cfrac{π}{n}\right)=\cos\left(\cfrac{π}{3}\right)\cos\left(\cfrac{π}{4}\right)\cos\left(\cfrac{π}{5}\right)\cdots$$	0.1149420448533	M_KEPLER_BOUWKAMP
Erdős–Borwein constant	$$E=\sum_{n=1}^{\infty}\cfrac{1}{2^n-1}=\cfrac{1}{1}+\cfrac{1}{3}+\cfrac{1}{7}+\cfrac{1}{15}\cdots$$	1.6066951524153	M_ERDOS_BORWEIN
Omega constant	$$Ω=\cfrac{1}{π}\int_{0}^{π}\log\left(1+\cfrac{\sin t}{t}e^{t \cot t}\right)dt$$	0.56714329040978	M_OMEGA
Gauss's constant	$$G=\cfrac{1}{\text{agm}\left(1, \sqrt{2}\right)}=\cfrac{1}{4π}\sqrt{\cfrac{2}{π}}Γ\left(\cfrac{1}{4}\right)^2=\cfrac{ϖ}{π}$$	0.83462684167407	M_GAUSS
Second Hermite constant	$$γ_2=\cfrac{2}{\sqrt{3}}$$	1.1547005383793	M_SECOND_HERMITE
Liouville's constant	$$L=\sum_{n=1}^{\infty}\cfrac{1}{10^{n!}}=\cfrac{1}{10^{1!}}+\cfrac{1}{10^{2!}}+\cfrac{1}{10^{3!}}+\cfrac{1}{10^{4!}}+\cdots$$	0.110001	M_LIOUVILLE
Ramanujan's constant	$${e}^{π\cdot\sqrt{163}}$$	2.6253741264077E+17	M_RAMANUJAN
Dottie number	$$D$$	0.73908513321516	M_DOTTIE
Meissel-Mertens constant	$$M=\lim_{n\rightarrow\infty}\left(\sum_{p\le n}\cfrac{1}{p}-\ln\left(\ln n\right)\right)=γ+\sum_{p}\left(\ln\left(1-\cfrac{1}{p}\right)+\cfrac{1}{p}\right)$$	0.26149721284764	M_MEISSEL_MERTENS
Universal parabolic constant	$$ \ln{\left(1+\sqrt{2}\right)}+\sqrt{2}$$	2.2955871493926	M_UNIVERSAL_PARABOLIC
Cahen's constant	$$C=\sum_{k=1}^{\infty}\cfrac{\left(-1\right)^k}{s_k-1}=\cfrac{1}{1}-\cfrac{1}{2}+\cfrac{1}{6}-\cfrac{1}{42}+\cfrac{1}{1806}\pm\cdots$$	0.64341054628834	M_CAHEN
Gelfond's constant	$${e}^π$$	23.140692632779	M_GELFOND
Gelfond-Schneider constant	$$2^{\sqrt{2}}$$	2.6651441426902	M_GELFOND_SCHNEIDER
Second Favard constant	$$K_2=\cfrac{π^2}{8}$$	1.2337005501362	M_SECOND_FAVARD
Golden angle	$$g=π\cdot\left(3-\sqrt{5}\right)$$	2.3999632297287	M_GOLDEN_ANGLE
Sierpiński's constant	$$K=π\left(2γ+\ln\cfrac{4π^3}{Γ\left(\cfrac{1}{4}\right)^4}\right)=π\left(2γ+4\ln Γ\left(\cfrac{3}{4}\right)-\ln π\right)=π\left(2\ln2+3\ln π+2γ-4\ln Γ\left(\cfrac{1}{4}\right)\right)$$	2.5849817595793	M_SIERPINSKI
Landau-Ramanujan constant	$$b=\cfrac{1}{\sqrt{2}}\prod_{p\equiv 3\ (\mod 4)}\left(1-\cfrac{1}{p^2}\right)^{-\cfrac{1}{2}}=\cfrac{π}{4}\prod_{p\equiv 1\ (\mod 4)}\left(1-\cfrac{1}{p^2}\right)^{\cfrac{1}{2}}$$	0.76422365358922	M_LANDAU_RAMANUJAN
First Nielsen-Ramanujan constant	$$a_1=\cfrac{π^2}{12}$$	0.82246703342411	M_FIRST_NIELSEN_RAMANUJAN
Gieseking constant	$$G=\cfrac{3\sqrt{3}}{4}\left(1-\sum_{n=0}^{\infty}\cfrac{1}{\left(3n+2\right)^2}+\sum_{n=1}^{\infty}\cfrac{1}{\left(3n+1\right)^2}\right)=\cfrac{\sqrt{3}}{4}\left(\cfrac{ψ_1\left(1/3\right)}{2}-\cfrac{π^2}{3}\right)$$	1.0149416064097	M_GIESEKING
Bernstein's constant	$$β=\lim_{n\rightarrow\infty}2nE_{2n}\left(f\right)$$	0.28016949902387	M_BERNSTEIN
Tribonacci constant	$$\cfrac{1+\sqrt[3]{19+3\cdot\sqrt{33}}+\sqrt[3]{19-3\cdot\sqrt{33}}}{3}$$	1.8392867552142	M_TRIBONACCI
Brun's constant	$$B_2=\sum_{p}\left(\cfrac{1}{p}+\cfrac{1}{p+2}\right)=\left(\cfrac{1}{3}+\cfrac{1}{5}\right)+\left(\cfrac{1}{5}+\cfrac{1}{7}\right)+\left(\cfrac{1}{11}+\cfrac{1}{13}\right)\cdots$$	1.902160583104	M_BRUN
Twin primes constant	$$C_2=\prod_{p\ \text{prime,}\ p\geq3}\left(1-\cfrac{1}{\left(p-1\right)^2}\right)$$	0.66016181584687	M_TWIN_PRIMES
