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Equation Reference

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Auto-generated from lib/equations.json \u2014 0 equations, 0 variables, 0 outputs.

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\n Compound Interest (Finance)\n

Formula: A = P(1 + r/n)^(nt)

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Growth of an investment with compound interest

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Inputs

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VariableSymbolRangeDefaultDescription
PrincipalP100 \u2013 100,000 $10,000 $Initial investment amount
Interest Rater0.1000 \u2013 20 %5 %Annual interest rate
Compounds Per Yearn1 \u2013 365 times12 timesHow often interest is compounded
Yearst1 \u2013 50 years10 yearsInvestment period
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Outputs

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OutputUnitDescription
Final Amount
principal * Math.pow(1 + rate / (100 * compounds), compounds * years)		$Total value after compounding
Interest Earned
principal * Math.pow(1 + rate / (100 * compounds), compounds * years) - principal		$Total interest earned
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\n Loan Amortization (Finance)\n

Formula: M = P \u00b7 [r(1+r)^n] / [(1+r)^n - 1]

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Monthly payment for a fixed-rate amortizing loan

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Inputs

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VariableSymbolRangeDefaultDescription
Loan AmountP1,000 \u2013 500,000 $200,000 $Principal loan amount
Annual Interest Rater0.5000 \u2013 15 %6 %Annual interest rate
Loan Termn1 \u2013 30 years30 yearsLoan term in years
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Outputs

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OutputUnitDescription
Monthly Payment
loanAmount * (annualRate / 1200 * Math.pow(1 + annualRate / 1200, loanYears * 12)) / (Math.pow(1 + annualRate / 1200, loanYears * 12) - 1)		$Fixed monthly payment (principal + interest)
Total Paid
(loanAmount * (annualRate / 1200 * Math.pow(1 + annualRate / 1200, loanYears * 12)) / (Math.pow(1 + annualRate / 1200, loanYears * 12) - 1)) * loanYears * 12		$Total amount paid over full loan term
Total Interest
(loanAmount * (annualRate / 1200 * Math.pow(1 + annualRate / 1200, loanYears * 12)) / (Math.pow(1 + annualRate / 1200, loanYears * 12) - 1)) * loanYears * 12 - loanAmount		$Total interest paid over loan term
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\n Net Present Value (Finance)\n

Formula: NPV = CF / (1+r)^t - I

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Present value of a future cash flow minus initial investment

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Inputs

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VariableSymbolRangeDefaultDescription
Initial InvestmentI-100,000 \u2013 100,000 $10,000 $Upfront cost or investment
Future Cash FlowCF0 \u2013 200,000 $15,000 $Expected future return
Discount Rater0 \u2013 30 %8 %Required rate of return
Time Periodst1 \u2013 30 years5 yearsNumber of years until cash flow
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Outputs

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OutputUnitDescription
Net Present Value
cashFlow / Math.pow(1 + discountRate / 100, periods) - initialInvestment		$NPV > 0: investment adds value
Present Value of CF
cashFlow / Math.pow(1 + discountRate / 100, periods)		$Future cash flow discounted to today
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\n Return on Investment (Finance)\n

Formula: ROI = (Gain - Cost) / Cost \u00d7 100

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Profit relative to cost, expressed as percentage

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Inputs

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VariableSymbolRangeDefaultDescription
Cost of InvestmentC100 \u2013 100,000 $5,000 $Total amount invested
Final ValueG0 \u2013 200,000 $7,500 $Total value received
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Outputs

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OutputUnitDescription
ROI
(finalValue - investmentCost) / investmentCost * 100		%Return on Investment (positive = profit)
Profit / Loss
finalValue - investmentCost		$Absolute gain or loss in dollars
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\n Ohm's Law (Physics)\n

Formula: V = I \u00d7 R

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Relationship between voltage, current, and resistance

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Inputs

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VariableSymbolRangeDefaultDescription
CurrentI0 \u2013 10 A2 AElectric current in Amperes
ResistanceR1 \u2013 1,000 \u03a9100 \u03a9Electrical resistance in Ohms
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Outputs

