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A Level Maths Formula Sheet
Levels & boards: AS & A Level Mathematics (Years 12-13) Cambridge International 9709; Pearson Edexcel IAL Mathematics. Pure, Mechanics and Statistics: component coverage varies.
Search this A Level Maths formula resource by topic, save useful equations and reveal original worked examples. Pure Mathematics, selected Mechanics and Probability & Statistics relationships are organised into individual cards with notation, conditions and common mistakes.
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39 formulas
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Arithmetic series sum
S_n = n*(2*a+(n-1)*d)/2
Notation / units: n positive integer.
Use when: Arithmetic sequence.
Worked example and common mistake
a=2,d=3,n=5 gives sum=40.
Watch out: Do not use a geometric ratio in an arithmetic formula.
Geometric term
u_n = a*r^(n-1)
Notation / units: a first term; r common ratio.
Use when: Repeated multiplication by r.
Worked example and common mistake
a=3,r=2 gives u_5=48.
Watch out: The exponent is n-1 when indexing begins at 1.
Finite geometric sum
S_n = a*(1-r^n)/(1-r)
Notation / units: r not equal to 1.
Use when: For r=1 the sum is na.
Worked example and common mistake
a=2,r=3,n=4 gives S_4=80.
Watch out: A negative common ratio must keep its sign throughout.
Infinite geometric sum
S_infinity = a/(1-r)
Notation / units: Requires abs(r)<1.
Use when: The geometric series converges only under this condition.
Worked example and common mistake
a=6,r=0.5 gives sum=12.
Watch out: The formula is invalid for r=2.
Binomial coefficient
nCr = n!/(r!*(n-r)!)
Notation / units: Integers 0<=r<=n.
Use when: Counts unordered choices without replacement.
Worked example and common mistake
5C2=10.
Watch out: Do not multiply by r! again when order does not matter.
Power differentiation
d(x^n)/dx = n*x^(n-1)
Notation / units: x in the expression's real domain.
Use when: n constant.
Worked example and common mistake
d(4x^3)/dx=12x^2.
Watch out: A constant term differentiates to zero.
Power integration
integral x^n dx = x^(n+1)/(n+1)+C
Notation / units: n not equal to -1.
Use when: Indefinite integral on a valid interval.
Worked example and common mistake
Integral 6x^2 dx = 2x^3+C.
Watch out: Use ln(abs(x))+C for x^-1.
Chain rule
dy/dx = dy/du * du/dx
Notation / units: Differentiable compositions.
Use when: Differentiate outer and inner functions.
Worked example and common mistake
d((3x+1)^4)/dx=12(3x+1)^3.
Watch out: Do not omit the derivative of the inside expression.
Product rule
(u*v)' = u'*v + u*v'
Notation / units: Differentiable u and v.
Use when: Derivative of a product.
Worked example and common mistake
d(x*e^x)/dx=e^x+x*e^x.
Watch out: The derivative is not simply u'*v'.
Quotient rule
(u/v)' = (u'*v-u*v')/v^2
Notation / units: v non-zero.
Use when: Derivative of a quotient.
Worked example and common mistake
d(x/(x+1))/dx=1/(x+1)^2.
Watch out: Reversing the numerator terms changes the sign.
Trapezoidal integration estimate
integral y dx approx h*(y0+yn+2*sum(interior y))/2
Notation / units: Equal spacing h.
Use when: Approximates area/integral from samples.
Worked example and common mistake
h=1, y=1,2,5 gives estimate 5.
Watch out: This is an approximation unless each segment is linear.
Logarithm product law
ln(a*b) = ln(a)+ln(b)
Notation / units: a>0,b>0 for real logs.
Use when: Same base logarithms.
Worked example and common mistake
ln(6)=ln(2)+ln(3).
Watch out: ln(a+b) is not ln(a)+ln(b).
Radians and arc length
s = r*theta
Notation / units: theta radians.
Use when: Circular arc.
Worked example and common mistake
r=4, theta=pi/3 gives s=4pi/3.
Watch out: Degree values must be converted before using s=r*theta.
Radian sector area
A = r^2*theta/2
Notation / units: theta radians.
Use when: Sector area.
Worked example and common mistake
r=4, theta=pi/3 gives A=8pi/3.