Plastic Ratio	$$ρ=\sqrt[3]{\cfrac{1}{2}+\cfrac{\sqrt{69}}{18}}+\sqrt[3]{\cfrac{1}{2}-\cfrac{\sqrt{69}}{18}}$$	1.3247179572447	M_PLASTIC_RATIO
Prouhet-Thue-Morse constant	$$τ=\sum_{n=0}^{\infty}\cfrac{t_n}{2^{n+1}}=\cfrac{1}{4}\left[2-\prod_{n=0}^{\infty}\left(1-\cfrac{1}{2^{2^n}}\right)\right]$$	0.41245403364011	M_PROUHET_THUE_MORSE
Golomb-Dickman constant	$$λ=\int_0^1e^{Li\left(t\right)}dt=\int_0^{\infty}\cfrac{ρ\left(t\right)}{t+2}dt$$	0.62432998854355	M_GOLOMB_DICKMAN
Lebesgue constant	$$c=\lim_{n\rightarrow\infty}\left(L_n-\cfrac{4}{π^2}\ln\left(2n+1\right)\right)$$	0.98943127383115	M_LEBESGUE
Feller-Tornier constant	$$C_{FT}=\cfrac{1}{2}\prod_{p\ \text{prime}}\left(1-\cfrac{2}{p^2}\right)+\cfrac{1}{2}$$	0.66131704946962	M_FELLER_TORNIER
Champernowne constant	$$C_{10}=0.1\ 2\ 3\ 4\ 5\ 6\ 7\ 8\ 9\ 10\ 11\ 12\ 13\ 14\cdots$$	0.12345678910111	M_CHAMPERNOWNE
Salem constant	$$σ_{10}=x^{10}+x^9-x^7-x^6-x^5-x^4-x^3+x+1$$	1.1762808182599	M_SALEM
Lévy's constant	$$β=\cfrac{π^2}{12\cdot\ln{2}}$$	1.1865691104156	M_LEVY
Copeland-Erdős constant	$$C_{CE}=0.2\ 3\ 5\ 7\ 11\ 13\ 17\ 19\ 23\ 29\ 31\ 37\cdots$$	0.23571113171923	M_COPELAND_ERDOS
Mills' constant	$$\left\lfloor A^{3^n}\right\rfloor$$	1.3063778838631	M_MILLS
Gompertz constant	$$δ=\int_0^{\infty}\cfrac{e^{-x}}{1+x}dx$$	0.59634736232319	M_GOMPERTZ
Van der Pauw constant	$$\cfrac{π}{\ln{2}}$$	4.5323601418272	M_VAN_DER_PAUW
Magic angle	$$θ_m=\tan^{-1}{\sqrt{2}}$$	0.95531661812451	M_MAGIC_ANGLE
Artin's constant	$$C_{Artin}=\prod_{p\ \text{prime}}\left(1-\cfrac{1}{p\left(p-1\right)}\right)$$	0.3739558136192	M_ARTIN
Porter's constant	$$C=\cfrac{6\ln 2}{π^2}\left(3\ln 2+4γ-\cfrac{24}{π^2}ζ'\left(2\right)-2\right)-\cfrac{1}{2}$$	1.467078079434	M_PORTER
Lochs constant	$$L=\cfrac{6\ln 2\ln 10}{π^2}$$	0.97027011439203	M_LOCHS
Lieb's square ice constant	$$\left(\cfrac{4}{3}\right)^{\cfrac{3}{2}}$$	1.539600717839	M_LIEB_SQUARE_ICE
Niven's constant	$$C=1+\sum_{n=2}^{\infty}\left(1-\cfrac{1}{ζ\left(n\right)}\right)$$	1.7052111401054	M_NIVEN
Stephens' constant	$$C_S=\prod_{p\ \text{prime}}\left(1-\cfrac{p}{p^3-1}\right)$$	0.57595996889295	M_STEPHENS
Zero	$$0$$	0	M_ZERO
Negative one	$$-1$$	-1	M_NEGATIVE_ONE
Square Root of 2	$$\sqrt{2}$$	1.4142135623731	M_SQRT2
Square Root of 3	$$\sqrt{3}$$	1.7320508075689	M_SQRT3
Square Root of 5	$$\sqrt{5}$$	2.2360679774998	M_SQRT5
Cube Root of 2	$$\sqrt[3]{2}$$	1.2599210498949	M_CURT2
Cube Root of 3	$$\sqrt[3]{3}$$	1.4422495703074	M_CURT3
Twelfth Root of 2	$$\sqrt[12]{2}$$	1.0594630943593	M_TWRT2
Natural Log of 2	$$\ln(2)$$	0.69314718055995	M_LN2
Natural Log of 10	$$\ln(10)$$	2.302585092994	M_LN10
Natural Log of Pi	$$\ln(π)$$	1.1447298858494	M_LNPI
Base 10 Log of e	$$\log10(e)$$	0.43429448190325	M_LOG10E
Base 2 Log of e	$$\log2(e)$$	1.442695040889	M_LOG2E
Half of Pi	$$π/2$$	1.5707963267949	M_PI_2
Quarter of Pi	$$π/4$$	0.78539816339745	M_PI_4
Inverse of Pi	$$1/π$$	0.31830988618379	M_1_PI
Two over Pi	$$2/π$$	0.63661977236758	M_2_PI
Square Root of Pi	$$\sqrt{π}$$	1.7724538509055	M_SQRTPI
Two over Square Root of Pi	$$2/\sqrt{π}$$	1.1283791670955	M_2_SQRTPI
Inverse of Square Root of 2	$$1/\sqrt{2}$$	0.70710678118655	M_SQRT1_2

