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Sum of Cubes Calculator

Compute the sum of the first n cubes or solve cubic binomials (a³ + b³). Features interactive factoring steps and a database of perfect cubes.

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Category Math & Algebra
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About

The summation of cubic numbers is a frequent problem in algebra and number theory, appearing in everything from structural stacking problems to volume approximation integration. This calculator offers two distinct modes: summation series (∑ i3) and the algebraic factorization of the sum of two specific cubes (a3 + b3).

Understanding the factorization pattern is critical for simplifying rational expressions in calculus. This tool visualizes the expansion a3 + b3 = (a + b)(a2 − ab + b2), replacing variables with your inputs to show exactly how the result is derived.

algebra summation cubes factoring polynomials

Formulas

For the sum of the first n cubes, the closed-form formula (Nicomachus's Theorem) is:

n∑i=1 i3 = n2(n + 1)24

For the sum of two specific cubes:

a3 + b3 = (a + b)(a2 − ab + b2)

Reference Data

Number (n)Cube (n3)Sum of first n cubes
111
289
32736
464100
5125225
6216441
7343784
85121296
97292025
1010003025

Frequently Asked Questions

Yes, specifically for the sequence starting from 1. A famous identity states that the sum of the first n cubes equals the square of the sum of the first n integers: (1+2+...+n)².
The pattern is very similar but signs are swapped: a³ − b³ = (a − b)(a² + ab + b²). This tool focuses on the sum, but remembering "SOAP" (Same, Opposite, Always Positive) helps recall the signs.
The database logic handles integers up to Javascript's safe integer limit, but the "Perfect Cube" lookup table is optimized for n values up to 100 resulting in cubes up to 1,000,000.