Watch out: Radians, not degrees, are required here.
Exponential derivative
d(e^(k*x))/dx = k*e^(k*x)
Notation / units: k constant.
Use when: Natural exponential.
Worked example and common mistake
d(e^(3x))/dx=3e^(3x).
Watch out: The factor k comes from the chain rule.
Logarithm derivative
d(ln(x))/dx = 1/x
Notation / units: x>0 in the real-valued form.
Use when: Natural logarithm.
Worked example and common mistake
At x=4, the derivative is 1/4.
Watch out: Logarithms to another base require a constant factor.
Sine and cosine derivatives
d(sin(x))/dx=cos(x); d(cos(x))/dx=-sin(x)
Notation / units: x measured in radians.
Use when: Standard trigonometric derivatives.
Worked example and common mistake
d(3cos(x))/dx=-3sin(x).
Watch out: These derivatives assume radian input.
Integration by parts
integral u dv = u*v – integral v du
Notation / units: Differentiable functions on the interval.
Use when: Choose u so differentiating it helps.
Worked example and common mistake
Integral x*e^x dx = x*e^x-e^x+C.
Watch out: The remaining integral is subtracted, not added.
Normal standardisation
z = (x-mu)/sigma
Notation / units: sigma>0.
Use when: For a normal variable, z follows the standard normal distribution.
Worked example and common mistake
x=74, mu=70, sigma=8 gives z=0.5.
Watch out: Use standard deviation, not variance, as the denominator.
Binomial probability
P(X=r)=nCr*p^r*(1-p)^(n-r)
Notation / units: 0<=p<=1; r integer 0..n.
Use when: Independent fixed-number trials with the same success probability.
Worked example and common mistake
n=3,p=0.5,r=2 gives 3/8.
Watch out: Sampling without replacement may violate independence.
Binomial moments
mean=n*p; variance=n*p*(1-p)
Notation / units: n integer; 0<=p<=1.
Use when: For X binomial(n,p).
Worked example and common mistake
n=20,p=0.3 gives mean 6 and variance 4.2.
Watch out: Standard deviation is the square root of variance.
Poisson probability
P(X=r)=e^(-lambda)*lambda^r/r!
Notation / units: lambda>0; r non-negative integer.
Use when: Model assumptions include a constant rate and independent events.
Worked example and common mistake
lambda=2,r=0 gives e^-2, about 0.1353.
Watch out: A good-looking histogram alone does not verify Poisson assumptions.
Expected value
E(X)=sum(x*p(x))
Notation / units: Probabilities must sum to 1.
Use when: Discrete random variable.
Worked example and common mistake
X=0 or 4 with probabilities 0.75 and 0.25 gives E(X)=1.
Watch out: The expected value need not be a possible single outcome.
Variance
Var(X)=E(X^2)-(E(X))^2
Notation / units: Squared units.
Use when: Use the distribution's probabilities.
Worked example and common mistake
For equally likely 1 and 3: E(X)=2, E(X^2)=5, variance=1.
Watch out: E(X^2) and E(X)^2 are different.
Friction limiting value
F_lim = mu*R
Notation / units: F and R N; mu dimensionless.
Use when: At impending slip; actual static friction can be smaller.
Worked example and common mistake
mu=0.3, R=20 N gives limiting friction=6 N.
Watch out: Static friction is not always equal to muR.
Power in linear motion
P = F*v
Notation / units: P W; v m/s.
Use when: Force and velocity parallel; otherwise use the component along velocity.
Worked example and common mistake
Driving force 500 N at 12 m/s gives 6000 W.
Watch out: Net force can be zero while the engine still supplies power.
Straight-line gradient
m = (y2-y1)/(x2-x1)
Notation / units: Vertical change per horizontal unit.
Use when: Requires different x-coordinates.
Worked example and common mistake
Through (1,3) and (5,11): m=8/4=2.
Watch out: Keep the point order the same in numerator and denominator.
Straight-line equation
y = m*x + c
Notation / units: m gradient; c y-intercept.
Use when: A non-vertical straight line.
Worked example and common mistake
m=3 through (2,7): 7=6+c, so y=3x+1.
Watch out: The y-intercept is not normally the x-intercept.