Function	Description	Sample example
ABS	Returns the absolute value of a number. The absolute value of a number is the same number without a sign.	ABS(-9)=9
ACCRINT	Returns the total interest on securities that pay interest on regular dates.	ACCRINT(39508, 39691, 39569, 0.1, 1000, 2, 0)=16.666666666667
ACCRINTM	Returns the accrued interest on a security that pays interest on the maturity date.	ACCRINTM(39539, 39614, 0.1, 1000, 3)=20.547945205479
ACOS	Returns the arccosine (the inverse of the cosine function) of the specified number. The arccosine is the angle whose cosine is the specified number. The resulting angle is given in radians in the range zero to pi.	ACOS(0.5)=1.0471975511966
ACOSH	Returns the inverse hyperbolic cosine of a number. The number must be greater than or equal to 1. The inverse hyperbolic cosine is a value whose hyperbolic cosine is a number, so ACOSH(COSH(number)) equals the number.	ACOSH(2)=1.3169578969248
ACOT	Returns the value of the arccotangent (inverse cotangent) of the specified number.	ACOT(3.14)=0.30831566219543
ACOTH	Returns the inverse hyperbolic cotangent of the specified number.	ACOTH(3.14)=0.32994497940173
AMORDEGRC	Returns the depreciation for one accounting period. This function is adapted to the French accounting model. If the asset was acquired during the accounting period, the depreciation is calculated on a straight-line basis. The AMORDEGRC function is similar to the AMORLINC function, but differs in that the depreciation coefficient depends on the useful life of the asset.	AMORDEGRC(2400, 39679, 39813, 300, 1, 0.15, 1)=776
AMORLINC	Returns the depreciation for one accounting period. This function is adapted to the French accounting model. If the asset was acquired during the accounting period, proportional depreciation is considered.	AMORLINC(2400, 39679, 39813, 300, 1, 0.15, 1)=360
ARABIC	Converts Roman numerals to Arabic numerals.	ARABIC(V)=5
ASIN	Returns the arcsine (the inverse of the sine) of the specified number. The arcsine is the angle whose sine is the specified number. The resulting angle is given in radians in the interval from pi/2 to pi/2.	ASIN(1)=1.5707963267949
ASINH	Returns the hyperbolic arcsine of the specified number. The hyperbolic arcsine is a value whose hyperbolic sine is the specified number, so ARCSINH(SINH(number)) is equal to the specified number.	ASINH(1)=0.88137358701954
ATAN	Returns the arctangent (the inverse of the tangent function) of the specified number. The arctangent is the angle whose tangent is the specified number. The resulting angle is given in radians in the range pi/2 to pi/2.	ATAN(2)=1.1071487177941
ATAN2	Returns the arctangent (the inverse of tangent) of the specified x and y coordinates. The arctangent is the angle between the x axis and the line containing the origin (0;0) and the point at coordinates (x_number, y_number). This angle is given in radians in the interval -pi to pi, except for the value -pi.	ATAN2(2, 3)=0.98279372324733
ATANH	Returns the arctangent (the inverse of the tangent function) of the specified number. The arctangent is the angle whose tangent is the specified number. The resulting angle is given in radians in the range pi/2 to pi/2.	ATANH(0.5)=0.54930614433405
AVEDEV	Returns the average of the absolute deviations of data points from their mean. STANDARD DEVIATION is a measure of the variability of a data set.	AVEDEV(10, 30, 2, 3)=9.375
AVERAGEA	Calculates the average (arithmetic mean) of the values in the argument list.	AVERAGEA(10, 20, 30)=20
BESSELI	Returns a modified Bessel function that is equivalent to the Bessel function calculated for purely imaginary arguments.	BESSELI(1, 2)=0.1357476665807
BESSELJ	Returns the Bessel function.	BESSELJ(1, 2)=0.1149034849319
BESSELK	Returns a modified Bessel function that is equivalent to the Bessel function calculated for purely imaginary arguments.	BESSELK(1, 2)=1.6248388844172
BESSELY	Returns the Bessel function, also known as the Weber or Neumann function.	BESSELY(1, 2)=-1.6506826133039
BIN2DEC	Converts a binary number to decimal.	BIN2DEC(1101)=13
CEILING	Rounds a number up to the nearest integer or to the nearest multiple of significance.	CEILING(1.2)=2
CEILING.MATH	The CEILING.MATH function rounds a number up to the nearest integer or, optionally, to the nearest multiple of significance.	CEILING.MATH(1.2, 1)=2
