Simplify Your Workflow: Search MiniWebtool.
Add Extension
Home Page > Math > Advanced Math Operations > Sum of Cubes Calculator

Sum of Cubes Calculator

Calculate the sum of consecutive cubes from n₁³ to n₂³ with step-by-step formula breakdown, visual cube representation, and mathematical analysis. Perfect for algebra, calculus, and number theory studies.

Sum of Cubes Calculator

Try quick examples:

First 10 Cubes 1³ to 100³ 5³ to 15³ Prime Cubes
Sum from n₁³ to n₂³

Embed Sum of Cubes Calculator Widget

About Sum of Cubes Calculator

Welcome to the Sum of Cubes Calculator, a powerful mathematical tool that computes the sum of consecutive cubic numbers using elegant closed-form formulas. Whether you need to calculate 1³ + 2³ + ... + n³, find the sum from n₁³ to n₂³, or compute cubes of custom numbers, this calculator provides instant results with step-by-step explanations and visual representations.

The Beautiful Sum of Cubes Identity

Nicomachus's Theorem

$$1^3 + 2^3 + 3^3 + \cdots + n^3 = \left(\frac{n(n+1)}{2}\right)^2 = (1 + 2 + 3 + \cdots + n)^2$$

The sum of the first n cubes equals the square of the sum of the first n natural numbers!

This remarkable identity, known as Nicomachus's Theorem, reveals a profound connection between cubic and linear sums. It means that adding cubes always produces a perfect square - specifically, the square of the nth triangular number.

Sum of Cubes Formulas

Sum of First n Cubes

First n Cubes
$$\sum_{k=1}^{n} k^3 = 1^3 + 2^3 + \cdots + n^3 = \left[\frac{n(n+1)}{2}\right]^2 = \frac{n^2(n+1)^2}{4}$$

Sum of Cubes from n₁ to n₂

Range Sum
$$\sum_{k=n_1}^{n_2} k^3 = n_1^3 + (n_1+1)^3 + \cdots + n_2^3 = S(n_2) - S(n_1-1)$$

Where S(n) = [n(n+1)/2]² is the sum of the first n cubes.

How to Use This Calculator

  1. Choose calculation mode:
    • Range mode: Calculate sum from n₁³ to n₂³
    • First n cubes: Calculate 1³ + 2³ + ... + n³
    • Custom numbers: Enter any list of numbers to cube and sum
  2. Enter your values: Input the required numbers based on your chosen mode.
  3. Calculate: Click the button to compute the sum using the optimal formula.
  4. Review results: Examine the sum, step-by-step calculation, and visual chart of individual cubes.

Quick Reference: Sum of First n Cubes

nSum FormulaSum of CubesVerification
1[1×2/2]² = 1²11³ = 1
2[2×3/2]² = 3²91 + 8 = 9
3[3×4/2]² = 6²361 + 8 + 27 = 36
4[4×5/2]² = 10²1001 + 8 + 27 + 64 = 100
5[5×6/2]² = 15²2251 + 8 + 27 + 64 + 125 = 225
10[10×11/2]² = 55²3,025Sum of 1³ to 10³
100[100×101/2]² = 5050²25,502,500Sum of 1³ to 100³

Why Sum of Cubes = Perfect Square?

The identity can be visualized geometrically: imagine building an L-shaped gnomon for each term. The first cube (1³=1) forms a 1×1 square. Each subsequent cube can be arranged as an L-shape that extends the square. The cube 2³=8 forms an L that makes the square 3×3, and so on. The pattern continues, always producing a perfect square with side length equal to the triangular number T(n) = 1+2+...+n.

Applications of Sum of Cubes

Calculus and Integration

The sum of cubes formula is essential when computing Riemann sums for cubic functions. When approximating ∫₀ⁿ x³dx, you need ∑k³. As n→∞, this helps derive that ∫x³dx = x⁴/4.

Number Theory

The sum of cubes identity connects to triangular numbers, perfect squares, and the relationship between different power sums. It is a fundamental result in additive number theory.