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OutputUnitDescription
Voltage
current * resistance		VElectrical voltage in Volts
Power Dissipated
current * current * resistance		WPower dissipated as heat (P = I\u00b2R)
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\n Projectile Motion (Physics)\n

Formula: R = (v\u2080\u00b2 \u00d7 sin(2\u03b8)) / g

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Range of a projectile launched at an angle with initial velocity

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Inputs

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VariableSymbolRangeDefaultDescription
Initial Velocityv\u20801 \u2013 100 m/s20 m/sLaunch velocity
Launch Angle\u03b80 \u2013 90 \u00b045 \u00b0Angle from horizontal
Gravitational Accelerationg1 \u2013 50 m/s\u00b29.8100 m/s\u00b2Gravity constant (Earth = 9.81)
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Outputs

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OutputUnitDescription
Range (Distance)
velocity * velocity * Math.sin(2 * angle * Math.PI / 180) / gravity		mHorizontal distance traveled
Max Height
velocity * velocity * Math.sin(angle * Math.PI / 180) * Math.sin(angle * Math.PI / 180) / (2 * gravity)		mMaximum height reached
Flight Time
2 * velocity * Math.sin(angle * Math.PI / 180) / gravity		sTotal time in the air
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\n Hooke's Law (Physics)\n

Formula: F = -k \u00b7 x

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Force exerted by an ideal spring: F = -kx

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Inputs

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VariableSymbolRangeDefaultDescription
Spring Constantk1 \u2013 500 N/m100 N/mStiffness of the spring
Displacementx-2 \u2013 2 m0.5000 mDistance stretched or compressed from equilibrium
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Outputs

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OutputUnitDescription
Restoring Force
-(springConstant * displacement)		NForce exerted by the spring (negative = opposite to displacement)
Force Magnitude
Math.abs(springConstant * displacement)		NAbsolute magnitude of the spring force
Elastic Potential Energy
0.5 * springConstant * displacement * displacement		JEnergy stored in the spring (PE = \u00bdkx\u00b2)
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\n Newton's Gravitation (Physics)\n

Formula: F = G \u00b7 m\u2081 \u00b7 m\u2082 / r\u00b2

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Gravitational force between two masses: F = G\u00b7m\u2081\u00b7m\u2082/r\u00b2

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Inputs

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VariableSymbolRangeDefaultDescription
Mass 1m\u208110 \u2013 1,000,000,000,000 kg1,000 kgMass of first object
Mass 2m\u208210 \u2013 1,000,000,000,000 kg1,000 kgMass of second object
Distancer1 \u2013 1,000 m10 mDistance between centers of mass
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Outputs

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OutputUnitDescription
Gravitational Force
6.674e-11 * mass1 * mass2 / (distance * distance)		NForce of gravitational attraction
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\n Simple Pendulum (Physics)\n

Formula: T = 2\u03c0\u221a(L/g)

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Period of a simple pendulum for small-angle oscillations

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Inputs

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VariableSymbolRangeDefaultDescription
LengthL0.1000 \u2013 10 m1 mLength of the pendulum
Gravitational Accelerationg0.5000 \u2013 25 m/s\u00b29.8100 m/s\u00b2Local gravity (Earth = 9.81)
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Outputs

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OutputUnitDescription
Period T
2 * Math.PI * Math.sqrt(pendulumLength / pendulumGravity)		sTime for one complete swing
Frequency f
1 / (2 * Math.PI * Math.sqrt(pendulumLength / pendulumGravity))		HzOscillations per second
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\n Quadratic Formula (Math)\n

Formula: x = (-b \u00b1 \u221a(b\u00b2 - 4ac)) / 2a

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Solutions to ax\u00b2 + bx + c = 0

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Inputs

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VariableSymbolRangeDefaultDescription
Coefficient aa-10 \u2013 101Coefficient of x\u00b2
Coefficient bb-20 \u2013 20-5Coefficient of x
Constant termc-20 \u2013 206Constant term
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Outputs

