What Is Area Model?
The area model, also called the box method, splits each factor into parts and multiplies every pair. The boxes are partial products, and their total is the answer.
Use it for whole numbers, decimals and for two binomials such as (2x + 3)(x + 5). The calculator splits numbers by place value, fills the grid, adds the boxes and shows the sum. In binomial mode it shows the x², x and constant boxes and combines like terms.
Below are the formulas, a worked example for 23 × 14 and answers on using the area model with decimals and algebra.
How to Use the Area Model Calculator
- Choose numbers or binomials.
- Enter the two factors.
- Read the box grid of partial products.
- Add the boxes for the final product.
Area Model Formula
Distributive property
Area Model Variables Explained
| Symbol | Meaning | Notes |
|---|---|---|
| a, b, c, d | Coefficients in (ax + b)(cx + d) | |
| Partial product | Area of one box | |
| Place value | Hundreds, tens, ones, tenths and so on | |
| Product | Sum of all boxes |
Area Model Worked Example
For 23 × 14, split 23 into 20 + 3 and 14 into 10 + 4. The four boxes are 20 × 10 = 200, 3 × 10 = 30, 20 × 4 = 80 and 3 × 4 = 12. Adding gives 200 + 30 + 80 + 12 = 322.
For (2x + 3)(x + 5) the boxes are 2x², 10x, 3x and 15, so the product is 2x² + 13x + 15.
With the example values loaded in the calculator above, the steps are:
- Split 23 into place-value parts: 20 + 3.
- Split 14 into place-value parts: 10 + 4.
- Multiply every top part by every side part to fill the boxes: 200, 30, 80, 12.
- Add all the partial products: 200 + 30 + 80 + 12 = 322.
Result: 23 × 14 = 322.
Where Area Model Is Used
Learning multiplication
See why the algorithm works.
Decimal multiplication
Split 4.5 into 4 + 0.5.
Algebra
Expand two binomials.
Checking homework
Compare with your own boxes.
Common Area Model Mistakes to Avoid
- Split by place value for numbers.
- Combine like terms for binomials.
- Check by estimating: 23 × 14 is about 20 × 15 = 300.
Area Model Terms and Definitions
- Partial product
- The product of one part of each factor, shown in one box.
Area Model FAQs
What is the area model for multiplication?
A rectangle split into boxes whose areas are partial products that add up to the full product.
How do you use the area model with decimals?
Split each decimal into place-value parts such as 4 + 0.5, then multiply every pair and add.
How do you use it with binomials?
Put each term on a side, multiply to fill four boxes and combine like terms.
Is it the same as the box method?
Yes. Area model and box method are two names for the same idea.