Computer Science

Algorithm analysis sometimes involves cube sums when analyzing nested loop complexity. Understanding the closed-form formula allows O(1) computation instead of O(n) iteration.

Physics and Engineering

Cube sums appear in problems involving three-dimensional scaling, volume calculations, and moment of inertia computations for certain geometric configurations.

Proof of the Sum of Cubes Formula

The formula can be proven multiple ways:

Related Formulas

Frequently Asked Questions

What is the sum of cubes formula?

The sum of the first n cubes has a beautiful closed-form formula: 1³ + 2³ + 3³ + ... + n³ = [n(n+1)/2]² = (1 + 2 + 3 + ... + n)². This remarkable identity shows that the sum of cubes equals the square of the triangular number.

How do you calculate the sum of cubes from n₁ to n₂?

To find the sum of cubes from n₁³ to n₂³, use the formula: S(n₂) - S(n₁-1), where S(n) = [n(n+1)/2]². This gives you n₁³ + (n₁+1)³ + ... + n₂³ without having to add each term individually.

Why is the sum of cubes equal to a perfect square?

The sum of the first n cubes equals [n(n+1)/2]², which is always a perfect square because it is the square of the nth triangular number. This elegant mathematical identity can be proven using induction or geometric visualization with stacked cubes.

What is the sum of the first 10 cubes?

The sum of the first 10 cubes is 3,025. Using the formula: [10×11/2]² = 55² = 3,025. Verified: 1 + 8 + 27 + 64 + 125 + 216 + 343 + 512 + 729 + 1000 = 3,025.

What is the relationship between sum of cubes and triangular numbers?

The nth triangular number T(n) = 1 + 2 + ... + n = n(n+1)/2. The sum of the first n cubes equals T(n)². For example, T(5) = 15, and 1³ + 2³ + 3³ + 4³ + 5³ = 225 = 15². This connection makes cube sums related to both linear and quadratic sequences.

How is the sum of cubes formula used in calculus?

In calculus, the sum of cubes formula is used to evaluate Riemann sums for cubic functions. When computing ∫x³dx using left or right Riemann sums, you need ∑k³ from 1 to n, which equals [n(n+1)/2]². This helps derive the antiderivative x⁴/4.

Additional Resources

Reference this content, page, or tool as:

"Sum of Cubes Calculator" at https://MiniWebtool.com/sum-of-cubes-calculator/ from MiniWebtool, https://MiniWebtool.com/

by miniwebtool team. Updated: Jan 19, 2026

You can also try our AI Math Solver GPT to solve your math problems through natural language question and answer.

Related MiniWebtools:

Sum of Positive Integers CalculatorFeatured
Sum of Squares Calculator

Advanced Math Operations:

Top & Updated:

Random PickerRandom Name PickerRelative Standard Deviation CalculatorLine CounterSort NumbersFPS ConverterMAC Address GeneratorMAC Address LookupBatting Average CalculatorRemove SpacesFeet and Inches to Cm ConverterERA CalculatorWord to Phone Number ConverterRandom Truth or Dare GeneratorPercent Off CalculatorRandom Quote GeneratorSum CalculatorFacebook User ID LookupSaturn Return CalculatorLog Base 10 CalculatorBitwise CalculatorText FormatterSHA256 Hash GeneratorNumber of Digits CalculatorCm to Feet and Inches ConverterRoman Numerals ConverterPhone Number ExtractorAI ParaphraserOn Base Percentage CalculatorSun, Moon & Rising Sign Calculator 🌞🌙✨Instagram User ID LookupVideo to Image ExtractorSquare Root (√) CalculatorSlugging Percentage CalculatorSalary Conversion CalculatorOutlier CalculatorCompound Growth CalculatorFirst n Digits of PiQuotient and Remainder CalculatorBinary to Gray Code ConverterMP3 LooperNumber to Word ConverterOctal CalculatorAI Punctuation AdderOPS CalculatorRandom Superpower GeneratorAntilog CalculatorPER CalculatorBCD to Decimal ConverterStandard Error CalculatorRandom Birthday GeneratorDay of Year CalendarAudio SplitterLeap Years ListDecimal to BCD ConverterRemove AccentRemove Audio from VideoMerge VideosTime Duration CalculatorLove Compatibility CalculatorLog Base 2 CalculatorRandom Credit Card GeneratorMedian CalculatorBase-N CalculatorScientific Notation to Decimal ConverterName Number CalculatorIP Address to Hex ConverterRandom Movie PickerPercent Growth Rate CalculatorMultiplication CalculatorRandom Group GeneratorCM to Inches ConverterGray Code to Binary ConverterNext Leap Year CalculatorDay of the Year Calculator - What Day of the Year Is It Today?SHA512 Hash GeneratorRemove Lines Containing...Bonus CalculatorAverage Deviation CalculatorList RandomizerField Goal Percentage CalculatorNumber ExtractorFirst n Digits of eRandom Emoji GeneratorBinary to BCD ConverterList of Prime NumbersModulo CalculatorCompare Two StringsVideo CompressorRatio to Percentage CalculatorMAC Address AnalyzerProportion CalculatorMaster Number CalculatorExponential Decay CalculatorRandom IMEI GeneratorVideo CropperRandom Tournament Bracket GeneratorComplex Number CalculatorPVIFA CalculatorInvisible Text GeneratorEmail ExtractorURL ExtractorAdd Prefix and Suffix to TextWHIP CalculatorSort Lines AlphabeticallyHex to BCD ConverterBCD to Binary ConverterLottery Number GeneratorBCD to Hex ConverterBreak Line by CharactersAverage CalculatorReverse VideoWAR CalculatorHypotenuse CalculatorActual Cash Value CalculatorAngel Number CalculatorRoot Mean Square CalculatorSum of Positive Integers CalculatorSHA3-256 Hash GeneratorAI Sentence Expander📅 Date CalculatorLbs to Kg ConverterHex to Decimal ConverterConvolution CalculatorRandom String GeneratorRemove Leading Trailing SpacesAmortization CalculatorMarkup CalculatorPVIF CalculatorDecimal to Hex ConverterDaily Time Savings CalculatorLorem Ipsum GeneratorReadability Score CalculatorKeyword Density CheckerBionic Reading ConverterText to Speech ReaderFancy Text GeneratorZalgo Text GeneratorUpside Down Text GeneratorASCII Art GeneratorList Difference CalculatorText Column ExtractorText to SQL List ConverterInvisible Character RemoverStock Average CalculatorPosition Size CalculatorMargin Call CalculatorShort Selling Profit CalculatorRisk of Ruin CalculatorBlack-Scholes Option Pricing CalculatorOption Greeks CalculatorImplied Volatility CalculatorOptions Profit CalculatorImpermanent Loss CalculatorCrypto Arbitrage CalculatorSatoshi to USD ConverterCrypto Leverage CalculatorPivot Point CalculatorFibonacci Extension CalculatorStop Loss & Take Profit CalculatorKelly Criterion CalculatorMartingale Strategy CalculatorCompound Daily Interest CalculatorPip Value CalculatorGaussian Distribution GeneratorRandom Meal GeneratorRandom Color Palette GeneratorRandom Country GeneratorRandom JSON GeneratorRandom User-Agent GeneratorRandom Coordinate GeneratorRandom Date GeneratorRandom Time