(a+b)² and (a+b)³ as Area and Volume

Also called

algebraic identities · (a+b)^2 · (a+b)² · square of a binomial · (a+b)^3 · (a+b)³ · cube of a binomial · (a-b)^2 · (a−b)² · a² + 2ab + b² · a³ + 3a²b + 3ab² + b³ · area model of an identity · volume model of an identity · expansion of a binomial · binomial cube · बीजीय सर्वसमिका · द्विपद का वर्ग · द्विपद का घन

Topics

• A square of side a + b has area (a + b)²; it splits into a square a², two rectangles ab and a square b².
• So (a + b)² = a² + 2ab + b²: the two ab rectangles are congruent, just turned.
• A cube of side a + b has volume (a + b)³ and splits into eight blocks.
• The eight blocks are one cube a³, three slabs a²b, three bars ab² and one small cube b³.
• So (a + b)³ = a³ + 3a²b + 3ab² + b³.
• Cutting two strips of width b from a square of side a leaves a square of side a − b.
• The first strip is ab, but the second is only (a − b)b: one b² less, so (a − b)² = a² − 2ab + b².
• An identity is true for every value of a and b, which is why the pieces always fit back together.

About this model

A GLB file of about 101 KB, shown in Lably's 3D viewer and, on a phone, placed in your room with AR. Credit: Model, science and animation: Lably (built procedurally with three.js; no downloaded meshes). | CC BY 4.0. Original geometry and animation made for Lably.

More in Mathematics

(a+b)² and (a+b)³ as Area and VolumeClass 93D model

Algebraic cuboid partition for distributivityClass 93D model

C

Circle Theorems in MotionClass 93D model

Circle Theorems in MotionClass 93D model

CoinClass 1–103D model

C

Coin (heads and tails)Class 1–103D model