CEILING.PRECISE	Returns a number that is rounded up to the nearest integer or to the nearest multiple of significance. Regardless of the sign of the number, the number is rounded up. However, if the number or the significance is zero, zero is returned.	CEILING.PRECISE(1.2, 1)=2
CHIDIST	Returns the right-tailed probability of a chi-square distribution. The χ2 distribution is related to the χ2 test. The χ2 test compares observed and expected values. For example, in a genetic experiment, you might predict that the next generation of plants will have flowers of a certain color. By comparing the observed results with the expected results, you can determine whether the original prediction is valid.	CHIDIST(15, 15)=0.45141721122575
CHIINV	Returns the inverse of the right-tailed probability distribution function of the chi-square distribution. If the probability is equal to CHIDIST(x;...), the value of CHIINV(probability;...) is equal to x. Use this function to compare observed results with expected results to determine whether the original assumption is valid.	CHIINV(0.05, 10)=18.307038053334
CODE	Returns the numeric code of the first character of a given text string.	CODE(1)=49
COMBIN	Returns the number of combinations for a specified number of elements. Use the COMBIN function to determine the total number of possible groups for a specified number of elements.	COMBIN(15, 9)=5005
COMBINA	Returns the number of combinations (with repetition) for a given number of items.	COMBINA(15, 6)=38760
COS	Returns the cosine of the specified angle.	COS(1)=0.54030230586814
COSH	Returns the hyperbolic cosine of the specified number.	COSH(1)=1.5430806348152
COT	Returns the cotangent of an angle specified in radians.	COT(1)=0.64209261593433
COTH	Returns the hyperbolic cotangent of a hyperbolic angle.	COTH(1)=1.3130352854993
COUNT	Returns the number of items in the argument list.	COUNT(1, 2)=2
CSC	Returns the cosecant of an angle specified in radians.	CSC(1)=1.1883951057781
CSCH	Returns the hyperbolic cosecant of an angle specified in radians.	CSCH(1)=0.85091812823932
DEGREES	Converts radians to degrees.	DEGREES(PI())=180
DELTA	Tests the equality of two numbers. Returns 1 if number1 = number2; otherwise, returns 0. This function can be used to filter a set of values. For example, when adding several DELTA functions, you can find the number of identical pairs. This function is also known as the Kronecker Delta function.	DELTA(2, 2)=1
DEVSQ	Returns the sum of the squares of the deviations of data points from their sample mean.	DEVSQ(10, 3, 2)=38
EXP	Returns e raised to the power of the argument number. The constant e is equal to 2.718281828459, the base of natural logarithms.	EXP(1)=2.718281828459
FACT	Returns the factorial of a number. The factorial of a number is equal to 123... number.	FACT(9.5)=362880
FACTDOUBLE	Returns the double factorial of the specified number.	FACTDOUBLE(12)=46080
FISHER	Returns the Fisher transformation value of x. This transformation produces a function with a normal distribution rather than a skewed distribution. You can use this function to test the hypothesis about the correlation coefficient.	FISHER(0.9)=1.4722194895832
FISHERINV	Returns the inverse of the Fisher transform. You can use this transform to analyze correlations between regions or matrices of data. If y = FISHER(x), FISHERINV(y) = x.	FISHERINV(10)=0.99999999587769
FLOOR	Rounds a number down to the nearest whole number or to the nearest multiple of the specified value.	FLOOR(1.2)=1
GAMMA	Returns the value of the gamma function.	GAMMA(12)=39916800.000056
GAMMALN	Returns the natural logarithm of the gamma function.	GAMMALN(12)=17.502307845875
GAUSS	Calculates the probability that a member of a standard population falls within the interval between the mean standard deviation and the standard deviation z from the mean.	GAUSS(12)=0.5
GCD	Returns the greatest common divisor of two or more integers.	GCD(12, 15, 1.5)=1
GEOMEAN	Returns the geometric mean of an array or range of positive data. For example, you can use the GEOMEAN function to calculate the average growth rate of a given compound interest with a variable interest rate.	GEOMEAN(5, 2)=3.1622776601684
GESTEP	It returns a value of 1 if the number ≥ the specified value, otherwise it returns 0 (zero).	GESTEP(5, 2)=1
HARMEAN	Returns the harmonic mean of a data set. The harmonic mean is the reciprocal of the arithmetic mean of the reciprocals.	HARMEAN(10, 3)=4.6153846153846
INDEX	Returns the value of the element in the array at the selected index.	INDEX((15, 20), 1)=15
INT	Rounds a number down to the nearest integer.	INT(1.2)=1