GeneratorRandom IP Address GeneratorRandom Playing Card GeneratorMagic 8-BallRock Paper Scissors GeneratorCoin FlipperDice RollerSpin the WheelAquarium Volume & Stocking CalculatorAquarium Substrate CalculatorScale Model Conversion CalculatorPlant Spacing CalculatorDice Roll Probability CalculatorDepth of Field (DoF) CalculatorGolden Hour / Blue Hour CalculatorPrint Size & Resolution Calculator (DPI/PPI)Soap Making Lye Calculator (SAP)Candle Wax & Fragrance Oil CalculatorHomebrew ABV Calculator (Alcohol by Volume)Coffee Brew Ratio CalculatorBrine and Salinity CalculatorYarn CalculatorFabric CalculatorCross-Stitch Size CalculatorBaker's Percentage CalculatorDough Hydration CalculatorBaking Pan Size ConverterYeast Conversion CalculatorJSON to YAML ConverterJWT DecodercURL to JSON ConverterSQL FormatterCrontab Expression GeneratorFavicon GeneratorSVG OptimizerHtaccess Redirect GeneratorResistor Color Code CalculatorOhm's Law CalculatorVoltage Drop CalculatorPCB Trace Width CalculatorBattery Life CalculatorRandom PIN GeneratorRandom NanoID GeneratorRandom Port Number GeneratorRandom Fake Address GeneratorRandom User Persona GeneratorRandom Writing Prompt GeneratorRandom Haiku GeneratorRandom Domain Name GeneratorRandom Cocktail Recipe GeneratorRandom Activity GeneratorRandom Excuse GeneratorRandom US State GeneratorLemonade Stand CalculatorPizza Value CalculatorVampire Apocalypse CalculatorChristmas Tree CalculatorParking Ratio CalculatorGooglebot Crawl Size CheckerZombie Survival Time CalculatorHelium Balloon Lift CalculatorKinetic Energy Chicken CookerTeleportation Error Rate CalculatorHamster Power CalculatorBanana Radiation CalculatorFalling Through Earth CalculatorPenny Drop Impact CalculatorPopcorn Fill Room CalculatorLego Brick House CalculatorWedding Alcohol CalculatorPizza Party PlannerCaffeine Overdose CalculatorBBQ CalculatorTaco Bar CalculatorBeer Chill Time CalculatorSpaghetti Portion CalculatorCocktail ABV CalculatorChocolate Fountain CalculatorCheese Board CalculatorSudoku Generator & Solver24 Game Solver & TrainerNonogram Generator (Picross)KenKen Generator (Calcudoku)Kakuro GeneratorPoop Salary CalculatorMeeting Cost TickerCommute Life Wasted CalculatorFIRE CalculatorEmail Reply Time CalculatorCoffee vs. Sleep CalculatorReal Hourly Wage CalculatorSick Day Probability CalculatorBuzzword Bingo GeneratorKeyboard Mileage CalculatorToilet Paper Value CalculatorElectricity Cost of Bitcoin Mining CalculatorShower Cost CalculatorLight Bulb Savings CalculatorCat Calorie CalculatorHuman to Ant Weight ConverterTruth Table GeneratorSet Theory CalculatorVenn Diagram Generator (3 Sets)Chinese Remainder Theorem CalculatorEuler's Totient Function CalculatorExtended Euclidean Algorithm CalculatorModular Multiplicative Inverse CalculatorContinued Fraction CalculatorShoelace Formula CalculatorTriangle Centroid CalculatorTriangle Orthocenter CalculatorPoint to Plane Distance CalculatorSphere Equation CalculatorGram-Schmidt CalculatorVector Projection CalculatorMatrix LU Decomposition CalculatorRadius of Convergence CalculatorCurvature CalculatorCollatz Conjecture CalculatorHappy Number CalculatorMagic Square GeneratorDijkstra's Shortest Path CalculatorMinimum Spanning Tree CalculatorGraph Degree Sequence ValidatorCatalan Number GeneratorDerangement (Subfactorial) CalculatorStirling Numbers CalculatorPigeonhole Principle CalculatorMarkov Chain Steady State CalculatorGini Coefficient CalculatorSingular Value Decomposition (SVD) CalculatorRandom RPG Character GeneratorRandom Chess Opening GeneratorRandom Poker Hand GeneratorRandom Loadout GeneratorRandom Chord GeneratorRandom Sound Frequency GeneratorMatrix Rank CalculatorMatrix Trace CalculatorWronskian CalculatorRunge-Kutta (RK4) Method CalculatorFourier Series Coefficients CalculatorFunction Odd Even Neither CheckerCone Flat Pattern (Template) GeneratorPolygon Diagonals CalculatorEuler Characteristic CalculatorRSA Encryption Step-by-Step SimulatorPrimitive Root Calculator