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OutputUnitDescription
Discriminant
b*b - 4*a*c		b\u00b2 - 4ac (negative = complex roots)
Root 1
(-b + Math.sqrt(Math.abs(b*b - 4*a*c))) / (2*a)		First solution (real part)
Root 2
(-b - Math.sqrt(Math.abs(b*b - 4*a*c))) / (2*a)		Second solution (real part)
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\n Pythagorean Theorem (Math)\n

Formula: c = \u221a(a\u00b2 + b\u00b2)

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Relationship between sides of a right triangle: a\u00b2 + b\u00b2 = c\u00b2

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Inputs

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VariableSymbolRangeDefaultDescription
Side aa0.1000 \u2013 203Length of first leg
Side bb0.1000 \u2013 204Length of second leg
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Outputs

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OutputUnitDescription
Hypotenuse c
Math.sqrt(sideA * sideA + sideB * sideB)		Length of the hypotenuse
Area
0.5 * sideA * sideB		u\u00b2Area of the right triangle
Perimeter
sideA + sideB + Math.sqrt(sideA * sideA + sideB * sideB)		Perimeter of the triangle
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\n 2D Distance Formula (Math)\n

Formula: d = \u221a((x\u2082-x\u2081)\u00b2 + (y\u2082-y\u2081)\u00b2)

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Distance between two points in the Cartesian plane

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Inputs

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VariableSymbolRangeDefaultDescription
Point 1 \u2014 xx\u2081-10 \u2013 100x-coordinate of first point
Point 1 \u2014 yy\u2081-10 \u2013 100y-coordinate of first point
Point 2 \u2014 xx\u2082-10 \u2013 103x-coordinate of second point
Point 2 \u2014 yy\u2082-10 \u2013 104y-coordinate of second point
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Outputs

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OutputUnitDescription
Distance
Math.sqrt((x2 - x1) * (x2 - x1) + (y2 - y1) * (y2 - y1))		Euclidean distance between points
Midpoint \u2014 x
(x1 + x2) / 2		x-coordinate of the midpoint
Midpoint \u2014 y
(y1 + y2) / 2		y-coordinate of the midpoint
Slope
(y2 - y1) / (x2 - x1)		Slope of the line (undefined if vertical)
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\n Standard Deviation (Math)\n

Formula: \u03c3 = \u221a(\u03a3(x-\u03bc)\u00b2 / n)

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Measure of spread in a small dataset (3 values)

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Inputs

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VariableSymbolRangeDefaultDescription
Value 1x\u2081-50 \u2013 5010First data point
Value 2x\u2082-50 \u2013 5020Second data point
Value 3x\u2083-50 \u2013 5030Third data point
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Outputs

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OutputUnitDescription
Mean (\u03bc)
(d1 + d2 + d3) / 3		Arithmetic mean of the dataset
Std Dev (\u03c3)
Math.sqrt(((d1 - (d1 + d2 + d3) / 3) * (d1 - (d1 + d2 + d3) / 3) + (d2 - (d1 + d2 + d3) / 3) * (d2 - (d1 + d2 + d3) / 3) + (d3 - (d1 + d2 + d3) / 3) * (d3 - (d1 + d2 + d3) / 3)) / 3)		Population standard deviation
Variance (\u03c3\u00b2)
((d1 - (d1 + d2 + d3) / 3) * (d1 - (d1 + d2 + d3) / 3) + (d2 - (d1 + d2 + d3) / 3) * (d2 - (d1 + d2 + d3) / 3) + (d3 - (d1 + d2 + d3) / 3) * (d3 - (d1 + d2 + d3) / 3)) / 3		Population variance
Range
Math.max(d1, d2, d3) - Math.min(d1, d2, d3)		Max - Min
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\n Law of Cosines (Math)\n

Formula: c\u00b2 = a\u00b2 + b\u00b2 - 2ab\u00b7cos(C)

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Find the third side of a triangle given two sides and the included angle

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Inputs

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VariableSymbolRangeDefaultDescription
Side aa0.1000 \u2013 205Length of side a
Side bb0.1000 \u2013 207Length of side b
Angle CC0 \u2013 180 \u00b060 \u00b0Included angle between sides a and b
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Outputs