LCM	Returns the least common multiple of integers. The least common multiple is the smallest positive integer that is a multiple of all integer arguments number1, number2, and so on. Use LCM to add fractions with different denominators.	LCM(10, 6, 3)=30
LN	Returns the natural logarithm of a number. The base of the natural logarithm is the constant e (2.71828182845904).	LN(10)=2.302585092994
LOG	Returns the logarithm of a number to a given base.	LOG(10, 3)=2.0959032742894
LOG10	Returns the base 10 logarithm of a number.	LOG10(10)=1
MAX	Returns the maximum value in a set of values.	MAX(10, -10)=10
MAXA	Returns the maximum value in a set of values.	MAXA(10, -10, TRUE)=10
MEDIAN	Returns the median of the specified numbers. The median is the number that lies in the middle of a set of numbers.	MEDIAN(10, 20, 15, 1)=12.5
MIN	Returns the minimum value in a set of values.	MIN(10, -10)=-10
MOD	Returns the remainder after dividing a number by a divisor.	MOD(20, 3)=2
MODE	Returns the most frequently occurring or recurring value in a range of data.	MODE(20, 3, 20, 3, 26, 3)=3
MROUND	Returns a number rounded to the desired multiple.	MROUND(20, 2)=20
MULTINOMIAL	Returns the factorial of the sum of values divided by the product of their factorials.	MULTINOMIAL(10, 20, 30)=3.553261127085E+24
PERMUT	Returns the number of permutations for a given number of objects that can be selected from number objects. A permutation is any set or subset of objects or events where internal order is significant. Permutations are different from combinations, for which the internal order is not significant. Use this function for lottery-style probability calculations.	PERMUT(10, 2)=90
PERMUTATIONA	Returns the number of permutations for a given number of objects (with repetitions) that can be selected from the total objects.	PERMUTATIONA(10, 2)=100
PHI	Returns the value of the density function for the standard deviation.	PHI(2)=0.053990966513188
PI	Returns the number 3.1415926535898, the mathematical constant pi.	PI()=3.1415926535898
POWER	Returns the power of a number.	POWER(3, 2)=9
QUOTIENT	Returns the whole part of a division. This function is used to remove the remainder after division.	QUOTIENT(3, 2)=1
RADIANS	Converts degrees to radians.	RADIANS(180)=3.1415926535898
RAND	RAND returns uniformly distributed random real numbers greater than or equal to 0 and less than 1. A new random real number is returned each time the worksheet is recalculated.	RAND()=0.91200778862089
RANDBETWEEN	Returns a random integer from the specified interval. A new random integer is returned each time the worksheet is recalculated.	RANDBETWEEN(180, 220)=208
ROUNDDOWN	Rounds a number down towards zero.	ROUNDDOWN(180.66669, 3)=180.666
ROUNDUP	Rounds a number up, away from zero.	ROUNDUP(180.66669, 3)=180.667
SEC	Returns the secant of an angle.	SEC(0.5)=1.1394939273245
SECH	Returns the hyperbolic secant of an angle.	SECH(1)=0.64805427366389
SIGN	Returns the sign of the argument. Returns 1 if the number is positive, 0 for 0, and -1 if the number is negative.	SIGN(1)=1
SIN	Returns the sine of a given angle.	SIN(PI()/6)=0.5
SINH	Returns the hyperbolic sine of a number.	SINH(PI()/6)=0.54785347388804
SKEW	Returns the skewness of the distribution of a random variable. Skewness indicates the degree of asymmetry of the distribution of the variable around the mean. Positive skewness indicates a distribution with an asymmetric side that is skewed toward more positive values. Negative skewness indicates a distribution with an asymmetric side that is skewed toward more negative values.	SKEW(10, 3, 4, 5)=1.5970779829308
SKEW.P	Returns the skewness coefficient of a distribution based on a set of values: a characteristic of the degree of skewness of the distribution around the mean value.	SKEW.P(12, 5, 6)=0.65201211704405
SMALL	Returns the kth smallest value in a data set. This function is used to determine the value that has a specific relative position in a data set.	SMALL((12, 15), 2)=15
SQRT	Returns the positive square root.	SQRT(100)=10
STDEVP	Calculates standard deviation based on the entire population given as arguments. The standard deviation is a measure of how widely values are dispersed from the average value (the mean).	STDEVP(10, 20)=5
SUM	Returns the sum of all numbers.	SUM(-2, 20)=18
TAN	Returns the tangent of the specified angle.	TAN(1)=1.5574077246549
TANH	Returns the hyperbolic tangent of the argument.	TANH(1)=0.76159415595576

