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.
How to Read an Area Model
Each box shows one partial product, and the boxes add up to the answer. For 23 × 14, the boxes are 200, 30, 80 and 12, which add to 322. The same layout works for binomials, where the boxes hold terms such as ax·cx and ax·d.
Looking at the boxes also shows where the digits of the standard algorithm come from. The 200 and 80 are the tens-times-tens and tens-times-ones products that the algorithm folds into one line.
Where Area Model Is Used
Multiplying whole numbers
Break each factor into place-value parts and see each partial product.
Decimals
Split each decimal into parts and multiply the same way.
Binomials
Expand (ax + b)(cx + d) in a box layout.
Teaching
Show why the standard algorithm works.
Common Area Model Mistakes to Avoid
- Split each number by place value so each box has a simple product.
- Add all the boxes to get the answer, and compare with an estimate.
- For binomials, combine like terms from the boxes that share the same power of x.
- Write each partial product in full, including zeros.
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?
It is a rectangle split into boxes, with each side labelled by the parts of a factor. Each box holds a partial product and the total of all boxes is the answer.
How do you use the area model with decimals?
Split each decimal into parts such as 2 + 0.3, multiply every pair of parts, and add the products. An estimate checks the position of the decimal point.
How do you use it with binomials?
Label the sides with the terms of each binomial, multiply to fill the boxes, then combine like terms. (2x + 3)(x + 4) gives 2x², 8x, 3x and 12, which sum to 2x² + 11x + 12.
Is it the same as the box method?
Yes. The box method and the area model are two names for the same layout.