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OutputUnitDescription
Side c
Math.sqrt(sideA * sideA + sideB * sideB - 2 * sideA * sideB * Math.cos(angleC * Math.PI / 180))		Length of side opposite angle C
Triangle Area
0.5 * sideA * sideB * Math.sin(angleC * Math.PI / 180)		u\u00b2Area using formula A = \u00bdab\u00b7sin(C)
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\n Shannon Entropy (Info Theory)\n

Formula: H = -p \u00b7 log\u2082(p) - (1-p) \u00b7 log\u2082(1-p)

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Information content of a binary source \u2014 maximum at p = 0.5

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Inputs

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VariableSymbolRangeDefaultDescription
Probability of 1p0.0100 \u2013 0.99000.5000Probability of the outcome being 1 (vs 0)
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Outputs

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OutputUnitDescription
Entropy H
-(p * Math.log2(p) + (1 - p) * Math.log2(1 - p))		bitsAverage information content per symbol
Redundancy
1 - (-(p * Math.log2(p) + (1 - p) * Math.log2(1 - p)))		Redundancy relative to maximum entropy (1 bit)
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\n Shannon-Hartley Theorem (Info Theory)\n

Formula: C = B \u00b7 log\u2082(1 + S/N)

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Maximum data rate over a noisy channel

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Inputs

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VariableSymbolRangeDefaultDescription
BandwidthB100 \u2013 20,000 Hz3,000 HzChannel bandwidth in Hertz
Signal-to-Noise RatioS/N1 \u2013 1,00031.6000Linear signal-to-noise ratio (not dB)
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Outputs

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OutputUnitDescription
Channel Capacity
bandwidth * Math.log2(1 + snrLinear)		bit/sMaximum theoretical data rate
SNR (dB)
10 * Math.log10(snrLinear)		dBEquivalent SNR in decibels
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\n Bayes' Theorem (Info Theory)\n

Formula: P(A|B) = P(B|A)\u00b7P(A) / (P(B|A)\u00b7P(A) + P(B|\u00acA)\u00b7(1-P(A)))

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Posterior probability given prior, likelihood, and false positive rate

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Inputs

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VariableSymbolRangeDefaultDescription
Prior Probability P(A)P(A)0.0010 \u2013 0.50000.0100Base rate of the condition
Likelihood P(B|A)P(B|A)0.5000 \u2013 10.9500Probability of positive test given condition
False Positive Rate P(B|\u00acA)P(B|\u00acA)0.0010 \u2013 0.50000.0500Probability of positive test given no condition
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Outputs

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OutputUnitDescription
Posterior P(A|B)
(likelihood * prior) / (likelihood * prior + falsePositive * (1 - prior))		Probability of condition given positive test
Positive Predictive Value
((likelihood * prior) / (likelihood * prior + falsePositive * (1 - prior))) * 100		%PPV \u2014 same as posterior, expressed as percentage
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\n Signal-to-Noise Ratio (Engineering)\n

Formula: SNR_dB = 10 \u00b7 log\u2081\u2080(S / N)

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Ratio of signal power to noise power in decibels

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Inputs

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VariableSymbolRangeDefaultDescription
Signal PowerS0.1000 \u2013 100 W10 WPower of the signal
Noise PowerN0.0100 \u2013 10 W0.1000 WPower of the noise floor
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Outputs

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OutputUnitDescription
SNR (dB)
10 * Math.log10(signalPower / noisePower)		dBSignal-to-noise ratio in decibels
SNR (Linear)
signalPower / noisePower		Signal-to-noise ratio as a linear ratio
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\n Ideal Gas Law (Engineering)\n

Formula: P = nRT / V

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Pressure of an ideal gas: PV = nRT \u2014 R = 8.314 J/(mol\u00b7K)

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Inputs

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VariableSymbolRangeDefaultDescription
Amount of Substancen0.1000 \u2013 10 mol1 molNumber of moles of gas
TemperatureT100 \u2013 1,000 K300 KAbsolute temperature in Kelvin
VolumeV0.0010 \u2013 1 m\u00b30.0250 m\u00b3Volume of the container
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Outputs