Thickness of shells under internal pressure for Torispherical Heads having values $t_s/L≥ 0.002$

Torispherical Head

Values for calculation

Material

General requirements

G1 - To these stress values a casting quality factor as specified in PG–25 of Section I; UG–24 of Section VIII, Division 1; or TM–190 of Section XII shall be applied.
G2 - These stress values include a joint efficiency factor of 0.60.
G3 - These stress values include a joint efficiency factor of 0.85.
G4 - For Section I applications, these stresses apply when used for boiler, water wall, superheater, and economizer tubes that are enclosed within a setting. A joint efficiency factor of 0.85 is included in values above 450°C.
G5 - Due to the relatively low yield strength of these materials, these higher stress values were established at temperatures where the shorttime tensile properties govern to permit the use of these alloys where slightly greater deformation is acceptable. The stress values in this range exceed 66²/₃% but do not exceed 90% of the yield strength at temperature. Use of these stresses may result in dimensional changes due to permanent strain. These stress values are not recommended for the flanges of gasketed joints or other applications where slight amounts of distortion can cause leakage or malfunction. For Section III applications, Table Y–2 lists multiplying factors that, when applied to the yield strength values shown in Table Y–1, will give allowable stress values that will result in lower levels of permanent strain.
G6 - Creep–fatigue, thermal ratcheting, and environmental effects are increasingly significant failure modes at temperatures in excess of 825°C and shall be considered in the design.
G7 - For Section VIII applications, these stress values are based on expected minimum values of 310 MPa tensile strength and yield strength of 140 MPa resulting from loss of strength due to thermal treatment required for the glass coating operation. UG–85 does not apply.
G8 - These stress values are established from a consideration of strength only and will be satisfactory for average service. For bolted joints where freedom from leakage over a long period of time without retightening is required, lower stress values may be necessary as determined from the flexibility of the flange and bolts and corresponding relaxation properties.
G9 - For Section III applications, the use of these materials shall be limited to materials for tanks covered in Subsections NC and ND, component supports, and for nonpressure–retaining attachments (NC/ND–2190).
G10 - Upon prolonged exposure to temperatures above 425°C, the carbide phase of carbon steel may be converted to graphite. See Nonmandatory Appendix A, A–201 and A–202.
G11 - Upon prolonged exposure to temperatures above 475°C, the carbide phase of carbon–molybdenum steel may be converted to graphite. See Nonmandatory Appendix A, A–201 and A–202.
G12 - At temperatures above 550°C, these stress values apply only when the carbon is 0.04% or higher on heat analysis.
G13 - These stress values at 575°C and above shall be used only when the grain size is ASTM No. 6 or coarser.
G14 - These stress values shall be used when the grain size is not determined or is determined to be finer than ASTM No. 6.
G15 - For Section I applications, use is limited to stays as defined in PG–13 except as permitted by PG–11.
G16 - For Section III Class 3 applications, these S values do not include a casting quality factor. Statically and centrifugally cast products meeting the requirements of NC–2570 shall receive a casting quality factor of 1.00.
G17 - For Section III Class 3 applications, statically and centrifugally cast products meeting the requirements of NC–2571(a) and (b), and cast pipe fittings, pumps, and valves with inlet piping connections of DN 50 and less, shall receive a casting quality factor of 1.00. Other casting quality factors shall be in accordance with the following:
(a) for visual examination, 0.80
(b) for magnetic particle examination, 0.85
(c) for liquid penetrant examination, 0.85
(d) for radiography, 1.00
(e) for ultrasonic examination, 1.00
(f) for magnetic particle or liquid penetrant plus ultrasonic examination or radiography, 1.00
G18 - See Table Y–1 for yield strength values as a function of thickness over this range. Allowable stresses are independent of yield strength in this thickness range.
G19 - This steel may be expected to develop embrittlement after service at moderately elevated temperature. See Nonmandatory Appendix A, A–207 and A–208.
G20 - These stresses are based on weld metal properties.
G21 - For Section I, use is limited to PEB–5.3. See PG–5.5 for cautionary note.
G22 - For Section I applications, use of external pressure charts for material in the form of bar stock is permitted for stiffening rings only.
G23 - For temperatures above the maximum temperature shown on the external pressure chart for this material, Figure CS–2 may be used for the design using this material.
G24 - A factor of 0.85 has been applied in arriving at the maximum allowable stress values in tension for this material. Divide tabulated values by 0.85 for maximum allowable longitudinal tensile stress.
G25 - For Section III applications, for both Class 2 and Class 3, the completed vessel after final heat treatment shall be examined by the ultrasonic method in accordance with NB–2542 except that angle beam examination in both the circumferential and the axial directions may be performed in lieu of the straight beam examination in the axial direction. The tensile strength shall not exceed 860 MPa.
G26 - Material that conforms to Class 10, 11, or 12 is not permitted.
G27 - Material that conforms to Class 11 or 12 is not permitted.
G28 - Supplementary Requirement S15 of SA–781, Alternate Mechanical Test Coupons and Specimen Locations for Castings, is mandatory.
G29 - For Section III applications, impact testing in accordance with the requirements of NC–2300 is required for Class 2 components and in accordance with ND–2300 for Class 3 components.
G30 - These stresses apply to all product forms (C, H, and P) as defined in SA/EN 10028–7.
G31 - The allowable stress value given for 250°C can be used up to the temperature of 260°C.
G32 - The allowable stress value given for 200°C can be used up to the temperature of 204°C.

Heat treatment requirements

H1 - For temperatures above 550°C, these stress values may be used only if the material is heat treated by heating to the minimum temperature specified in the material specification, but not lower than 1040°C, and quenching in water or rapidly cooling by other means.
H2 - For temperatures above 550°C, these stress values may be used only if the material is heat treated by heating to a minimum temperature of 1095°C, and quenching in water or rapidly cooling by other means.
H3 - Quenched and tempered at 650°C.
H4 - Solution treated and quenched.
H5 - For Section III applications, if heat treatment is performed after forming or fabrication, it shall be performed at 825°C to 1000°C for a period of time not to exceed 10 min at temperature, followed by rapid cooling.
H6 - Material shall be solution annealed at 1100°C to 1170°C, followed by a rapid cooling in water or air.