Midpoint
M = ((x1+x2)/2, (y1+y2)/2)
Notation / units: Coordinate units.
Use when: Average corresponding coordinates.
Worked example and common mistake
Midpoint of (2,4) and (8,10) is (5,7).
Watch out: Do not average an x-coordinate with a y-coordinate.
Distance between points
d = sqrt((x2-x1)^2 + (y2-y1)^2)
Notation / units: Same coordinate scale on both axes.
Use when: Euclidean distance on a flat coordinate plane.
Worked example and common mistake
Between (1,2) and (4,6): sqrt(9+16)=5.
Watch out: Square both coordinate differences before adding.
Pythagoras theorem
c^2 = a^2 + b^2
Notation / units: c is the hypotenuse.
Use when: Right-angled triangles only.
Worked example and common mistake
Legs 5 and 12 give c=sqrt(169)=13.
Watch out: Do not use the longest side as a leg in the equation.
SOHCAHTOA
sin(theta)=opposite/hypotenuse; cos(theta)=adjacent/hypotenuse; tan(theta)=opposite/adjacent
Notation / units: Sides relative to the chosen angle.
Use when: Right-angled triangle; check degree/radian mode.
Worked example and common mistake
Opposite 3, hypotenuse 5 gives theta=asin(0.6), about 36.87 degrees.
Watch out: The opposite side changes when the reference angle changes.
Sine rule
a/sin(A) = b/sin(B) = c/sin(C)
Notation / units: Match each side to its opposite angle.
Use when: Any triangle; an ambiguous sine case can give two solutions.
Worked example and common mistake
A=30, B=45, a=10 gives b=10*sin45/sin30, about 14.14.
Watch out: Pairing a with the wrong angle gives a plausible but wrong result.
Cosine rule
c^2 = a^2+b^2-2*a*b*cos(C)
Notation / units: C is the included angle between a and b.
Use when: Any triangle.
Worked example and common mistake
a=3, b=4, C=60 gives c=sqrt(13).
Watch out: Use the angle opposite c, not an arbitrary angle.
Constant-acceleration velocity
v = u+a*t
Notation / units: u and v m/s; a m/s^2.
Use when: Acceleration is constant.
Worked example and common mistake
u=3, a=2, t=4 gives v=11 m/s.
Watch out: This equation is not a general variable-acceleration model.
Constant-acceleration displacement
s = u*t + a*t^2/2
Notation / units: s m; t s.
Use when: Acceleration constant along the chosen axis.
Worked example and common mistake
u=0, a=3, t=4 gives s=24 m.
Watch out: Only the second term includes t squared.
Velocity-displacement relation
v^2 = u^2 + 2*a*s
Notation / units: SI units; sign convention consistent.
Use when: Constant acceleration; time is eliminated.
Worked example and common mistake
u=2, a=3, s=10 gives v^2=64; forward speed=8 m/s.
Watch out: The square-root sign alone does not determine direction.
Resultant force
F = m*a
Notation / units: F N; m kg; a m/s^2.
Use when: Constant mass; F is the net force.
Worked example and common mistake
A 4 kg object at 2.5 m/s^2 has resultant force 10 N.
Watch out: One applied force is not necessarily the resultant force.
Momentum
p = m*v
Notation / units: p kg m/s.
Use when: Non-relativistic motion.
Worked example and common mistake
m=0.6 kg, v=5 m/s gives p=3 kg m/s.
Watch out: Include signs when opposite directions are compared.
How to use the A Level Maths Formula Sheet
- Choose a topic or enter a formula name, symbol or unit in the search field.
- Open the worked example and check the conditions before substituting values.
- Save difficult formulas for a focused list, test recall, then print or export that list.
A worked example
Differentiate y=(3x+1)^4 by the chain rule: dy/dx=4(3x+1)^3 multiplied by 3, giving 12(3x+1)^3. The final factor comes from differentiating the inner expression.
What to check before using your result
- This is a curated reference, not a complete official formula booklet. Select formulas that belong to your AS/A Level component route.
- Use consistent units before substitution; a correct formula with incompatible units still gives a wrong result.
- A saved formula is a revision prompt, not evidence that you can apply it to an unfamiliar problem.
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