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OutputUnitDescription
Pressure
moles * 8.314 * temperature / volume		PaPressure in Pascals
Pressure (atm)
moles * 8.314 * temperature / volume / 101325		atmPressure in atmospheres
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\n RC Circuit Charging (Engineering)\n

Formula: V(t) = V\u2080 \u00b7 (1 - e^(-t / RC))

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Voltage across a capacitor as it charges through a resistor

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Inputs

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VariableSymbolRangeDefaultDescription
Source VoltageV\u20801 \u2013 100 V12 VApplied voltage
ResistanceR100 \u2013 100,000 \u03a910,000 \u03a9Series resistance
CapacitanceC0.0000 \u2013 0.0010 F0.0001 FCapacitor value
Time Elapsedt0 \u2013 5 s0.5000 sTime since charging began
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Outputs

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OutputUnitDescription
Capacitor Voltage
sourceVoltage * (1 - Math.exp(-time / (resistance * capacitance)))		VVoltage across the capacitor
Time Constant \u03c4
resistance * capacitance		sRC time constant (time to reach 63.2%)
Percent Charged
(1 - Math.exp(-time / (resistance * capacitance))) * 100		%Percentage of final voltage reached
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\n Kinetic Energy (Engineering)\n

Formula: KE = \u00bd \u00b7 m \u00b7 v\u00b2

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Energy of a moving object: KE = \u00bdmv\u00b2

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Inputs

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VariableSymbolRangeDefaultDescription
Massm0.1000 \u2013 1,000 kg10 kgMass of the object
Velocityv0 \u2013 100 m/s20 m/sSpeed of the object
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Outputs

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OutputUnitDescription
Kinetic Energy
0.5 * mass * velocity * velocity		JKinetic energy in Joules
Momentum
mass * velocity		kg\u00b7m/sLinear momentum
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\n Doppler Effect (Engineering)\n

Formula: f' = f \u00b7 (v_sound + v_observer) / v_sound

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Observed frequency when observer moves toward a stationary source

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Inputs

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VariableSymbolRangeDefaultDescription
Source Frequencyf100 \u2013 2,000 Hz440 HzFrequency emitted by the source
Speed of Soundv100 \u2013 500 m/s343 m/sSpeed of sound in the medium
Observer Speedv\u20800 \u2013 100 m/s15 m/sSpeed of observer moving toward source
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Outputs

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OutputUnitDescription
Observed Frequency
sourceFreq * (soundSpeed + observerSpeed) / soundSpeed		HzPerceived frequency (higher than source when approaching)
Frequency Shift
sourceFreq * (soundSpeed + observerSpeed) / soundSpeed - sourceFreq		HzChange in frequency (\u0394f)
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\n Thermal Linear Expansion (Engineering)\n

Formula: \u0394L = \u03b1 \u00b7 L\u2080 \u00b7 \u0394T

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Change in length of a material due to temperature change

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Inputs

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VariableSymbolRangeDefaultDescription
Expansion Coefficient\u03b11 \u2013 50 \u00d710\u207b\u2076/K12 \u00d710\u207b\u2076/KLinear thermal expansion coefficient (\u00d710\u207b\u2076)
Initial LengthL\u20801 \u2013 100 m10 mOriginal length of the material
Temperature Change\u0394T-100 \u2013 500 K50 KChange in temperature
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Outputs

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OutputUnitDescription
Length Change
coefficient * 0.000001 * initialLength * tempChange		mChange in length (\u0394L)
Length Change
coefficient * 0.000001 * initialLength * tempChange * 1000		mmChange in length in millimeters
Final Length
initialLength + coefficient * 0.000001 * initialLength * tempChange		mLength after thermal change
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Interactive calculators for finance, physics, math, information theory, and engineering.

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The Equation Visualizer in the sidebar lets you interact with all 0 equations using sliders. This page is a static reference generated from the same data.

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See also: Interactive demo on the projects page \u2192

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