Size requirements

S1 - For Section I applications, stress values at temperatures of 450°C and above are permissible but, except for tubular products 75 mm O.D. or less enclosed within the boiler setting, use of these materials at these temperatures is not current practice.
S2 - For Section I applications, stress values at temperatures of 475°C and above are permissible but, except for tubular products 75 mm O.D. or less enclosed within the boiler setting, use of these materials at these temperatures is not current practice.
S3 - For Section I applications, stress values at temperatures of 550°C and above are permissible but, except for tubular products 75 mm O.D. or less enclosed within the boiler setting, use of these materials at these temperatures is not current practice.
S4 - For Section I applications, stress values at temperatures of 625°C and above are permissible but, except for tubular products 75 mm O.D. or less enclosed within the boiler setting, use of these materials at these temperatures is not current practice.
S5 - Material that conforms to Class 10, 11, or 12 is not permitted when the nominal thickness of the material exceeds 19 mm.
S6 - Material that conforms to Class 10, 11, or 12 is not permitted when the nominal thickness of the material exceeds 32 mm.
S7 - The maximum thickness of unheat–treated forgings shall not exceed 95 mm. The maximum thickness as–heat–treated may be 100 mm.
S8 - The maximum section thickness shall not exceed 75 mm for double–normalized–and–tempered forgings, or 125 mm for quenched–and–tempered forgings.
S9 - Both DN 200 and larger, and schedule 140 and heavier.
S10 - The maximum pipe size shall be DN 100 and the maximum thickness in any pipe size shall be Schedule 80.
S11 - Either DN 200 and larger and less than schedule 140 wall, or less than DN 200 and all wall thicknesses.

Time–dependent properties

T1 - Allowable stresses for temperatures of 370°C and above are values obtained from time–dependent properties.
T2 - Allowable stresses for temperatures of 400°C and above are values obtained from time–dependent properties.
T3 - Allowable stresses for temperatures of 455°C and above are values obtained from time–dependent properties.
T4 - Allowable stresses for temperatures of 480°C and above are values obtained from time–dependent properties.
T5 - Allowable stresses for temperatures of 510°C and above are values obtained from time–dependent properties.
T6 - Allowable stresses for temperatures of 540°C and above are values obtained from time–dependent properties.
T7 - Allowable stresses for temperatures of 565°C and above are values obtained from time–dependent properties.
T8 - Allowable stresses for temperatures of 595°C and above are values obtained from time–dependent properties.
T9 - Allowable stresses for temperatures of 620°C and above are values obtained from time–dependent properties.
T10 - Allowable stresses for temperatures of 425°C and above are values obtained from time–dependent properties.
T11 - Allowable stresses for temperatures of 350°C and above are values obtained from time–dependent properties.
T12 - Allowable stresses for temperatures of 650°C and above are values obtained from time–dependent properties.

Welding requirements

W1 - Not for welded construction.
W2 - Not for welded construction in Section III.
W3 - Welded.
W4 - Nonwelded, or welded if the tensile strength of the Section IX reduced section tension test is not less than 690 MPa.
W5 - Welded, with the tensile strength of the Section IX reduced tension test less than 690 MPa but not less than 655 MPa.
W6 - This material may be welded by the resistance technique.
W7 - In welded construction for temperatures above 450°C, the weld metal shall have a carbon content of greater than 0.05%.
W8 - Welding and oxygen or other thermal cutting processes are not permitted when carbon content exceeds 0.35% by heat analysis.
W9 - For Section I applications, for pressure retaining welds in 21/4Cr–1Mo materials, other than circumferential butt welds less than or equal to 89 mm in outside diameter, when the design metal temperatures exceed 450°C, the weld metal shall have a carbon content greater than 0.05%.
W10 - For Section III applications, material that conforms to Class 10, 13, 20, 23, 30, 33, 40, 43, 50, or 53 is not permitted for Class 2 and Class 3 construction when a weld efficiency factor of 1.00 is used in accordance with Note W12.
W11 - For Section VIII applications, Section IX, QW–250 Variables QW–404.12, QW–406.3, QW–407.2, and QW–409.1 shall also apply to this material. These variables shall be applied in accordance with the rules for welding of Part UF.
W12 - These S values do not include a longitudinal weld efficiency factor. For Section III applications, for materials welded without filler metal, ultrasonic examination, radiographic examination, or eddy current examination, in accordance with NC–2550, shall provide a longitudinal weld efficiency factor of 1.00. Materials welded with filler metal meeting the requirements of NC–2560 shall receive a longitudinal weld efficiency factor of 1.00. Other longitudinal weld efficiency factors shall be in accordance with the following:
(a) for single butt weld, with filler metal, 0.80
(b) for single or double butt weld, without filler metal, 0.85
(c) for double butt weld, with filler metal, 0.90
(d) for single or double butt weld, with radiography, 1.00
W13 - For Section I applications, electric resistance and autogenous welded tubing may be used with these stresses, provided the following additional restrictions and requirements are met:
(a) The tubing shall be used for boiler, waterwall, superheater, and economizer tubes that are enclosed within the setting.
(b) The maximum outside diameter shall be 89 mm.
(c) The weld seam of each tube shall be subjected to an angle beam ultrasonic inspection per SA–450.
(d) A complete volumetric inspection of the entire length of each tube shall be performed in accordance with SA–450.
(e) Material test reports shall be supplied.
W14 - These S values do not include a weld factor. For Section VIII, Division 1, and Section XII applications using welds made without filler metal, the tabulated tensile stress values shall be multiplied by 0.85. For welds made with filler metal, consult UW–12 for Section VIII, Division 1, or TW–130.4 for Section XII, as applicable.
W15 - The Nondestructive Electric Test requirements of SA–53 Type E pipe are required for all sizes. The pipe shall be additionally marked "NDE" and so noted on the material specification.

Joint efficiency

Type No.	Joint Description	Limitations	Joint Category	Extent of Radiographic or Ultrasonic Examination [Note (1), Note (2), Note(3)]
Type No.	Joint Description	Limitations	Joint Category	(a) Full [Note (4)]	(b) Spot [Note (5)]	(c) None
(1)	Butt joints as attained by double‐welding or by other means that will obtain the same quality of deposited weld metal on the inside and outside weld surfaces to agree with the requirements of UW-35. Welds using metal backing strips that remain in place are excluded.	None	A, B, C, and D	1.00	0.85	0.70
(2)	Single‐welded butt joint with backing strip other than those included under (1)	(a) None except as in (b) below	A, B, C, and D	0.9	0.8	0.65
(2)		(b) Circumferential butt joints with one plate offset; see UW-13(b)(4) and Figure UW-13.1, sketch (i)	A, B, and C	0.9	0.8	0.65
(3)	Single‐welded butt joint without use of backing strip	Circumferential butt joints only, not over ⁵/₈ in. (16 mm) thick and not over 24 in. (600 mm) outside diameter	A, B, and C	NA	NA	0.6
(4)	Double full fillet lap joint	(a) Longitudinal joints not over ³/₈ in. (10 mm) thick	A	NA	NA	0.55
(4)	Double full fillet lap joint	(b) Circumferential joints not over ⁵/₈ in. (16 mm) thick	B and C [Note (6)]	NA	NA	0.55
(5)	Single full fillet lap joints with plug welds conforming to UW-17	(a) Circumferential joints [Note (7)] for attachment of heads not over 24 in. (600 mm) outside diameter to shells not over ¹/₂ in. (13 mm) thick	B	NA	NA	0.50
(5)		(b) Circumferential joints for the attachment to shells of jackets not over ⁵/₈ in. (16 mm) in nominal thickness where the distance from the center of the plug weld to the edge of the plate is not less than 1¹/₂ times the diameter of the hole for the plug.	C	NA	NA	0.50
(6)	Single full fillet lap joints without plug welds	(a) For the attachment of heads convex to pressure to shells not over ⁵/₈ in. (16 mm) required thickness, only with use of fillet weld on inside of shell; or	A and B	NA	NA	0.45
(6)	Single full fillet lap joints without plug welds	(b) for attachment of heads having pressure on either side, to shells not over 24 in. (600 mm) inside diameter and not over ¹/₄ in. (6 mm) required thickness with fillet weld on outside of head flange only	A and B	NA	NA	0.45
(7)	Corner joints, full penetration, partial penetration, and/or fillet welded	As limited by Figure UW-13.2 and Figure UW-16.1	C and D [Note (8)]	NA	NA	NA
(8)	Angle joints	Design per U-2(g) for Category B and C joints	B, C and D	NA	NA	NA

General note: E = 1.00 for butt joints in compression.
Notes:
(1) Some welding processes require ultrasonic examination in addition to radiographic examination and other processes require ultrasonic examination in lieu of radiographic examination. See UW-11 for some additional requirements and limitations that may apply.
(2) Joint efficiency assignment rules of UW-12(d) and UW-12(e) shall be considered and may further reduce the joint efficiencies to be used in the required thickness calculetions.
(3) The rules of UW-12(f) may be used in lieu of the rules of this Table at the Manufacturer´s option.
(4) See UW-12(a) and UW-51.
(5) See UW-12(b) and UW-52.
(6) For Type No. 4 Category C joint, limitation not applicable for bolted flange connection.
(7) Joints attaching hemispherical heads to shell are excluded.
(8) There is no joint efficiency E in the design equations of this Division for Category C and D corner joints. When needed, a value of E not greater than 1.00 may be used.

$ E $

Internal design pressure

$ P $ $ \mathrm{MPa} $

Temperature

$ T $ $ \mathrm{°C} $

Inside length of the major axis of an torispherical head

$ D $ $ \mathrm{mm} $

Minimum specified thickness of head after forming

$ t_s $ $ \mathrm{mm} $

Crown radius

$ L $ $ \mathrm{mm} $

Inside knuckle radius

$ r $ $ \mathrm{mm} $

Maximum allowable stress value

$ S $ $ \mathrm{MPa} $

Calculation

Factor $ M $

$$M=\cfrac{1}{4}\cdot\left(3+\sqrt{\cfrac{L}{r}}\right)$$

Minimum required thickness of head after forming

$$t=\cfrac{P\cdot L\cdot M}{2\cdot S\cdot E-0.2\cdot P}$$

$$t_s\geq t$$

Requirements

$$ t_s/L \geq 0.002 $$ $$ L \le D+2\cdot t_s $$ $$ r \geq 0.06\cdot\left(D+2\cdot t_s\right) $$ $$ r > 3\cdot t_s $$

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